`Existence of Critical Points for the Ginzburg-Landau Functional on Riemannian ManifoldsbyJeff MesaricA thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of TorontoCopyright c 2009 by Jeff MesaricAbstractExistence of Critical Points for the Ginzburg-Landau Functional on Riemannian Manifolds Jeff Mesaric Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2009 In this dissertation, we employ variational methods to obtain a new existence result for solutions of a Ginzburg-Landau type equation on a Riemannian manifold. We prove that if N is a compact, orientable 3-dimensional Riemannian manifold without boundary and  is a simple, smooth, connected, closed geodesic in N satisfying a natural nondegeneracy condition, then for every &gt; 0 sufficiently small,  a critical point u  H 1 (N ; C) of the Ginzburg-Landau functional E (u) := 1 2| ln | | u|2 +N(|u|2 - 1)2 22and these critical points have the property that E (u )  length() as  0. To accomplish this, we appeal to a recent general asymptotic minmax theorem which basically says that if E -converges to E (not necessarily defined on the same Banach space as E ), v is a saddle point of E and some additional mild hypotheses are met, then there exists0&gt; 0 such that for every (0, 0 ), E possesses a critical point u andlim 0 E (u ) = E(v). Typically, E is only lower semicontinuous, therefore a suitable notion of saddle point is needed. Using known results on R3 , we show the Ginzburg-Landau functional E defined above -converges to a functional E which can be thought of as measuring the arclength of a limiting singular set. Also, we verify using regularity theory for almost-minimal currents that  is a saddle point of E in an appropriate sense. iiDedicationFor my fianc´e, Melissa. eAcknowledgementsI would like to thank my supervisor Robert Jerrard for his guidance and patience over the past few years.iiiContents1 Introduction 2 Definitions 2.1 2.2 2.3 Basic Geometric Measure Theory . . . . . . . . . . . . . . . . . . . . . . Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -Convergence and Saddle Point . . . . . . . . . . . . . . . . . . . . . . .1 10 10 14 15 17 21 28 39 52 54 54 66 68 733 Some Known Results 4 Saddle Points of Mass 4.1 4.2 4.3-1 Flat Local Minimizers of Mass in PW V (0) . . . . . . . . . . . . . . . . . .Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Construction of QV W . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 Applications 5.1 5.2 -convergence of EU on N . . . . . . . . . . . . . . . . . . . . . . . . . . Existence of critical points for EU on N . . . . . . . . . . . . . . . . . . .6 Appendix BibliographyivChapter 1 IntroductionOver the past 30 years, there has been an extensive study of the equation -u + where u : N  Rk , k = 1, 2, and N is typically an open, bounded subset of Rk+m , m  1, with smooth boundary. (1.2) Of course, solutions of (1.1) are critical points of the energy functional E (u) = 1 2| ln | | u|2 +N12(|u|2 - 1)u = 0,(1.1)(|u|2 - 1)2 dX. 22(1.3)When k = 1, (1.3) is usually referred to as the Allen-Cahn functional and the GinzburgLandau functional when k = 2. Many problems regarding these functionals involve making a connection between critical points of (1.3) and m-dimensional minimal surfaces in certain geometric settings. One method for establishing this connection uses the notion of -convergence. Since being introduced in the 1970's by De Giorgi, the concept of -convergence of a family of functionals to a limiting functional E has proven to be a useful vehicle in describing the asymptotic behaviour of sequences of minimizers as  0. The definition of -convergence (see Section 2.3) is designed to guarantee that, if a sequence of minimizers converges, then the limit has got to be a minimizer of E. Basic 1Chapter 1. Introduction2-convergence results for (1.3) are proved in [15],[14] (k = 1) and [10],[1] (k = 2), and these results automatically lead to a description of the limiting behaviour of a sequence {u } as  0, where u is a minimizer of E . In particular, the energy of u concentrates around a minimal surface of dimension m. -convergence was originally believed to only be useful for describing the asymptotic behaviour of sequences of minimizers. The first result to go beyond this was that of Kohn and Sternberg [12], which did not appear until more than 10 years after the basic definitions and examples of -convergence had been developed. They show that local minimizers of E (with k = 1) exist provided E -converges to E and an isolated local minimizer of E exists. A similar type of result for k = 2 is proved in [16]. Recently, existence of more general critical points of (1.3) for k = 1 has been proved in [13] (N satisfying (1.2)) and [17] (N a compact Riemannian manifold of dimension k + m) via linearization techniques that require precise control over the spectrum of E . These techniques have not yet been extended to cover the k = 2 case, as the GinzburgLandau energy defined for vector-valued functions has worse spectral properties than its scalar counterpart. The first existence result for general critical points of (1.3) with k = 2 and N satisfying (1.2) (m = 1) is established in [11] by -convergence arguments. The intent of this thesis is to do the same for N a compact 3-dimensional manifold. It should be noted that these -convergence arguments merely yield existence and information about critical values whereas in the k = 1 case, the authors are able to obtain a precise description of the critical points near given m-dimensional minimal surfaces satisfying natural nondegeneracy conditions. Hence the results obtained for k = 2 are strictly weaker than those for k = 1. The above history regarding the functional E (u) = 1 2| ln | | u|2 +N(|u|2 - 1)2 , 22u : N  Rk , k = 1, 2 can be summarized as follows:Chapter 1. Introduction N  Rk+m open, bounded with smooth boundary Result basic -convergence results, description of sequences of minimizers as  0 existence of local minimizers of E for sufficiently small existence of critical points of E for sufficiently small N a compact (k + m)-dimensional manifold Result k=1 k=2 Kowalczyk (2005) (m = 1) Jerrard,Sternberg (to appear) (m = 1) Kohn,Sternberg (1989) Montero,Sternberg,Ziemer (2004) (m = 1) Mortola (1987) Alberti,Baldo,Orlandi (2005) k=1 Modica,Mortola (1977) k=2 Jerrard,Soner (2002)3existence of critcal points of E for Picard,Ritor´ (2003) Mesaric (2009) e sufficiently small (m = 1)In [11], Jerrard and Sternberg formulate and prove an abstract theorem (Theorem 3.1) which implies the existence of critical points of E assuming E -converges to a limiting functional E which possesses a nondegenerate critical point and other mild hypotheses are met. Since E is typically only lower semicontinuous, one difficulty for them was to come up with a suitable notion of a nondegenerate critical point. In practice, it is not an easy task to show that a candidate for a critical point in the sense of JS (Jerrard and Sternberg) satisfies the conditions of the definition (Definition 2.1). JS use their abstract theorem to prove new existence results for the Ginzburg-Landau equation in 3 dimensions. This is accomplished by finding a reasonable candidate for a critical point of E (the -limit of the Ginzburg-Landau functional) and then verifying that it is indeed one via ad hoc arguments depending heavily on the specific setting. In this thesis, we useChapter 1. Introduction4a more robust method to identify critical points in the sense of JS of certain functionals which merely lower semicontinuous. Let U, V be Banach spaces, EU , EV : U, V  (-, ],  (0, 1] and V0 = {v  V : EV (v) &lt; }. We say that EU -converges to EV as  0 if there exists a continuousmap PV U : U  V and a map QU V : V0  U (not necessarily continuous) satisfying lower bound: if v  V0 and {u }  U is a sequence such that PV U (u ) - v then lim inf EU (u )  EV (v) and upper bound: for every v  V0 , PV U (QU V (v)) - vV V 0 as  0, 0 andlim sup EU (QU V (v))  EV (v) as  0. This definition is not exactly standard but is equivalent to other definitions. As mentioned above, JS take EU to be the Ginzburg-Landau functional EU (u) = 1 2| ln | | u|2 +(|u|2 - 1)2 dX, 22where u  U = H 1 (; C),  is a bounded domain in R3 with smooth boundary. It is known (see [1],[10]) that EU -converges as  0 to a limiting functional EV thatcan be thought of as measuring the arclength of a limiting singular set. More precisely, V0 = R1 (), which is the space consisting of elements T that are unions of oriented Lipschitz curves in  equipped with a norm that basically gives the minimum area over all 2-d surfaces in  with boundary T and EV (T ) = M(T ), which can be interpreted as the length of the union of curves. In the language of geometric measure theory (GMT), V is the space of 1-currents T that are boundaries with finite flat norm and M(T ) is referred to as the `mass of T '. The abstract theorem of JS roughly says that if EU is a sequence of functionals that -converges to a limiting functional EV and vs is a saddle point of EV (see DefinitionChapter 1. Introduction52.1), then, under certain mild additional hypotheses, EU has a critical point u for every sufficiently small and EU (u )  EV (vs ) as points converge in any sense to vs ). To use the abstract theorem, a saddle point vs is needed. Their candidate for a saddle point is the 1-current associated with an oriented line segment M   joining points x0 , y0  . To be consistent with our notation below, we will label this 1-current as TM (they label it as T ). They fix open sets - , + containing x0 , y0 respectively and assume the distance function d0 : - × +  R+ defined by d0 (x, y) = |x - y| has a non-degenerate critical point at (x0 , y0 ), i.e., d0 (x0 , y0 ) = 0 and detD2 d0 (x0 , y0 ) = 0.  0 (it need not be true that the criticalThe second condition implies that 0 is not an eigenvalue of D2 d0 (x0 , y0 ). To prove that TM is a saddle point of EV in the sense of Definition 2.1, continuous maps PW V : V  Rl , QV W : W  V0 are constructed (here, l denotes the number of negative eigenvalues of D2 d0 (x0 , y0 )) and a number 0 &gt; 0 is found to satisfy PW V (vs ) = 0, QV W (0) = vs , PW V  QV W (w) = w for all w  W , supwW,|w|a(1.4)EV (QV W (w)) &lt; EV (vs ) for all a  (0, r1 )(1.5)and EV (vs ) &lt; EV (v) for v  {v  V : 0 &lt; v - vsV 0 , PW V (v) = 0}(1.6)where W is the open l-dimensional ball of radius r1 centered at the origin for some r1 &gt; 0 appropriately chosen. The construction of these maps relies solely on the presence of a boundary. The most difficult property to verify is (1.6); that TM is a strict local minimizer of EV in the flat norm topology among all T  V with PW V (T ) = 0. This isChapter 1. Introduction accomplished by assuming M(T )  M(TM ) = L, PW V (T ) = 0 and the flat norm of T - TM is less than some number 0 &gt; 0 to be chosen6(1.7)and then showing T = TM . To do this, they first show the existence of a `piece of T ', labeled T , whose support consists of a single Lipschitz curve   that runs from - to + and stays in a cylinder of radius r &lt; R about M , assuming 0 &lt; cr2 for some absolute constant c &gt; 0. It then suffices to show T = TM . This is done by showing the endpoints of the support of T coincide with those of TM , taking r sufficiently small, and using the fact that M(T )  L. The following dissertation develops a more systematic approach of the above that works directly with the spectrum of the Jacobi operator J associated with M (J acts on normal vectorfields on M and is defined through the second variation of EV ). We take EU to be EU (u) =11 2| ln || u|2 +N(|u|2 - 1)2 , 22where u  U = H (N ; C), N is a 3-dimensional Riemannian manifold, V = F1 (N ) and EV (T ) = M(T ). In Chapter 5, we show that EU -converges to EV along with a compactness result (see Theorem 5.1). Here, PV U is independent of and is the 1-current associated withthe jacobian of u. The compactness property (2.8) and lower bound property are verified using local coordinates along with Theorem 3.2 in [11] and Theorem 5.2 in [10] respectively, and a map QU V : ImQV W  U is constructed satisfying the upper bound property (the abstract theorem does not require that the full -limit hold; it suffices that QU V be defined for every v of the form v = QV W (w), w  W ). We then verify that the additional hypotheses of the abstract theorem are satisfied to obtain our main result: there exists0&gt; 0 such that if  (0, 0 ), EU possesses a critical point u and lim 0 EU (u ) = E(vs ),where vs a saddle point of EV in the sense of Definition 2.1. To use the abstract theorem, we need a saddle point vs . Our candidate for vs is anyChapter 1. Introduction7smooth, closed, oriented geodesic in N that does not intersect itself with index l and nullity 0 (this means that the associated Jacobi operator J has l negative eigenvalues 1  · · ·  l and 0 is not an eigenvalue of J). This generalizes the assumptions in [11]. We label such a curve as  and the associated 1-current as TM . The proof that TM is a saddle point in the sense of Definition 2.1 is the content of Chapter 4. To do this, we need to construct maps PW V : V  Rl , QV W : W  V0 and find a number 0 &gt; 0 satisfying properties (1.4)-(1.6), where, as in [11], we take W to be the open l-dimensional ball of radius r1 centered at the origin for some appropriately chosen r1 &gt; 0. The intuition behind the construction of PW V comes from the following: if a closed, oriented, Lipschitz curve  can be written as the graph of a normal vectorfield u over M with uW 1,sufficiently small, then the length of  can be computed in terms ofu through a Taylor expansion. Since  is a geodesic, the first-order term vanishes and, letting L := M(TM ) = lengthM , 1 length() = L + (Ju, u)L2 + u 2 Sturm-Liouville theory allows us to express u =W 1, O(u2 H 1 ). i i=1 ci z ,where {z i } is an orthonormal(in L2 ) basis of eigenfunctions of J for the space of L2 normal vectorfields on M , and thus, letting n =l i i=1 ci z , p= i i=l+1 ci z ,length()  L -1 c - 1 2 2n2 H1+c p 42 H1for some small positive constant c. In this situation, T will be close to TM in the flat norm, where T is the 1-current associated with , so we would like to say that if PW V (T ) = 0, then length()  L with equality if and only if  = M . This can be accomplished by setting PW V (T ) to be (c1 , ..., cl ) so that n is identically 0. PW V (T ) is thought of as the projection of T onto the `unstable directions' of the functional EV near TM . The definition of PW V is then extended to all of V . The argument above suggests that for each w  W , we associate QV W (w) with the curve that is the graph of the normal vectorfield uw =l i=1wi z i over M . Since uw2 H1Chapter 1. Introduction8goes to 0 as r1 goes to 0, QV W (0) will be a strict local maximum of M if r1 is small enough (this is (1.5)). {QV W (w) : w  W } represents roughly the unstable manifold of M near M . Sections 4.1 and 4.2 are spent verifying that there exists a 0 &gt; 0 so that (1.6) holds. As in [11], we assume (1.7) and show T = TM for some 0 &gt; 0. First, we find a number r0 &gt; 0 so that the exponential map is well-defined on a thin tube Kr0 of radius r0 about3 M and write Kr0 in coordinates through a map . Then, assuming 0  r0 , we showthat there is a `piece of T ', again labeled T , that consists of a single Lipschitz curve with no boundary that lies in Kr0 /4 and is homologous to TM in Kr0 (this means T - TM is a boundary in Kr0 ), along with some other good properties (see Lemma 4.4). The proof of this relies strongly on the fact that  is a geodesic. It then suffices to show T = TM . The tactics of JS can no longer be used since the support of T has no endpoints. To do this, we introduce an auxiliary functional M = M + C |PW V |2 , where C &gt; 0 is to be chosen, and we show T = TM assuming TM is the unique minimizer of M among all T such that the support of T lies in Kr0 /4 and T is homologous to TM in Kr0 . (1.8) is verified in section 4.2 and this is where the ideas of Brian White [20] are introduced and expanded upon. This is done by analyzing Qmin =  -1 Tmin , where Tmin is any minimizer of M among all T such that the support of T lies in Kr0 /4 and T is homologous to TM in Kr0 . We show that Qmin is (F, , )-minimal, where F is the pullback of M by . This means that for any closed T with support in a ball of radius   (0, ], F (Qmin ) may be bigger than F (Qmin + T ), but no bigger than F (Qmin + T ) + ()M(Qmin + T ), where  : (0, ]  R+ satisfies lim0+ () = 0. Using regularity theory, we conclude that the support of Tmin can be written as the graph of a C 1 normal vectorfield umin over M whose H 1 -norm goes to 0 as r0 goes to 0, and then we use a Taylor expansion as described above to show umin is identically 0, taking r0 smaller if necessary. In [20], Brian White develops a minmax characterization of minimal surfaces. Through-(1.8)Chapter 1. Introduction9out his paper, F denotes a smooth, parametric, elliptic integrand (see section 2.2) on a Riemannian manifold N and M denotes a smooth, compact embedded submanifold (with or without boundary) of N . His main result states that if M is a critical point of F and is strictly stable for F (the eigenvalues of the associated Jacobi operator are all positive), then there exists a neighbourhood of M such that if S = M is homologous to M in this neighbourhood, then F (M ) &lt; F (S). This is used to show that if M has index l and nullity 0, then M is strictly minimizing in a neighbourhood of M for an auxiliary functional F  essentially of the form F  (S) = F (S) + C |Sf (x)dx|2 for suitablef , C &gt; 0. Also, he shows that there exists a family of surfaces {Mw }, indexed by w in a neighbourhood of the origin in Rl , such that F (Mw ) has a unique local maximum at w = 0. The maps PW V , QV W constructed in this thesis are very similar to the maps SSf (x)dx, w  Mw respectively constructed by Brian White. The overall approachtaken in section 4.2 is very similar to that taken in [20], however, it is not clear that his work can be cited to suit our needs. Moreover, his paper is written in a way that makes it difficult for readers who are not extremely well-versed in GMT regularity theory to extract all of the details. Therefore, the arguments are reconstructed in section 4.2.Chapter 2 Definitionsd d Throughout this paper, Ua (p) (Ba (p)) will denote the open (closed) d-dimensional ball d d d d of radius a centered at p. We will make the convention Ua := Ua (0), Ba := Ba (0).2.1Basic Geometric Measure TheoryLet  denote an open set of a Riemannian manifold N of dimension n. Assume also that  is smooth. Define Dk () := {continuous, linear functionals on Dk ()} where Dk () denotes the space of C  k-forms with compact support in . If T  Dk (), then T is referred to as a k-current in . The boundary of a k-current T , denoted T , is the (k - 1)-current defined by T () := T (d),   Dk-1 ().The mass and flat norm of a k-current T  Dk () are given by M(T ) := supDk (): L () 1|T ()|(2.1)10Chapter 2. Definitions and F(T ) := inf{M(S) : S  Dk+1 (), S = T }11(2.2)respectively. We set F(T ) = + if there does not exist S  Dk+1 () with finite mass such that S = T . Let Rk () denote the space of rectifiable, integer multiplicity k-currents in . If T  Rk (), we will write T =  (, m, T ) to indicate that for   Dk (), T () =(x), T (x) m(x)dHk (x) =(x), T (x) d T (x),where  = sptT is a k-rectifiable set (a union of countably many Lipschitz k-submanifolds of  and a set of Hk -measure zero), T (x) is, at Hk a.e. x, a simple k-vector of unit length orienting the approximate tangent space apTx  and m is a non-negative integer-valued Hk -integrable function called the multiplicity of the current. Set Rk () := {T  Rk () : M(T ) &lt; , T = S for some S  Rk+1 ()}, Fk () := {T  Dk () : T = S for some S  Dk+1 (), F(T ) &lt; }. The restriction of T =  (, m, T )  Rk () to a set A  , denoted T A, is the k-current defined by (T A)() :=A(x), T (x) m(x)dHk (x),  Dk ().For general T  Dk () with finite mass, define (T A)() = T (A ). In this case, the right-hand side can be understood by representing T by a measure  and restricting  to A. Suppose  = (O) for some diffeomorphism  and open set O  Rn . Then any   D1 () can be expressed asn(x) =i=1 i (x)dXiChapter 2. Definitions where X = -1 (x). If f : N  R is Lipschitz, then for a.e. x,n12df (x) =i=1d (f  )|-1 (x) dXi . dXiLet f : 1  2 be a Lipschitz function, where i is an open subset of a Riemannian manifold Ni of dimension ni , i = 1, 2. For T  Dk (1 ), define f T  Dk (2 ) by f T () = T (f ),   Dk (2 ).Here, f  is the pullback of  by f . If i can be written in coordinates through a map i and  =n2 i=1 i (x)dXi  D1 (2 ), X = -1 (x), then 2n1 n2f (~) = xi=1 j=1 i (f (~)) xd ~ ~ h (X)dXi ~ i dXj~ where h = -1  f  1 and X = -1 (~) 2 1 x If I  R is an interval and  : I   is a Lipschitz curve, we say that a 1-current T corresponds to integration over  if for any   D1 (), T () =I i ((t))i (t)dtwhere (t) = -1 ((t)). Let Ik () = {T  Rk () : M(T ) &lt; }. If T  Ik (), then we say that T is an integral k-current in . Any integral 1-current can be written as a finite or countable sum T =iTi ,where each Ti corresponds to integration over a Lipschitz curve i , M(T ) =iM(Ti ) =iH1 (i )Chapter 2. Definitions and M(T ) =i13M(Ti ).This can be seen by isometrically embedding N into Rn+m for some m  0 (see [11], p.6). Note that if T = 0, then Ti = 0 for every i. If T is a k-current in  such that M(T ) + M(T ) &lt;  and f :   R is Lipschitz, then for a.e. s define T, f, s := (T ) {x : f (x)  s} - (T {x : f (x)  s}). The (k - 1)-currents T, f, s are referred to as `the slices of T by level sets of f '. If T is a 1-current corresponding to integration over a Lipschitz curve  : I  , where I is an interval, then there is an explicit formula for T, f, s , namely, T, f, s =tI:(t)f -1 (s)signd f ((t)) (t) . dt(2.3)Here, we use the convention sign(0)=0. This is a special case of Theorem 4.3.8(2) in [7]. Suppose now that N is 3-dimensional. For u = u1 + iu2  H 1 (), let J(u) denote the 2-form du1  du2 and j(u) denote the 1-form1 (udu 2i- udu). We can identify J(u) witha 1-current, denoted J(u), defined through its action on 1-forms   D1 () by J(u)() :=   J(u).Similarly, j(u) can be associated with a 2-current, denoted j(u), that acts on 2-forms   D2 () via j(u)() :=   j(u).One can check through integration by parts that J(u) = 1 ( j(u)). Thus, 2 1 1 F( J(u))  M( j(u)) = j(u) 2 2L1 () .(2.4)If  is a diffeomorphism that takes some open subset O of R3 onto an open subset  of N , then  ( Ju) = J(u  -1 ). (2.5)Chapter 2. Definitions142.2RegularityIn this section, assume N = Rn and  is an open subset of Rn . This terminology is used in Sections 4.1,4.2 and Chapter 6. We encourage the reader to bypass this section for now until needed. A parametric integrand of degree k on an open set  is a continuous real-valued function  = (X, ) defined for X   and   k (Rn ) which is homogeneous of degree 1 in the second variable, i.e., (X, c) = c(X, ) We will assume that  is non-negative. For T =  (, m, T )  Rk () and X  , define (T ) :=for c &gt; 0.(X, T (X))m(X)dXand X (T ) :=~ ~ ~ (X, T (X))m(X)dX.Note that if (X, ) = ||, then (T ) is just the mass of T . For the rest of this section,  denotes a parametric integrand of degree 1. Let ( 1 , ...,n)be an orthonormal reference frame in Rn centered at X0  . Wewrite c = (c1 , ..., , cn ) = (~, cn ) for the associated coordinates of a point X  , i.e., c X = X0 +n i=1 ci i= X0 + AcT for some n × n matrix A satisfying AAT = AT A = I.n)Given , X0 , the reference frame ( 1 , ...,and p = (p1 , ..., pn-1 ), letn-1§ (~, cn , p) := (X, ci=1 §pi i +n ). is called the non-parametric integrand associated with  and the frame ( 1 , ...,n ). is said to be -elliptic in  if for every flat T1 =  (1 , m1 , T1 ) (this means T1 is constant on 1 ) and X  , -1 (M(T2 ) - M(T1 ))  X (T2 ) - X (T1 )Chapter 2. Definitions for all rectifiable T2 with compact support and T2 = T1 .15Suppose K is a compact subset of , Q is a rectifiable 1-current with compact support in K and  is a positive function defined for   (0, ] such that lim0+ () = 0. We say that Q is (, , )-minimal if (Q K)  (Q K + T ) + ()M(Q K + T ) for all rectifiable T with compact support in K, T = 0 and diam(sptT )    . For y  Rn-1 , t  R and X = (y, t), let p(X) := t. Define E(T, t, a) := M(T C(t, a)) - M(p (T C(t, a))) , a1 where C(t, a) := Rn-1 ×Ba (t), a &gt; 0. E(T, t, a) is referred to as the excess of T in C(t, a).Let Ca := C(0, a).2.3-Convergence and Saddle PointSuppose U, V are Banach spaces, EU , EV : U, V  (-, ],  (0, 1] and V0 = {v  V : EV (v) &lt; }. We say that EU -converges to EV as  0 if for all  (0, 1], there exists a continuous map PV U : U  V and a map QU V : V0  U (not necessarily continuous) satisfying lower bound: if v  V0 and {u }  U is a sequence such that PV U (u ) - v then lim inf EU (u )  EV (v) and upper bound: for every v  V0 , EU (QU V (v))  EV (v) and PV U (QU V (v)) - v  0 as  0. (2.7)V 0 as  0,(2.6)VChapter 2. Definitions16We will only be interested in -limits for which the following compactness condition is satisfied: if sup (0,1] EU (u ) &lt; , then {PV U (u )} (0,1] is precompact in V . (2.8)Given a C 1 functional E : U  R, a sequence {uj } is said to be a Palais-Smale j=1 sequence if E(uj )U 0 as j  and{E(uj )} is bounded. j=1The functional E is said to satisfy the Palais-Smale condition if every Palais-Smale sequence is precompact in U .Definition 2.1. We say that EV has a saddle point at vs  V0 if there exists an integer j  0, a number 0 &gt; 0, a neighbourhood W  Rj of 0, a continuous map PW V : V  Rj such that PW V (vs ) = 0 and a continuous map QV W : W  V0 satisfying the conditions EV (vs ) &lt; EV (v) for v  {v  V : 0 &lt; v - vs QV W (0) = vs , PW V  QV W (w) = w for all w  W and for every a &gt; 0, supwW,|w|a V 0 , PW V (v) = 0},(2.9) (2.10) (2.11)EV (QV W (w)) &lt; EV (vs ).(2.12)Chapter 3 Some Known ResultsRecall that we are assuming that N is a Riemannian manifold of dimension n and  is an open subset of N with smooth boundary. The following Sturm-Liouville type existence result can be deduced from the theory of bounded, linear, compact, symmetric operators (see Appendix D, Theorem 7 in [6]). Lemma 3.1. For y  C 1 ([0, L]; Rn-1 ), consider the boundary value problem S(y) := -y + Ay = y, y(L) = B T y(0), d d y(L) = B T y(0) dt dtwhere y T = (y1 , ..., yn-1 ), A : [0, L]  Mn-1,n-1 (R) is a smooth function such that (A(t))T = A(t) for all t  [0, L] and B  Mn-1,n-1 (R) with BB T = B T B = I. There exists a non-decreasing, unbounded sequence {i }  R and {Y i }  C 1 ([0, L]; Rn-1 ) such that {Y i } forms an orthornomal basis for L2 ([0, L]; Rn-1 ) and for each i, Y i satisfies S(Y i ) = i Y i , Y i (L) = B T Y i (0), d i d Y (L) = B T Y i (0). dt dtA useful inequality related to slices is the following (see [18], p.158). Refer to Section 2.1 for the definition of T, f, s , the slice of T by f -1 (s). Lemma 3.2. Suppose T  Dk (), M(T ) + M(T ) &lt;  and f :   R is Lipschitz. Then M( T, f, s )ds  sup | f (x)|M(T ).x17Chapter 3. Some Known Results18The following is a consequence of the convex hull property (see [20], p.207), which says that the support of a minimizer of a frozen elliptic integrand with boundary constraint B must lie in the convex hull of sptB. Lemma 3.3. Let  be an elliptic, parametric integrand of degree k on   Rn , K be a compact, convex subset of  and X  . Suppose Q is a rectifiable k-current supported in K and T minimizes X among all rectifiable k-currents with boundary equal to Q. Then sptT  K.Below is an isoperimetric type inequality. Lemma 3.4. Suppose N is compact and T  R1 (N ). There exist constants cN , CN &gt; 0 such that if M(T ) &lt; cN , then we can write T = S for some rectifiable 2-current S with M(S)  CN (M(T ))2 . Lemma 3.4 can be deduced as follows: first, cover N by finitely many open setsn {j }m where each j is diffeomorphic to a ball URj . Let cN be minimum of diam(j k ) j=1over all j, k such that j  k = . Writing T as a sum of indecomposables T =iTi ,we must have for each i that sptTi  j for some j. Applying 4.2.10 in [7] to the pullback of Ti and using Lemma 3.3, we can find Si  R2 (j ) such that Si = Ti and M(Si )  Cj M(Ti )2 . Setting S =iSi gives the desired result.Next, we state a cone type inequality for integral 1-currents on Rn (see Proposition 3.4 in [19]). Basically, for a given closed curve, this result is obtained by considering the `cone' swept out by contracting the curve to a suitable point. Lemma 3.5. If T  R1 (Rn ), then we can write T = S for some S  I2 (Rn ) satisfying M(S)  2diam(sptT )M(T ).Chapter 3. Some Known Results19The following is the important homotopy formula for 1-currents; see [18], p.139 for a proof. Given 2 non-self-intersecting Lipschitz curves f1 , f2 supported in an open cylindern-1 n-1 n-1  = UR × (a, b) that run from UR × {t = a} to UR × {t = b}, we use this to writethe difference of the curves as a boundary in  and also to get an upper bound for the mass of this surface in terms of f1 , f2 . In fact, Lemma 3.5 can probably be deduced from Lemma 3.6 below. Lemma 3.6. Let f1 , f2 : I   be Lipschitz functions, where I  R is an open interval and  is an open, convex subset of Rn . Define h : [0, 1] × I   by h(s, t) = sf2 (t) + (1 - s)f1 (t). If T  D1 (I) and h|[0,1]×sptT is proper, then h (E 1 [0, 1] × T )  D2 () and (f2 ) T - (f1 ) T = h (E 1 [0, 1] × T ) + h (E 1 [0, 1] × T ). Moreover, M(h (E 1 [0, 1] × T ))  sup |f1 - f2 | sup(|f1 | + |f2 |)M(T ).sptT sptTHere, E 1 is the standard 1-current obtained by integration of 1-forms over R.Finally, we state the aforementioned abstract theorem due to Jerrard and Sternberg [11], which is the backbone to proving our main theorem (Theorem 5.2). Theorem 3.1. Suppose that U, V are Banach spaces and that {EU } (0,1] is a family of C 1 functionals mapping U to R that -converge to a limiting functional EV : V0  R via maps PV U : U  V and QU V : V0  U . Assume also that the compactness condition (2.8) holds. Let vs  V be a saddle point in the sense of Definition 2.1. Assume also that PW V is uniformly continuous in {v  V : v - vsV 20 },(3.1) (3.2)QU W := QU V  QV W : W  U is continuous for all ,Chapter 3. Some Known Results PV U  QU W (w) - QV W (w) and EU (QU W (w))  EV (QV W (w)) uniformly in w  W as  0.V20  0 uniformly in w  W as  0, (3.3)(3.4)Then given  &gt; 0, there exists ~ &gt; 0 and a Palais-Smale sequence {uj } for every j=1  (0, ~) such that sup |EU (uj ) - EV (vs )|  .jIn particular, if EU satisfies the Palais-Smale condition for every , then there exists0&gt; 0 and a critical point u of EU for every  (0, 0 ) such that lim 0 EU (u ) = EV (vs ).Remark 3.1. Jerrard and Sternberg remark in their paper [11] that an inspection of the proof shows that we do not need the full -limit to hold. In particular, QU V need only be defined on ImQV W .Chapter 4 Saddle Points of MassThroughout this section, (N, g) will be a compact, orientable, n-dimensional Riemannian manifold with N =  that is isometrically embedded in Rn+m , n  2, m  1. For x  N , we identify Tx N as an n-dimensional subspace of Rn+m in the natural way. M will denote a 1-dimensional, connected, smooth submanifold of N without boundary. We can write M = {(t)} for some   C  (R/LZ; N ). Suppose also that  is a geodesic, i.e., D ( (t)) = 0 for all t  R/LZ. dt Here,D dt(4.1)denotes the covariant derivative. Since  is a geodesic, | | is constant. We shalltake this constant to be 1 so that the length of  is L. Let TM denote the multiplicity 1 current corresponding to integration over . Assume TM  R1 (N ). For p  M , we write Tp M to denote that normal space at M , characterized by  Tp M  Tp M = Tp N .Choose ui  C  (R/LZ) so that {u1 (t), ..., un-1 (t)} forms an orthonormal basis for T(t) M for all t  R/LZ. Therefore, for any t  R/LZ,{ (t), u1 (t), ..., un-1 (t)} is an orthonormal basis for T(t) N . 21(4.2)Chapter 4. Saddle Points of Mass For x1 , x2  N , define d(x1 , x2 ) := inf{length :  : [0, 1]  N is piecewise differentiable, (0) = x1 , (1) = x2 }. Set d(x) := dist(x, M ) = inf{d(x, x) : x  M } and ~ ~ Kr := {x  N : d(x) &lt; r}, r  (0, r0 ), 0 &lt; r0 &lt; 1.22n-1 For r0 sufficiently small, define  : Br0 × R/LZ  Kr0 by n-1(y, t) := exp (t),i=1yi ui (t) . Here, exp: {(x, v) : x  M, v  Tx M, |v| &lt; r0 }  Kr0 is a smooth map given byexp(x, v) := (1, x, v), where s  (s, x, v) is the unique geodesic of N which, at the instant s = 0, passes through x with velocity v. Let [t]  R/LZ denote the conjugacyn-1 class of t  R. From now on, when thinking of  as a map defined on Br0 × R, we willwrite (y, t) to mean (y, [t]). Note that for any t  R/LZ and i = 1, ..., n - 1,  (y, t) t and  (y, t) yi = d(0,t) (ei ) = d(exp(t) )0 (ui (t)) = ui (t).(0,t)=  (t)(0,t)(4.3)(4.4)Therefore, using (4.2), it follows from the inverse function theorem that for every t  R/LZ, there exists a neighbourhood t of (0, t) such that |t is a smooth diffeomorphism. Covering {0} × R/LZ by m ti for some {ti }  R/LZ and selecting r0 smaller i=1n-1 n-1 if necessary so that Br0 × R/LZ  m ti , we conclude that  : Br0 × R/LZ  Kr0 i=1is a smooth diffeomorphism. For x  Kr0 , let p(x) := the unique point of M closest to x in N-1 = (n (x))Chapter 4. Saddle Points of Mass and v(x) := the unique vector  Tp(x) such that expp(x) (v(x)) = x n-1 i=1 -1 -1 i (x)ui (n (x)).23= Set A := {u  C 1 ([0, L]) : u(t)  T(t) M for every t  [0, L],u(L) = u(0) andD u(L) dt=D u(0)}. dtFor u  A, there exists a positive number su such that |su(t)| &lt; r0 for all t  [0, L] and s  (-su , su ). Consider the variation hu : (-su , su ) × [0, L]  Kr0 defined by hu (s, t) := exp((t), su(t)). Let Tu,s denote the multiplicity 1 current corresponding to integration over hu (s, ·). Note that TM = Tu,0 for any u  A. The Jacobi operator J is a linear, second-order, differential operator defined through the relation d2 M(Tu,s ) ds2 Explicitly, Ju = -(u + R( , u) ), where R is the curvature of N (see chapter 9 in [5]). This relies on the assumptions that  is a geodesic and an arclength parametrization. Let {u1 (t), ..., un-1 (t)} be an orthonormal basis for T(t) M such that, for each i = p p= (Ju, u)L2 (0,L) .s=01, ..., n - 1, D i u (t) = 0 for all t  [0, L]. dt p Here, we use the subscript `p' because each ui is a `parallel' vectorfield on M . Since p  {u1 (0), ..., un-1 (0)} and {u1 (L), ..., un-1 (L)} are orthonormal bases for T(0) M = T(L) M , p p p pthere exists a matrix B = (bij ) satisfyingn-1ui p=j=1bij uj pandBB T = B T B = I.Chapter 4. Saddle Points of Mass If u  A and u =n-1 i=124yi ui , it follows that y(L) = B T y(0), y (L) = B T y (0) and pn-1 n-1Ju =i=1-yi +j=1aij yjui , pwhere y T = (y1 , ..., yn-1 ) and aij = -(R( , ui ) , uj ). Note that aij = aji . Let A = (aij ) p p and S(y) = -y + Ay. Using Lemma 3.1, there exists a non-decreasing, unbounded sequence {i }  R and {Y i }  C 1 ([0, L]; Rn-1 ) such that {Y i } forms an orthonormal basis for L2 ([0, L]; Rn-1 ) and for each i, Y i satisfies S(Y i ) = i Y i , Setting z i =n-1 j=1Y i (L) = B T Y i (0),d i d Y (L) = B T Y i (0). dt dtYji uj , it follows that p Jz i = i z i , z i (L) = z i (0), D D i z (L) = z i (0) dt dtand {z i } forms an orthonormal basis for U := {u  L2 (0, L) : u(t)  T(t) M for a.e. t  [0, L)}.Lemma 4.1. For any u  U  H 1 (0, L), we can write u = and p = i i=l+1 ci z . i i=1 ci z .Let n =l i i=1 ci zThen (Jn, n)L2 (0,L)  1 n2 H 1 (0,L) .Moreover, there exists a positive constant c, independent of u, such that n and (Jp, p)L2 (0,L)  c p2 H 1 (0,L) . 2 L2 (0,L)c n2 H 1 (0,L)Proof. The first claim is vacuous unless 1 &lt; 0. If 1 &lt; 0, we computel l(Jn, n)L2 (0,L) =i,j=1i ci cj ij =i=1i c 2  1 n i2 L2 (0,L) 1 n2 H 1 (0,L) .Chapter 4. Saddle Points of Mass For the second claim, notel l25n2 L2 (0,L)Ci,j=1|ci ||cj |  C/2i,j=1(c2 + c2 ) = lC n i j2 L2 (0,L) ,where C &gt; 0 depends on z 1 , ..., z l . Thus, n2 L2 (0,L) (1 + lC)-1 n2 H 1 (0,L) .Finally, to prove the third claim, first noteL (|p (t)| + (a(t), p(t)))dt = (Jp, p)L2 (0,L) =0 i,j=l+12i ci cj ij  l+1 p2 L2 (0,L)where a(t) = -R( (t), p(t)) (t). Now, for   (0, 1),  p2 L2 (0,L)  p2 L2 (0,L)+ (1 - )[(Jp, p)L2 (0,L) - l+1 pL 02 L2 (0,L) ]= (Jp, p)L2 (0,L) -((a(t), p(t)) + (1 - )l+1 |p(t)|2 )dt.~ ~ Since |a|  C|p| for some constant C depending on  and the Christoffel symbols of the connection on M , we can select  small enough so that ~ (a, p) + (1 - )l+1 |p|2  (l+1 - (C + l+1 ))|p|2  0. Therefore, for this value of ,  p and (Jp, p)L2 (0,L)  l+1 p  + l+12 H 1 (0,L) . 2 L2 (0,L) (Jp, p)L2 (0,L)Define V := F1 (N ) and EV : V  [0, ] by    M(T ) if T  V0 := R (N ) 1 EV (T ) :=   + if not.(4.5)Chapter 4. Saddle Points of Mass26Theorem 4.1. Suppose the Jacobi operator J associated with M has finite index and 0 nullity. Then TM is a saddle point of EV in the sense of Definition 2.1.Note that our assumptions imply 1  · · ·  l &lt; 0 &lt; l+1  · · · if l := index of J &gt; 0 and 0 &lt;  1  2  · · · if l = 0. To prove Theorem 4.1, we need to construct maps PW V : V  Rj , QV W : W  V0 satisfying the conditions of Definition 2.1 for some nonnegative integer j and neighbourhood W of 0 in Rj . We claim that this is possible with j = l. If l = 0, we adopt the convention R0 = {0} and set PW V (v) = 0 for all v  V . Our arguments will show TM is a local minimizer of mass in the flat norm topology. Now we define PW V for l &gt; 0. Keeping in mind that we need to verify (2.9), we would like to define PW V so that, if T corresponds to integration over a Lipschitz curve and sptT can be written as the graph of a normal vectorfield u over M with u sufficiently small, then, writing u = i i=1 ci z , W 1,we have PW V (T ) = (c1 , ..., cl ). The reasonfor this is that such a current T as described above will be close to TM in the flat norm and thus, if PW V (T ) = 0, we want to be able to say M(T )  L with equality if and only if T = TM . To see this, first note that we can take su = 2. Then, since  is a geodesic, 1 M(T ) = M(Tu,1 ) = L + (Ju, u)L2 + 2 1 = L + (Jn, n)L2 + 2 where n = uW 1, l i i=1 ci z , pu W 1, O( u 2 1 ) H 1 (Jp, p)L2 + u W 1, O( u 22 H1 )= i i=l+1 ci z .If c &gt; 0 is the constant from Lemma 4.1 and u2 H 1 )|is small enough so that u M(T )  L -W 1, |O(c 8u2 H1 ,we have1 c - 1 2 2n2 H1+c p 42 H1 .Chapter 4. Saddle Points of Mass27Thus, if PW V (T ) = 0, n is identically 0 and M(T )  L with equality if and only if T = TM . In this case, we can think of PW V as being the projection of T onto the negative eigenspace associated with J. Before defining PW V , we first need to make some definitions. Let Z(x) be the l-vector (z 1 ( (x)) · v(x), ..., z l ( (x)) · v(x)) and define  : N  (D1 (N ))l by    (|v(x)|)Z(x)d (x) for x  Kr 0 (x) :=   0 for x  N \Kr0 , where-1  (x) :=  -1 (p(x)) = n (x)  and   Cc ([0, r0 ); [0, 1]) with (s) = 1 for every s  [0, r0 /2]. Note that dLC/r0 . Now, for T  V , set PW V (T ) := T ()  Rl . The desired property of PW V described above is verified in Lemma 4.3. First, we show that PW V is continuous with respect to the F-norm. Lemma 4.2. PW V is uniformly continuous in the flat norm topology. Proof. Given T1 , T2  V , we can find S  D2 (N ) such that T1 - T2 = S and M(S)  2F(T1 - T2 ). Then |PW V (T1 ) - PW V (T2 )| = |(T1 - T2 )()| = |S()| = |S(d)|  d M(S)  F(T1 2 d which implies the above assertion.- T2 ),Lemma 4.3. For u  U and t  [0, L), let (t) = exp((t), u(t)). Suppose  is Lipschitz, Im  Kr0 /2 and T is the multiplicity 1 current corresponding to integration over .Chapter 4. Saddle Points of Mass Then, writing u = i i=1 ci z ,28we have PW V (T ) = (c1 , ..., cl ).Proof. Set (t) =  -1 ((t)). Since T corresponds to integration over  and Im  Kr0 , PW V (T ) = T () = = d (  )|(t) i (t)dt dXi 0 L d (|v((t))|)Z((t))  ((t))dt. dt 0 (|v((t))|)Z((t))LBy definition of p, v and  , we have p((t)) = (t), v((t)) = u(t) and  ((t)) = t. Therefore, since |u(t)| &lt; r0 /2 for every t  [0, L) and {z i } are orthonormal in L2 (0, L), i=1LPW V (T ) =0(z 1 (t) · u(t), ..., z l (t) · u(t))dtL L=0z 1 (t) · u(t)dt, ...,0z l (t) · u(t)dt= (c1 , ..., cl ).Note that if T = TM above, then u(t) = 0 for every t  [0, L) and PW V (TM ) = 0.4.1-1 Flat Local Minimizers of Mass in PW V (0)The goal of this section is to prove (2.9), i.e., to show that TM is a strict local minimizer of mass in the flat norm topology among currents T  R1 (N ) with PW V (T ) = 0. The conclusion of the following proposition is equivalent to (2.9). Proposition 4.1. There exists 0 &gt; 0 such that if T  R1 (N ), F(T - TM ) &lt; 0 , PW V (T ) = PW V (TM ) = 0 and M(T )  M(TM ) = L, (4.8) (4.6) (4.7)Chapter 4. Saddle Points of Mass then T = TM .29First we would like to show that if T is close to TM in the flat norm, then there exists a `piece' of T that is uniformly close to TM . This is a partial result of the following Lemma, which relies heavily on the fact that T is 1-dimensional. Basically, this is due to the fact that if a Lipschitz curve stretches between two sets A and B, then the length of the curve has to be at least the distance between A and B, whereas if an n-dimensional surface, n  2, stretches between two sets, the n-dimensional surface area can be arbitrarily small.3 Lemma 4.4. Suppose T  R1 (N ) satisfies (4.6) and (4.8). If 0  r0 and r0 is takensufficiently small, then there exists a 1-current T  R1 (N ) such that  := sptT consists of a single Lipschitz curve with no boundary,   Kr0 /4 and    -1 (t) =  for all t  R/LZ. In addition, M(T - T ) = M(T ) - M(T ) and T - TM = S for some 2-current S with sptS  K2r0 /3 and M(S ) &lt; . Proof. 1. Assumption (4.6) implies there exists S1  D2 (N ) such that S1 = T - TM in N and M(S1 ) &lt; 0 . (4.12) (4.11) (4.10) (4.9)~ We would like to replace T by a current T with support in Kr0 satisfying (4.6) and a slightly weaker form of (4.8). This can be accomplished by finding a slice of S1 by d with sufficiently small mass.Chapter 4. Saddle Points of Mass30Let's compute | d(x)| for x  Kr0 . Set G to be the 3 × 3 matrix with entries gij = (Xi , Xj ) and G-1 = (g ij ). By definition, (y, t) = y,t (1), where y,t is the unique geodesic which, at the instant s = 0, passes through (t) with velocityn-1 i=1yi ui (t).Since d((y, t)) must be attained by a path connecting (y, t) and p((y, t)) = (t) and geodesics locally minimize arclength (see Proposition 3.6 in [5]),1 n-1d((y, t)) =0|y,t (s)|ds = |y,t (0)| =i=1yi ui (t) = |y|.~-1 ~-1 Therefore, letting Y (x) = (1 (x), ..., n-1 (x), 0), we have d(x) = d ~ ~ g ij ( -1 (x)) (d  )|-1 (x) Xi ( -1 (x)) ~ dXj i,j=1 n-1 ~-1 j (x) ij ~-1 ~ = g ( (x)) X ( -1 (x)) |Y (x)| i i,j=1 ~ Y (x)G-1 ( -1 (x))(Y (x))T . |Y (x)|2nand | d(x)|2 = Using Lemma 3.2, it follows thatr0M( S1 , d, r )dr r0 /2supxKr0 \Kr0 /2| d(x)| M(S1 Kr0 \Kr0 /2 )  C1 0 ,which implies the existence of a number r  (r0 /2, r0 ) such that ~2 M( S1 , d, r )  2C1 0 /r0  2C1 r0 . ~~ ~ Let S1 = S1 Kr and T = T Kr + S1 , d, r . Then ~ ~ ~ ~ ~  S1 = T - TM , and2 ~ M(T )  L + 2C1 r0 .~ M(S1 )  M(S1 ) &lt; 0(4.13)2. Now we would like to estimate |  | in Kr0 . First note that  D gnn = 2 t , t yi dyi = 2 t , D y dt i =2 d (t , yi ) - 2 dt D t , yi , dtChapter 4. Saddle Points of Mass which implies  d gnn (0, t) = 2 ( (t), ui (t)) - 2 yi dt using (4.1) and (4.2). Sincen-1  ((y, t)) = t for all (y, t)  Ur0 × R/LZ,31D  (t), ui (t) dt=0we have (x) =n i=1~ ~ g in ( -1 (x))Xi ( -1 (x)) and g nn (X)2 1 nn (g nn (0, t), ..., gyn-1 (0, t)) g nn (0,t) y1|  ((X))| == g nn (0, t) + · y + O(|y|2 )  = 1 - 1 ( y1 gnn (0, t), ..., yn-1 gnn (0, t)) · y + O(|y|2 ) 2= 1 + O(|y|2 ). 3. Define ~ 0 := {t  (0, L) : M( T , , t Kr0 /8 ) = 0}, ~ 1 := {t  (0, L) : M( T , , t ) = M( T, , t Kr0 /8 ) = 1} and ~ 2 := {t  (0, L) : M( T , , t )  2}. Clearly, |0 | + |1 | + |2 |  L. Also, using Lemma 3.2,L(4.14)~ M( T , , t )dt 0~ sup |  (x)| M(T )xKr02 2  (1 + C2 r0 )(L + 2C1 r0 ) ()(4.15) 4L, ~ which implies M( T , , t ) &lt;  for a.e. t  (0, L). Here, (*) indicates that we are taking ~ r0 sufficiently small. We claim that, most of the time, T stays in Kr0 /8 and intersectsChapter 4. Saddle Points of Mass32level sets of  only once, i.e., |1 | is big. This will be verified in Steps 4 and 5 by showing |0 | and |2 | are small. 4. Note that for a.e. t  (0, L), ~ ~  S1 , , t =  S1 , , t = = ~ T , , t - TM , , t ~ T , , t - (t) .~ Hence, for a.e. t  0 ,  S1 , , t Kr0 /8 = -(t) . Using this, it follows that r0 ~ C1 M( S1 , , t )  8 for a.e. t  0 . (4.16) To see this, let f (x) =  (d(x)) where { } (0,r0 /16)  Cc ([0, r0 /8); [0, r0 /8)) satisfies (0)  r0 /8 - ,  (0) = 0 and  t  0 , r0 - 8 1. Then df=f C1 and for a.e.  (0) = f ((t)) ~ = - S1 , , t (f ) ~ = - S1 , , t (df ) ~  C1 M( S1 , , t ).Let  0 to obtain (4.16). Now, again using Lemma 3.2, r0 |0 |  C1 8  C10 2  C1 (1 + C2 r0 )0 ()~ M( S1 , , t )dt0 L~ M( S1 , , t )dt 2C1 0 , which gives2 |0 |  16C1 0 /r0  16C1 r0 .(4.17)5. Similar to (4.15), we compute ~ M(T )  1 2 1 + C2 r0L~ M( T , , t )dt0 (1 -2 C2 r0 )(|1 |+ 2|2 |).Chapter 4. Saddle Points of Mass332 Combining this with (4.13),(4.14) and (4.17), we see that |2 |  3(6C1 + LC2 )r0 , whichimplies2 |1 |  L - (34C1 + 3LC2 )r0 .(4.18) i=1~ ~ 6. Writing T as a sum of indecomposable currents T = corresponds to integration over a Lipschitz curve i , we claim that ~Ti , where each Ti~ ~ if for some i, one has sptTi  Kr0 /8 = , then sptTi  Kr0 /4 .(4.19)To verify this, first note that, once again using Lemma 3.2 and recalling the definition of 1 , ~ M(T Kr0 /8 )  1 2 1 + C2 r0 ~ M( T , , t Kr0 /8 )dt1 (1 -2 C2 r0 )|1 |2  L - 2(17C1 + 2LC2 )r0 .Thus, if the support of i intersects both Kr0 /8 and N \Kr0 /4 , then i must have arclength ~ ~ at least r0 /8 and r0 /8  =(4.13)~ M(T (N \Kr0 /8 )) ~ ~ M(T ) - M(T Kr0 /8 )2 4(9C1 + LC2 )r0 .Taking r0 sufficiently small leads to a contradiction. Therefore, (4.19) holds. 7. For any positive integer j, let 1,j := We claim that2 |1,1 |  L - 10(7C1 + LC2 )r0 .t  1 :iH0 (~i   -1 (t)) = j .(4.20)The explicit formula (2.3) for the slice of an indecomposable 1-current implies ~ M( T , , t ) i~ M( Ti , , t ) =iH0 (~i   -1 (t)) Chapter 4. Saddle Points of Mass for a.e. t. It follows thati34H0 (~i   -1 (t))  1 for a.e. t  1 and hence that |1,j | = |1 |.j=1 2 Since |J |(x) = |  (x)|  1 + C2 r0 , it follows from the coarea formula that L(4.21)H0 (~i   -1 (t))dt = 0 i ~2 |J |(x)dH1 (x)  (1 + C2 r0 )H1 (~i ). Using this, we have2 L + 2C1 r0 (4.13) =~ M(T ) ~ M(Ti )i=iH1 (~i )  1 2 1 + C2 r0 1- (1 - (1 - H0 (~i   -1 (t))dt i  = (4.21)1 2 C2 r02 1 + C2 r0j|1,j |j=1 2 C2 r0 )|1,1 | + 2j=2|1,j |=2 C2 r0 )(2|1 |- |1,1 |).This together with (4.18) implies (4.20). 8. Taking r0 sufficiently small, we have |1,1 | &gt; L/2 &gt; 0. Fix t  1,1 at which the Lebesgue density is 1. Let i be the unique closed curve that intersects  -1 (t ). The ~ point of intersection must be in Kr0 /8 since t  1 . Using (4.19), we conclude i is ~ entirely contained in Kr0 /4 . As the Lebesgue density at t  1,1 is 1, we can find points in 1,1 arbitrarily close to t . Since i is a closed, Lipschitz curve, we must have ~ i   -1 (t) =  for all t  R/LZ, ~ which verifies (4.10).Chapter 4. Saddle Points of Mass359. Now set T to be the current corresponding to integration over the curve i chosen ~ in the step above. Writing T as a sum T = Ti of indecomposable currents, we must~ have T = Tk for some k since T Kr0 /4 = T Kr0 /4 . Thus, M(T - T ) = Mi=kTi=i=kM(Ti ) =iM(Ti ) - M(Tk ) = M(T ) - M(T ),which is (4.11). 10. Finally, we need to show that T is homologous to TM in K2r0 /3 . To do this, parametrize i with respect to t  [t , t + L] and let (t) =  -1 (~i (t)). Using Lemma ~ n-1 (3.6) with the convex set  = Ur0 /2 × (t , t + L) and f1 (t) = (t), f2 (t) = (0, t) for n-1 t  (t , t + L), we can find a 2-current S supported in Ur0 /2 × (t , t + L) such that -1 T - T{0}×(t ,t +L) = S - T(t +L),(0,t +L) + T(t ),(0,t ) with M(S)  L( L+ L)( L+ 1) &lt; .Here, TX1 ,X2 denotes the multiplicity 1 current corresponding to integration over the line segment lX1 ,X2 , where lX1 ,X2 : [0, 1]  Rn is defined by lX1 ,X2 (s) := (1 - s)X1 + sX2 . Therefore, T - TM =  S with spt S  K2r0 /3 and M( S)  C M(S) &lt; , as required. Note that C = 1 + O(r0 ) since |J|(0, t) = 1 for every t  R/LZ.Our goal now is to show that T = TM . To do this, we first introduce a functional F =  M, the pullback of M induced by , then state and verify some good properties satisfied by F .n-1 For X  Ur0 × R and   Rn , letF (X, ) := |D(X)|,Chapter 4. Saddle Points of Mass36which is a parametric integrand of degree 1 (see Section 2.2). Here, we take D(X) to representn i=1 i Xi (X) T(X) N .n-1 From this, we can define a functional on R1 (Ur0 × R) byF (T ) :=F (X, T (X))m(X)dH1 (X),n-1 where T =  (, m, T ). Also, for X  Ur0 × R and T =  (, m, T )  R1 (Rn ), definethe `frozen' integrand FX by FX (T ) :=~ ~ ~ F (X, T (X))m(X)dH1 (X).One can easily verify F (T ) = M( T ) (4.22)n n-1 and the existence of a constant C3 = C3 () &gt; 0 such that if sptT  B (X)  Ur0 × R,then |FX (T ) - F (T )|  C3 M(T ). (4.23)n-1 Lemma 4.5. For any X  Ur0 × R and   Rn ,(1 - C3 r0 )||  F (X, )  (1 + C3 r0 )||.(4.24)Proof. Using (4.3),(4.4) and recalling { (t), u1 (t), ..., un-1 (t)} is an orthonormal basis for T(t) N , we have |D(0, t)| = || Therefore, letting X = (y, t), |F (X, ) - ||| = MVTfor any t  R and   Rn .||D(y, t)| - |D(0, t)|| |(D(y, t) - D(0, t))| C3 r0 ||.Chapter 4. Saddle Points of Mass From now on, we will assume that r0 is small enough so that ||/2  F (X, )  3||/2n-1 for all X  Ur0 × R and   Rn .37(4.25)n-1 Lemma 4.6. If T =  (, m, T )  R1 (Ur0 × R), then(1 - C3 r0 )M(T )  F (T )  (1 + C3 r0 )M(T ).n-1 In addition, for every X  Ur0 × R,(4.26)(1 - C3 r0 )M(T )  FX (T )  (1 + C3 r0 )M(T ).n-1 ~ ~ Proof. Using (4.24) with  = T (X), X    Ur0 × R, we have(4.27)~ 1 - C3 r0  F (X, T (X))  1 + C3 r0(4.28)n-1 ~ for every X  Ur0 × R. Multiplying through by m(X) and integrating over  gives~ ~ (4.27). Setting X = X in (4.28), multiplying through by m(X) and integrating over  gives (4.26).We now show that M(T - T ) is small. Set 1 := L - M(T ) = L - M(T ) - M(T - T )  0 and Q :=  -1 T . From Step 10 in the proof of Lemma 4.4, we have M(Q ) = H1 ()  L. Hence,(4.26) (4.11) (4.8)L  M(Q )  which implies1 (4.22) F (Q ) = 1 - C3 r0 C3 Lr0 . 1 - C3 r01+C3 r0 1 - C3 r0M(T ),1 (4.29)Chapter 4. Saddle Points of Mass This means, recalling (*) indicates that we are taking r0 sufficiently small, M(T - T )(4.11)38=M(T ) - M(T ) L - M(T ) 1 2C3 Lr0 . (4.30)(4.8) =(4.29),()Using Lemma 3.4 and taking r0 sufficiently small, there exists a 2-current S2 such that S2 = T - T and M(S2 )  CN (M(T - T ))2 . Thus,2 M(S2 )  CN (M(T - T ))2 = CN (M(T ) - M(T ))2  CN 1 . (4.11) (4.8)(4.31)For r  (0, 2r0 /3), let Hr := {T  R1 (N ) : sptT  Kr and T - TM = S for some S  D2 (N ) with support in K2r0 /3 and M(S) &lt; }. Recalling (4.9) and (4.12), T  Hr0 /4 . Also note that 0  Hr for any r  (0, 2r0 /3). / This is true because we can find   D1 (N ) with (x) = d (x) for x  Kr0 , which implies TM () = TM (d ) = L &gt; 0 and S() = S(d2  ) = 0 for any S  D2 (N ) with sptS  K2r0 /3 . For T  V , define M (T ) := M(T ) + C |PW V (T )|2 , where C &gt; 0 is to be chosen. We claim that there exists a C &gt; 0 such that TM is the unique minimizer of M in Hr for r  (0, r0 /2) and r0 sufficiently small. Let's assume this for now and complete the proof of Proposition 4.1. Since TM minimizes M in Hr0 /4 , M(TM ) = L and PW V (TM ) = 0, we have L = M (TM )  M (T ) = M(T ) + C |PW V (T )|2 = L - 1 + C |PW V (T )|2 , which implies 1  C |PW V (T )|2 .(4.32)Chapter 4. Saddle Points of Mass Using this, we have 1 (4.7)39C |PW V (T )|2 C |PW V (T ) - PW V (T )|2 C |(T - T )()|2 C |S2 ()|2 C |S2 (d)|2 C d C d 1 /2.2 2  (M(S2 )) 2 2 4  CN 1== = = (4.31)(4.29),()Since 1  0, this implies 1 = 0. Looking at the above string of inequalities, 1 = 0 implies PW V (T ) = 0. Thus, M (T ) = L = M (TM ). Since we are assuming (4.32), we have T = TM . Finally, recalling (4.30), 1 = 0 implies T = T = TM .4.2RegularityThis section consists of lemmas which lead to (4.32).Lemma 4.7. For r  (0, 2r0 /3), there exists a minimizer of M in Hr . Proof. First note that TM  Hr and mr := inf M (T )  M (TM ) = L &lt; .T HrSelect a minimizing sequence {Ti }  Hr . Then i=1 M (Ti )  mr as i  .Define the norm Fr0 by Fr0 (T ) = inf{M(S) : S  D2 (N ), sptS  K3r0 /2 , S = T }. Since sptTi  Kr , M(Ti )  C and Ti = 0 for all i, the Federer-Fleming Compactness Theorem implies that there exists a subsequence {ij } and Tr  R1 (N ) such that theChapter 4. Saddle Points of Mass support of Tr lies in Kr , M(Tr )  C, Tr = 0 and Fr0 (Tij - Tr )  0 as j  .40(4.33)Using (4.33), there exists a j such that Tij - Tr = Sr,j for some Sr,j  D2 (N ) with sptSr,j  K3r0 /2 and M(Sr,j ) &lt; 1. Since Tij  Hr , Tij - TM = Sj for some Sj  D2 (N ) with sptSj  K3r0 /2 and M(Sj ) &lt; . For this j, set Sr = Sr,j + Sj  D2 (N ). Then Tr - TM = Sr ,sptSr  K3r0 /2 and M(Sr ) &lt; . Therefore, Tr  Hr . Since M is weakly lower semicontinuous, PW V is weakly continuous and (4.33) implies Tij Tr , we have M (Tr )  lim inf M (Tij ) = mr .jIt follows that M (Tr ) = mr = min M (T ).T HrLemma 4.8. Suppose T  Hr , r  (0, 2r0 /3), and T has finite mass. Then M( Tr , , t )  1 for a.e. t  (0, L). In particular, Tr satisfies (4.34). Proof. As T  Hr , we can write T - TM = S for some 2-current S with M(S) &lt; . Suppose M( T, , t ) = 0 on a set of positive measure A  (0, L). Since  S, , t = T, , t - (t) for a.e. t  (0, L), we would have  S, , s = -(s) for some s  A. If  is any smooth function with compact support in N such that  = 1 in a neighbourhood of  -1 (s), then 1 = ((s)) = (s) () = - S, , s () = - S, , s (d) = 0, (4.34)Chapter 4. Saddle Points of Mass which is nonsense. Therefore, (4.34) holds.41We now show that Tr satisfies some good properties. First, since M(Tr )  M (Tr )  M(TM ) = L and Tr = 0, we can write Tr as a sum of indecomposable currents Tr = (4.35)iTr,i ,where each Tr,i corresponds to integration over a closed, Lipschitz curve r,i , M(Tr ) =iM(Tr,i ) and M(Tr ) =iM(Tr,i ).Similar to the definitions in the proof of Lemma 4.4, define r := {t  (0, L) : M( Tr , , t ) = 0}, 0 r := {t  (0, L) : M( Tr , , t ) = 1} 1 and r := {t  (0, L) : M( Tr , , t )  2}. 2 Note that these sets are disjoint and (4.34) implies |r | = 0. Also, using Lemma 3.2 and 0 the estimate for |  | derived in Step 2 in the proof of Lemma 4.4, |r | + 2|r | 1 2 r 1 LM( Tr , , t )dt +r 2M( Tr , , t )dt=0M( Tr , , t )dt2 (1 + C2 r0 )M(Tr ) 2 L + C2 Lr0 2 |r | + |r | + C2 Lr0 , 1 2(4.35) =2 2 which implies |r |  C2 Lr0 and |r | = L - |r |  L - C2 Lr0 . Now, letting 2 1 2r := 1,jt  r : 1iH0 (r,i   -1 (t)) = jand using an argument similar to the one in Step 7 in the proof of Lemma 4.4, we have2 |r |  2|r | - L - 2C2 Lr0 , 1,1 1Chapter 4. Saddle Points of Mass()422 which implies |r |  L - 4C2 Lr0 &gt; L/2 &gt; 0. From now on, t will denote a point 1,1 rin r at which the Lebesgue density is 1. Let r,i be the unique closed curve that 1,1 intersects  -1 (t ). It follows that r r,i   -1 (t) =  for all t  R/LZ. For r  (0, 2r0 /3), setn-1 Qr to be the 1-current compactly supported in Br × [t - 2L, t + 2L] r r n-1 defined on cylinders of height L by  -1 Tr  R1 (Ur0 × R/LZ).Note that M(Qr ) = 2 for every r  (0, 2r0 /3). Letn-1 K = B3r0 /4 × [-2L, 3L]so that sptQr  K for all r  (0, 2r0 /3). Since Qr is an integral current in Rn , we can write Qr as a sum of indecomposable currents Qr =iQr,i , where each Qr,i corresponds to integration over a closed (inn-1 n-1 Ur0 × (t - 2L, t + 2L)) Lipschitz curve r,i : [0, Lr,i ]  Br × [t - 2L, t + 2L]. Note r r r rthat the above discussion implies that we must have 4Tr,i =  Qr,j for some j, where r,j is the only curve among {r,i } satisfying r,i (Lr,i ) = r,i (0) ± (0, 4L). The curvesn-1 {r,i : i = j} correspond to closed loops in Br × [t - 2L, t + 2L]. Thus, recalling that r rp(X) = t for X = (y, t)  Rn-1 × R, sptQr  p-1 (t) =  Also, M(Qr C(t, L/2)) = M(Qr C(t + L/2, L/2)) rn-1  H1 (r,j |Br ×[t ,t +L] ) r rfor all t  (t - 2L, t + 2L). r r(4.36)(4.37) L and, using (4.22),(4.35)F (Qr C(t, L/2)) = M(Tr )  L(4.38)Chapter 4. Saddle Points of Mass for any t  (t - 3L/2, t + 3L/2). r r43Our goal now is to show that Qr is an almost minimizer for F ; this means that small perturbations of Qr by closed, rectifiable 1-currents T supported in a tiny ball of radius  may make F (Qr + T )  F (Qr ), but no smaller than F (Qr ) - ()M(Qr + T ) for some positive function () - 0. Our proof will require the use of Lemma 3.3, therefore, we must first verify that F satisfies the ellipticity condition described in Section 2.2.n-1 Lemma 4.9. There exists  &gt; 0 such that F is -elliptic in Ur0 × R. n-1 ~ Proof. Set FX,c () := FX () - c||, X  Br0 × R,   Rn , c &gt; 0. 0~ First, we will show that if there exists a c &gt; 0 such that FX,c is convex for alln-1 n-1 X  Ur0 × R, then F is c-1 -elliptic in Ur0 × R. Suppose Ti =  (i , mi , Ti )  R1 (Rn )has compact support,i = 1, 2, T1 = T2 and T1 is flat. We can find a number a &gt; 0 suchn that T2 - T1 = S for some 2-current S with sptS  Ua .Note thatT2 d T2=T1 d T1= M(T1 )T1 . To see this, let  : Rn  R be~ a continuous linear function and define   D1 (Rn ) so that (X),  = () for alln n ~ X  Ua . This implies d = 0 on Ua . Hence,T2 d T2 -T1 d T1=(T2 )d T2 -(T1 )d T1= (T2 - T1 )() = S() = S(d) = 0. Since  is arbitrary, we have Now, ~ ~ FX (T1 ) - cM(T1 ) = FX,c (M(T1 )T1 ) = FX,c T2 d T2 . T2 d T2 = T1 d T1 .Using Jensens's Inequality and the fact that FX,c is homogeneous of degree 1, we have FX (T1 ) - cM(T1 )  ~ FX,c (T2 )d T2 = FX (T2 ) - cM(T2 ),Chapter 4. Saddle Points of Massn-1 which means F is c-1 -elliptic in Ur0 × R.44n-1 ~ We claim that there exists c &gt; 0 such that FX,c is convex for all X  Ur0 × R. LetlX () = D(X), f () = || and G be the n × n matrix with entries gij = Xi · Xj . For any   Rn , ~  T D2 FX,c () = i k gik(X) |lX ()|2 - j l gij (X)gkl (X) - c T D2 f () |lX ()|3 |lX ()|2 |lX ()|2 - (lX () · lX ())2 = - c T D2 f () 3 |lX ()| |lX ()  lX ()|2 |  |2 -c  lX 3 ||3 ||3 |  |2 -1  ( lX -3 lX -4 - c) . ||3n i=1-1 Here, lX is a map that takes the vectorXi  T(X) N to   Rn . Let-3 -1 lX -4c=1 min lX n-1 2 XBr0 ×R/LZ. = 1 for every t  R andNote c &gt; 0 for r0 sufficiently small since l(0,t)-1 = l(0,t)-1 n-1 ~ lX , lX are continuous functions of X. Then FX,c is convex for all X  Ur0 × R and n-1 F is -elliptic in Ur0 × R with  = c-1 .Before stating Lemma 4.10, we note that the definition of (F, , )-minimality given in Section 2.2 is that of Bombieri [4] since his paper gives the regularity results that we require. In [20], Brian White proves a result similar to Lemma 4.10 below, however, since he uses a different definition of almost-minimality, his argument does not exacly apply here. Lemma 4.10. Let () =  . There exists a positive constant , independent of r, suchthat Qr is (F, , )-minimal for all r  (0, 2r0 /3) and r0 sufficiently small. Proof. Set  := r01/4 ()&lt; L/2.n-1 For X  Ur0 × R and   (0, ], letS := {T  R1 (Rn ) : sptT is compact in K,n sptT  B (X) and T = 0}.Chapter 4. Saddle Points of Mass To prove the lemma, it suffices to shown n ~ F (Qr B (X))  (1 + C)FX (Qr B (X) + T )45(4.39)n ~ for some absolute constant C &gt; 0, where T minimizes FX (Qr B (X) + ·) in S. Indeed,if the above holds, then for any T  S,n F (Qr B (X)) (4.23)n ~ (1 + C)FX (Qr B (X) + T ) n (1 + C)FX (Qr B (X) + T ) n n (1 + C)(F (Qr B (X) + T ) + C3 M(Qr B (X) + T )) n F (Qr B (X) + T )+(4.26)C3 (1 + Cr0 ) +()1/4C 1 - C3 r0n M(Qr B (X) + T ) n F (Qr B (X) + T ) + n F (Qr B (X) + T ) + n M(Qr B (X) + T )M(Qr + T ).n ~ Adding F (Qr K\B (X)) to both sides gives the desired result. We assume that T =0 since otherwise (4.39) follows easily from (4.23) and (4.27). Also, we assume thatn ~ ~ H1 (sptT  sptQr ) &gt; 0 otherwise (4.39) again follows easily since FX (Qr B (X) + T ) = n ~ FX (Qr B (X)) + FX (T ).Sincen n ~ ~ ~ ~ FX (Qr B (X) + T ) = FX (Qr sptT + T ) + Fx (Qr B (X)\sptT ) n ~ and T minimizes FX (Qr B (X) + ·) in S, it follows that^ ~ ~ T = Qr sptT + T minimizes FX among all T  R1 (Rn )n ~ with compact support in K, sptT  B (X) and T = (Qr sptT ).(4.40)n ~ To see this, let A = B (X)\sptT and suppose T  R1 (Rn ) has compact support in K, n n ~ sptT  B (X) and T = (Qr sptT ). Then (Qr A + T - Qr B (X)) = 0 and n ^ ~ FX (T ) + FX (Qr A) = FX (Qr B (X) + T ) FX (Qr A + T )  FX (Qr A) + FX (T ),Chapter 4. Saddle Points of Mass ^ which implies FX (T )  FX (T ), as claimed.46n-1 n ^ ~ Using Lemma 3.3, we conclude sptT  Br × [-2L, 3L]  B (X) as spt(Qr sptT ) n-1 n is contained in the convex set Br × [-2L, 3L]  B (X). Thus,n-1 n ~ sptT  Br × [-2L, 3L]  B (X).(4.41)~ ~ Let S minimize M among all 2-currents with boundary equal to T . Using (4.41),n-1 n ~ Lemma 3.3 implies sptS  Br × [-2L, 3L]  B (X). Now we would like to show n ~ ~ M(S) &lt; . First, as sptT  B (X), Lemma 3.5 implies that there exists a 2-current S~ such that S = T and ~ M(S)  4M(T ). Next, note that  Qr = 4Tr and ~ M(T )  24L. To verify (4.44), we estimate ^ M(T )(4.27) ()(4.42)(4.43)(4.44)(4.40)^ (1 - C3 r0 )-1 FX (T ) ~ (1 - C3 r0 )-1 FX (Qr sptT ) 1 + C3 r0 ~ M(Qr sptT ), 1 - C3 r0(4.27) which implies ~ M(T )  (4.26)()^ ~ M(T ) + M(Qr sptT ) 1 + C3 r0 ~ + 1 M(Qr sptT ) 1 - C3 r0 1 + C3 r0 ~ + 1 (1 - C3 r0 )-1 F (Qr sptT ) 1 - C3 r0 6F (Qr ) 24M(Tr ) 24L.(4.22),(4.43)=(4.35)Chapter 4. Saddle Points of Mass Therefore, ~ ~ M(S)  M(S)  4M(T ) &lt; 96L.47(4.45)n-1 n ~ Since T is homologous to 0 in Br × [-2L, 3L]  B (X) and Tr is homologous to~ TM in K2r0 /3 , it follows that Tr +  T is homologous to TM in K2r0 /3 . This, (4.41) and ~ (4.45) imply Tr +  T  Hr . Thus, using the fact that Tr minimizes M in Hr , M(Tr ) =  = M (Tr ) - C |PW V (Tr )|2 ~ M (Tr +  T ) - C |PW V (Tr )|2 ~ ~ M(Tr +  T ) + C (PW V (Tr +  T ) + PW V (Tr ))· ~ (PW V (Tr +  T ) - PW V (Tr )) =(4.22)~ ~ ~ M(Tr +  T ) + C (2Tr +  T )() ·  T () ~ ~ M(Tr +  T ) + C (2M(Tr ) + F (T ))  |~ T ()|.~ Note that, using (4.35), (4.26) and the above estimate for M(T ), ~ 2M(Tr ) + F (T )  50L. Recalling (4.42), ~ | T ()| = =   (4.26),() ()| S()| | S(d)| d d M(S) C M(S)  M(T )  F (T ).4C d 8C d~~Combining the above 2 estimates and letting C4 = 400LC C d,we have~ ~ F (Qr C(s, L/2))  F (Qr C(s, L/2) + T ) + C4 F (T ) for any s  [t - 3L/2, t + 3L/2]. r rChapter 4. Saddle Points of Mass48n ~ Since    &lt; L/2, sptT  B (X)  C(s, L/2) for some s  [t - 3L/2, t + 3L/2], r rand thusn n ~ ~ F (Qr B (X))  F (Qr B (X) + T ) + C4 F (T ) n n n ~ ~  F (Qr B (X) + T ) + C4 (F (Qr B (X) + T ) + F (Qr B (X))).This impliesn F (Qr B (X))()1 + C4  n ~ F (Qr B (X) + T ) 1 - C4 n ~ (1 + 4C4 )F (Qr B (X) + T ) n n ~ ~ (1 + 4C4 )(FX (Qr B (X) + T ) + C3 M(Qr B (X) + T )) n ~ (1 + C)FX (Qr B (X) + T ),(4.23)(4.27) which is (4.39).We are now in a position to show that Qr can be written as the graph of a C 1 function with H 1 norm on the order of r0 for some   (0, 1).Lemma 4.11. For r  (0, 2r0 /3) and r0 sufficiently small, sptTr = {exp((t), ur (t)), t  [0, L]}, for some C 1 function ur such that ur (t)  T(t) M for all t  [0, L],ur (0) = ur (L),D D ur (0) = ur (L) dt dtand urW 1, ([0,L]) 0 as r0  0+ .Proof. We would like to use Lemmas 6.1 and 6.2. To start, we show that the excess of Qr over thin cylinders is small (refer to Section   2.2 for the definition of excess). Take r0 smaller if necessary so that (2 r0 )-1 (L/2 - r0 )  ~ ~ is an integer I = I(r0 ). Fix t  (0, L) and let ti = t + 2i r0 , i = -I, ..., -1, 0, 1..., I.Chapter 4. Saddle Points of Mass Since the excess is always non-negative, ~ E(Qr , t, r0 )I49i=-I E(Qr , ti , r0 )=(4.26)(4.38) =~ (M(Qr C(t, L/2)) - L) ~ F (Qr C(t, L/2)) -1/2 r0 -L 1 - C3 r0 L -1/2 -L r0  1 - C3 r0 C3 L r0 . 1 - C3 r0 r0-1/2~ ~ ~ Recalling (4.36) and assuming X = 0 for some X  p-1 (t) (after a translation), it suffices to show the assumptions made in the Appendix hold with T = Qr ,  = Fn-1 n-1  = Ur0 × R, K = B3r0 /4 × [-2L, 3L]  () = , 0 &lt;     () R = r0 &lt; L.First, (T1) holds due to Lemma 4.10. Next, (T2) and (T4) follow from the discussion preceeding Lemma 4.10. Recall that  from Lemma 4.10 was chosen to be r0 diam(sptQr CR )  2 r2 + r0  () 1/4and thus13r0  /4,which is (T3). If we take r0 sufficiently small, then (T5) and (6.2) will hold. Property (P2) follows from (4.25). Since F is smooth on  × Rn \{0}, we can find a positive  constant so that (P3)-(P6) are satisfied with 0 =  F (X0 , 0 ) and () = . Finally, (P1) is a result of Lemma 4.9 and we will assume that the constant  obtained in Lemma 4.9 is larger than the constants found to satisfy (P2)-(P6). Therefore, using (6.4), we have1 ~ ~ spt(Qr C(t, R/2)) = {(yr (s), s), s  BR/2 (t)}for some C 1 function yr with yrL C5 r0 .1/8Chapter 4. Saddle Points of Mass50~ Since t  (0, L) was chosen arbitrarily, it follows that spt(Qr C(L/2, L/2)) coincidesn-1 with the graph of a function yr  C 1 ([0, L]; Br ) satisfying yr (0) = yr (L), yr (0) = yr (L)and yrW 1, (1 + C5 )r0 .n-11/8Setting ur (t) :=i=1yr,i (t)ui (t),we have sptTr = {exp((t), ur (t)), t  [0, L]}D with ur (0) = ur (L), dt ur (0) = D u (L) dt randW 1,ur C6 r0 .1/8(4.46)Finally, we complete the proof of (4.32). Lemma 4.12. There exists a positive constant C such that if Tr is a minimizer of M in Hr , r  (0, r0 /2), and r0 is sufficiently small, then Tr = TM . Proof. Write ur = nr + pr , nr =l i i=1 cr,i z . lLemma (4.3) implies c2 = nr r,i2 L2 .|P (Tr )| =i=12(4.47)Assume sr = urC([0,L])= yrC([0,L])&gt; 0, otherwise there is nothing to prove. Con-sider the variation hr : (-2sr , 2sr ) × [0, L)  Kr0 defined by hr (s, t) := ( ssr yr (t), t) and let Tr,s denote the multiplicity 1 current corresponding to integration over the Lipschitz curve hr (s, ·). Since |(syr (t), 1)|  1,  (syr (t), t) t(4.24) () 1 - C3 r0  1/2.Expanding M(Tr,s ) in a Taylor series about s = 0, we have M(Tr,s ) = L + 1 (Jur , ur )s2 + Rr,2 (s). 2s2 rChapter 4. Saddle Points of Mass Note that d M(Tr,s ) ds since  is a geodesic. Also note that |Rr,2 (s)|  o( ur (Y1 , ..., Y2n ) where Y = Yr (s, t) = (ss-1 yr (t), t, ss-1 yr (t), 1), r r we have d3 M(Tr,s ) = ds3 which implies |Rr,2 (s)|  C7 r0 s-3 ur r1/8 2 3 H 1 |s| L 3 2 H 1 )|s| .51=0s=0Indeed, viewing F as a function of Y =YsT0d 2 D F (Y ) Ys dt, ds(4.48)using Taylor's Theorem, (4.46) and the fact that ur · uj = yr,j . Therefore, 1 M(Tr ) = M(Tr,sr ) = L + (Jur , ur ) + Rr,2 (sr ) 2 and, using Lemma 4.1, L = (4.49),(4.47)(4.49)M (TM ) M (Tr ) L + 1 (Jur , ur ) + Rr,2 (sr ) + C nr 22 L2 2 L2== (4.48),()L + 1 (Jnr , nr ) + 1 (Jpr , pr ) + C nr 2 2 L + ( 1 + C c) nr 22 H1+ Rr,2 (sr )+c 2pr +2 H1 c 4+ Rr,2 (sr ) pr2 H1 .L + ( 1 + c(C - 1 )) nr 2 42 H1If we choose C &gt;c-21 , 4cthen nr and pr are identically 0, which implies Tr = TM .Chapter 4. Saddle Points of Mass524.3Construction of QV Wl i=1l l Let W = Ur1 where r1 &gt; 0 is to be chosen. For w  Br1 , define zw (t) :=wi z i (t).Assume r1 is small enough so thatl |zw (t)| &lt; r0 /2 for all t  [0, L] and w  Br1 .(4.50)Define w : [0, L]  Kr0 /2 by w (t) := exp((t), zw (t)) and QV W (w) to be the multiplicity 1 current corresponding to integration over the Lipschitz curve w .Lemma 4.13. QV W is continuous. Proof. For X1 , X2  Rn , let TX1 ,X2 denote the multiplicity 1 current corresponding to integration over lX1 ,X2 , where lX1 ,X2 : [0, 1]  Rn is defined by lX1 ,X2 (s) := (1-s)X1 +sX2 .l Fix w1 , w2  Ur1 . Using Lemma 3.6 with fi (t) =  -1 (wi (t)) for i = 1, 2, t  (0, L), n-1 we can find Sw1 ,w2  D2 (Ur0 × (0, L)) such thatSw1 ,w2 =  -1 QV W (w2 ) -  -1 QV W (w1 ) + Tf1 (0),f2 (0) - Tf1 (L),f2 (L) and M(Sw1 ,w2 )  L f1 - f2Lf1 + f2L C8 |w1 - w2 |. Therefore,  Sw1 ,w2 = QV W (w2 ) - QV W (w1 ) and F(QV W (w2 ) - QV W (w1 ))  M( Sw1 ,w2 )  C C8 |w2 - w1 |, which implies QV W is continuous.Chapter 4. Saddle Points of Mass To complete the proof of Theorem 4.1, we need to verify (2.10)-(2.12). · Verification of (2.10): Clearly, z0 is identically 0 and 0 = , which implies QV W (0) = TM . · Verification of (2.11): This follows directly from (4.50) and Lemma 4.3. · Verification of (2.12): For a  (0, r1 ) and w  W with |w|  a, sw = zwC([0,L])53&gt; 0. Consider the variationhw : (-2sw , 2sw ) × [0, L]  Kr0 defined by hw (s, t) = exp((t), ss zw (t)). Let Tw,s denote w the multiplicity 1 current corresponding to integration over the Lipschitz curve hw (s, ·). Then, arguing as in the proof of Lemma 4.12, M(QV W (w)) = M(Tw,sw ) = L + (Jzw , zw ) + o(|w|2 ). Note thatl(Jzw , zw ) =i=12 i wi  l |w|2 .Taking r1 small enough so thato(|w|2 ) |w|2 -l /2, we haveM(QV W (w))  L + l |w|2 /2. Thus, supl wUr1 ,|w|aM(QV W (w))  L + l a2 /2 &lt; L.Chapter 5 ApplicationsThroughout this chapter, (N, g) will be a 3-dimensional compact, orientable Riemannian manifold without boundary, U = H 1 (N ; C), V = F1 (N ) and EV will be given by (4.5). Define the Ginzburg-Landau energy EU : U  R by EU (u) := 1 2| ln | | u|2 +N(|u|2 - 1)2 . 22(5.1)5.1-convergence of EU on NTheorem 5.1. (2.6) and (2.8) are satisfied with PV U = PV U (u) = Ju/. Furthermore, there exists a family of maps QU V : ImQV W  U such that (2.7) is satisfied for every v  ImQV W . We will verify the claims made in Theorem 5.1 through Lemmas 5.1,5.3,5.4 and 5.5. Lemma 5.1. If {u }  U is a sequence of functions such that sup EU (u ) &lt; , then { Ju /} is precompact in V . Proof. For every x  N , there exists an open set x  N , a positive number Rx and a   diffeomorphism x : BRx  x such that x (0) = x and { X1 x (0), X2 x (0), X3 x (0)}forms an orthonormal basis for Tx N . Letting Gx be the 3 × 3 matrix with entries 54Chapter 5. Applications55  ( Xi x , Xj x ), we have G-1 (X) = I + Ox (|X|). Take Rx smaller if necessary so that x|Ox (|X|)|  1/2 for all X  BRx . Since N is compact, N = m xi for some {xi }  N . i=1 Let i = xi , i = xi , Ri = Rxi and Gi = Gxi . Set v i = u  i . Note that EH 1 (UR;C) (v i i) =1 2| ln | 1 | ln | 1 | ln || v i (X)|2 +URi1 (|v i (X)|2 - 1)2 dX 22 1 (|v i (X)|2 - 1)2 dX 22v i (X)G-1 (X)( v i (X)) + iURi=iv i |-1 (x) G-1 (-1 (x))( v i |-1 (x) ) i i i i 1 (|u (x)|2 - 1)2 |J-1 |(x) i 22+ CEU (u ). ~ ~ In the remainder of the proof, we will denote (2.2) by F . Also, define F by F (T ) = inf{M(R) + M(S) : T = R + S, R  D1 (), S  D2 ()}. Using the above estimate for EH 1 (URi;C) (vi) and Theorem 3.2 in [11], there existsQ1  F1 (UR1 ) and a subsequence { 1 }  (0, 1] such that j FUR1 ( Jv 11 - Q1 )  0jas j  .Since sup EH 1 (URi;C) (vi) &lt;  for all i = 1, ..., m, we can use Theorem 3.2 in [11]m - 1 more times to conclude that for i = 2, ..., m, there exists Qi  F1 (URi ) and a subsequence { i }  { ji-1 j }such that as j  .FURi ( Jv ii - Qi )  0jSettingj=m j ,we have as j   for all i = 1, ..., m.FURi ( Jv ij - Qi )  0Chapter 5. Applications Thus, using (2.5), Fi ( Ju j - (i ) Qi ) = Fi ( J(v ij  -1 ) - (i ) Qi ) i = Fi ((i ) ( Jv ij - Qi )  CFURi ( Jv ij - Qi )  0 as j   for all i = 1, ..., m.56(5.2)Let Tji = Ju j - (i ) Qi . Since Tji is a boundary in i , (5.2) implies ~ Fi (Tji )  0 as j   for all i = 1, ..., m.m i=1 (i )(5.3) Qi iLet {i } be a partition of unity subordinate to the cover {i }, T = and Tj =m i=1Tji i = Ju j - T . It follows from (5.3) that ~ FN (Tj )  ~  C  0m i=1~ FN (Tji i ) ~ Fi (Tji ) (5.4)m i=1as j  .~ for some constant C &gt; 0 depending on {1 , ..., m }. In particular, Tj converges weakly to 0 as j  . From this, we can conclude T is a boundary in N using Theorem 7 in Section 5.3.2 of [8]. Thus, T  F1 (N ). Now we need to show that FN (Tj )  0 as j  . First, choose Rj , Sj such that ~ Tj = Rj + Sj and M(Rj ) + M(Sj )  2FN (Tj ). Owing to (5.4), we can find J  Z+ ~ such that j  J implies FN (Tj )  cN /2, where cN &gt; 0 is the constant mentioned in the assumptions of Lemma 3.4. Now, since Rj = Tj = 0 j, we can find Sj  D2 (N ) such that Rj = Sj and M(Sj )  CN (M(Rj ))2 for j  J. Therefore, if j  J, FN (Tj )  FN (Tj - Rj ) + FN (Rj )  M(Sj ) + M(Sj ) ~ ~  2FN (Tj ) + 4CN (FN (Tj ))2 , which implies, using (5.4), FN ( Ju j - T )  0 as j  , as required.Chapter 5. Applications57We now state and prove a lemma that we will need in order to verify the lower bound property of -convergence. Lemma 5.2. If || is a Radon measure on R3 supported on a 1-dimensional curve and apTX0 || exists, then3 ||(Br (X0 )) = 0. 3 r0 ||(Br (X0 ))limBy definition, the above hypotheses mean there exists a line l  R3 passing through X0 and  &gt; 0 such that ||r H1 l = ||0 as r  0 where ||r (A) := 1 ||(X0 + rA). r3 3 Proof. Without loss of generality, assume X0 = 0. Since ||0 ((B1 )) = ||0 (B1 ) = 0and ||r||0 , we have3 3 lim ||r (B1 ) = ||0 (B1 ) = 0r0andr03 3 lim ||r (B1 ) = ||0 (B1 ) &gt; 0.Therefore,3 3 3 ||(Br ) ||r (B1 ) ||0 (B1 ) = lim = = 0. 3 3 3 r0 ||r (B1 ) r0 ||(Br ) ||0 (B1 )limWith this, we proceed to show (2.6) is satisfied with PV U = PV U (u) = Ju/. The idea of the proof is this: first, we use local coordinates  and consider Q = -1 T on a ball in R3 . Then, since T is rectifiable, we can identify a set of `good points' X of full T measure on which we can approximate Q Br (X) in a certain sense by a straightline l for r sufficiently small. By slicing Q Br (X) orthogonal to l and using known 2-d results in [10], we are able to deduce a local version of the required result. To complete the proof, we apply the local result to suitable disjoint open sets {i } which cover sptT except for a small set of T -measure. Lemma 5.3. If T  R1 (N ) and {u }  U is a sequence of functions such that Ju - T then lim inf EU (u )  M(T ).V 0 as 0,(5.5)Chapter 5. Applications Proof. If lim inf EU (u ) = +, then there is nothing to prove. Therefore, assume lim inf EU (u ) &lt; .58Let x0  sptT and  be an open subset of N such that x0   and  is dif3 feomorphic to BR for some R &gt; 0 and diffeomorphism . Assume (0) = x0 and    { X1 (0), X2 (0), X3 (0)} forms an orthonormal basis for Tx0 N . Let G = (gij ) be the3 × 3 matrix with entries gij = (Xi , Xj ), G-1 = (g ij ) and { j } be a subsequence such thatjlim Ej (u j ) = m := lim inf E (u ) &lt; .(5.6)Set |v(X)|G := v(X)T G(X)v(X),3 where v is any vectorfield defined on BR taking values in R3 . Taking R small enough, wehave 3 1 |v(X)|e  |v(X)|G  |v(X)|e 2 2 where | · |e denotes length with respect to the Euclidean inner product.3 If T =  (, m, T ) and  =  i dXi  D1 (UR ) where {dX1 , dX2 , dX3 }  1 (R3 ) denotes(5.7)the standard orthonormal basis of covectors on R3 , then -1 T () =  i (X)d-1 ((X)), T ((X)) m((X)) detG(X)dX. i Letting T  = i Xi ,  = D-1 (T ) = (1 , 2 , 3 ), || = H1 m G and  = || , we have -1 T () = Clearly, ||(A) = M(T (A))3 for any A  BR . As T is rectifiable, it follows that for ||-a.e. X0 , i di .(5.8)|(X0 )|G = 1,(5.9)Chapter 5. Applications apTX0 || exists and ||(Br (X0 )) &gt; 0 for all r &gt; 0.3 Also, since T has finite mass and (5.7) holds,   L1 (UR ; d||), which implies59 (5.10)(5.11)r0lim1 3 ||(Br (X0 ))|(X0 ) - (X)|e d|| = 0 for ||-a.e. X0 .3 Br (X0 )(5.12)If X0 satisfies (5.9)-(5.12), then we say that X0 is a good point.3 Define µG for A  BR byµG (A) :=1 2| ln |Xv A(X)G-1 (X)(Xv(X)) + detG(X)dX,1 (|v (X)|2 - 1)2 22where v = u  . Recalling (5.6), there exists a subsequence of { j }, still denoted { j },j and a nonnegative Radon measure µ such that µG converges to µ weakly as measures.Assume, without loss of generality, that X0 = 0 is a good point and (X0 ) = e3 (otherwise we can analyze the pullback of -1 T by an isometry R3  R3 that takes (0, 0, 0) to X0 and the z-axis onto the line through X0 in the direction of (X0 )). We claim that3 d|| ||(Br (X0 )) (X0 ) = lim   for µ-a.e. good point X0 . 3 r0 µ(Br (X0 )) dµ(5.13)3 From this point on, assume r  (0, R). Since Br (X0 ) denotes a closed ball andG(X0 ) = I,j 3 3 3 µ(Br (X0 ))  lim inf µG (Br (X0 ))   -1 (1 - Cr) lim inf µ j (Br (X0 ))(5.14)3 where, for A  BR ,µ (A) := Define µ ,t by µ ,t (S) := 1 2| ln |1 2| ln ||AXv(X)|2 +1 (|(v (X)|2 - 1)2 dX. 22|SXv(X1 , X2 , t)|2 +1 (|v (X1 , X2 , t)|2 - 1)2 dX1 dX2 . 22Chapter 5. Applications3 where S  BR  p-1 (t).60Since we are assuming (5.5), Theorem 5.2 in [10] implies 3 can be represented byt 3 slices 3 . This means in particular that for any open ball U  BR , t 3 (U  p-1 (t))dt3 (U ) =(5.15) di (t)ai (t) fort where p(X) = p(X1 , X2 , t) = t. For almost every t, 3 has the form some integers di (t) and points ai (t)  U  p-1 (t). The theorem also impliest lim inf µ j ,t (U  p-1 (t))  3 (U  p-1 (t))(5.16)3 for any open ball U  BR . Therefore, 3 µ(Br (X0 )) (5.14) =Fatou3 (1 - Cr) lim inf µ j (Ur (X0 ))(1 - Cr) lim inf (1 - Cr) (1 - Cr)3 µ j ,t (Ur (X0 )  p-1 (t))dt3 lim inf µ j ,t (Ur (X0 )  p-1 (t))dt t 3 3 (Ur (X0 )  p-1 (t))dt(5.16)(5.15)=3 (1 - Cr)3 (Ur (X0 ))=(1 - Cr)3 Ur (X0 )d3 .Recalling that (X0 ) = e3 and G(X0 ) = I, d3 =3 Ur (X0 ) 3 Ur (X0 )e3 · (X)d|| |(X) - (X0 )|e d||.3 Br (X0 )3  ||(Ur (X0 )) -Combining the above 2 estimates and using Lemma 5.2 as well as (5.12), we conclude3 3 µ(Br (X0 ))  (1 - o(1))||(Br (X0 )).(5.17)Here, o(1) is a quantity which  0 as r  0+ . Since (5.11) holds, we have for r sufficiently small,3 ||(Br (X0 ))  (1 + o(1)). 3 µ(Br (X0 ))(5.18)Chapter 5. Applications Basic theorems on differentiation of measures guarantee limr0 Hence, letting r  0+ in (5.18) we obtain (5.13). Now we wish to show M(T )  m . From (5.17), we can deduce || using (5.8), M(T ) = 1 3 ||(UR )  = =(5.13)3 ||(Br (X0 )) 3 (X )) µ(Br 061 exists µ-a.e..µ. Thus,1 3 ||({X  UR : X is a good point})  1 d|| (X)dµ 3  {XUR :Xis a good point} dµ3 µ(UR ).3 Since UR denotes an open ball,j 3 3 M(T )  µ(UR )  lim inf µG (UR ) = lim inf Ej (u j ) = m .jjFinally, we need to show lim inf EU (u )  M(T ). Fix  &gt; 0. We can cover sptT3 by disjoint open sets {i = i (URi )}, where i is a diffeomorphism as  above, andM(T i )  M(T ) - . Using the above result, lim inf EU (u )  Let   0+ to obtain lim inf EU (u )  M(T ), as required. lim inf Ei (u )  M(T i )  M(T ) - .Now we need to construct a map QU W = QU V  QV W : W  U satisfying (2.7). We will denote QU W by uw . First we need to recall and make some definitions.l First recall that W = Ur1 , where r1 &gt; 0 is the number chosen in Section 4.3 andQV W (w) is the multiplicity 1 current corresponding to integration over w , where w (t) = exp((t), zw (t)), zw (t) =l i=1wi z i (t), t  [0, L].l Let ui (t) = ui (t, w), i = 1, 2, be smooth functions on R/LZ × Br1 such that, for every wt  R/LZ, {u1 (t), u2 (t), w (t)} forms an orthonormal basis for Tw (t) N and ui (t) = ui (t). w w 0Chapter 5. Applications Let262Kw,r := for r  (0, r0 /2] andexp w (t),i=1yi ui (t) w2 : y  Ur , t  [0, L)Mw := {w (t) : t  [0, L)}.2 Recall that for X = (y, t)  Br0 × R/LZ, (X) = exp((t), -1 i (x) for i = 1, 2 and y(x) = y1 (x) + iy2 (x). 2 i=1yi ui (t)). Set yi (x) :=Define Ow : N  N by     exp exp(( (x)), (|v(x)|)zw ( (x))), Ow (x) :=    x where2 i yi (x)vw (x) i=1for x  Kr0 , for x  N \Kr0i vw (x) = ui ( (x)) + (|y(x)|)(ui ( (x)) - ui ( (x))) w  and   Cc ([0, r0 ); [0, 1]) with (s) = 1 for every s  [0, r0 /2]. Note that if r1 is smallenough, then Ow is a smooth diffeomorphism that takes M onto Mw , K onto Kw, and Kr0 \K onto Kr0 \Kw, for every  (0, r0 /2). Let v 0 (x) :=    y(x)/|y(x)| for x  Kr , 0   y 0 (x) for x  N \Kr0where y 0 is any smooth function taking values in S 1 such that y 0 (x) = y(x)/|y(x)| in a neighbourhood of Kr0 . We refer the reader to the Appendix for the justification of the existence of such a function y 0 .-1 Let w = Ow   and u0 = v 0  Ow . We claim that wJu0 = QV W (w). w(5.19)To see this, first note that Ju0 (N \Kw,r0 /2 ) = 0 since u0 is smooth and takes values in w w S 1 away from Mw . Hence it suffices to show Ju0 Kw,r0 /2 = QV W (w). w (5.20)Chapter 5. Applications Using (2.5) and setting u(X) = u(y, t) =y |y|63 for X  R2 × R/LZ, we have-1 2 (w ) ( Ju0 Kw,r0 /2 ) = Ju (Ur0 /2 × R/LZ). wIt is well-known that Ju = T{0}×R/LZ (see Example 4 in [9]). Thus,-1 (w ) ( Ju0 Kw,r0 /2 ) = T{0}×R/LZ w=  -1 TM-1 =  -1 (Ow ) QV W (w) -1 = (w ) (QV W (w)),which implies (5.19). For  (0, r0 /4), define    |y(x)|/  -1 and uw := v  Ow .v (x) := v 0 (x)for x  K , for x  N \K1Lemma 5.4. As 0, PV U (uw ) - QV W (w)V 0 uniformly on W .d d ij Proof. Let Gw be the 3 × 3 matrix with entries ( dXi w , dXj w ), G-1 = (gw ) and waw,i (x) = uw (x) ×d (u  w ) dXi w-1 w (x)- u0 (x) × wd (u0  w ) dXi w-1 w (x)where × denotes the complex cross product. Using (5.19), PV U (uw ) - QV W (w) =(2.4) 1 F(  1 2 VJuw - Ju0 ) wL1 (N ) =j(uw ) - j(u0 ) wKw, L 0 U21 2 1 = 2ij -1 gw (w (x))aw,i (x)aw,j (x) ij gw (X)aw,i (w (X))aw,j (w (X))||Jw |(X)dydtChapter 5. Applications642 l Note that since w is a smooth function in both X  Br0 × R/LZ and w  Br1 , thereexists an absolute constant C9 &gt; 0 such that |G-1 (X)| + |Jw |(X)  C9 w2 l for all X  Br0 × R/LZ and w  Br1 . Also, aw,3 (w (X)) = 0 and aw,i (w (X))  3 |y|forall i = 1, 2, X  U 2 × R/LZ, w  W and  (0, r0 /4). Therefore, PV U (uw ) - QV W (w)V 2 6C9  L 0 U21 dydt |y|2 = 12C9 Lwhich implies PV U (uw ) - QV W (w)V- 0 uniformly on W .0Lemma 5.5. As 0, EU (uw )  M(QV W (w)) uniformly on W .Proof. First we break EU (uw ) into pieces and then analyze each piece: EU (uw ) = 1 2| ln | | uw |2 +Kw, Kr0 \Kw,| uw |2 + +N \Kr0| uw |2 +Kw,(|uw |2 - 1)2 221 = (E1 + E2 + E3 + E4 ). 2| ln | · Estimate of E1 : Note that for X  U 2 × R/LZ, uw (w (X)) = E1 =Kw, L y1 +iy2. Using this,-1 -1 -1 (uw  w )|w (x) G-1 (w (x))( (uw  w )|w (x) ) w=0 U2 L(uw  w )(X)G-1 (X)( (uw  w )(X)) |Jw |(X)dydt w | (uw  w )(X)|2 dydt |(1, i, 0)T / |2 dydt C90 L U2= C90 U2= 2LC9 .Chapter 5. Applications65d · Estimate of E2 : Note that for any t  R/LZ, dXi w (X)|(0,t) = ui (t) for i = 1, 2 and w d  (X)|(0,t) dt w= w (t), which implies  0  1 0  Gw (0, t) =  0 1 0   0 0 |w (t)|2    ,  and |Jw |(0, t) =2 detGw (0, t) = |w (t)|. Also, for X  Ur0 \U 2 × R/LZ,(u0  w )(X) = w Thus,L1 2 2 (y - iy1 y2 , -y1 y2 + iy1 , 0)T . |y|3 2E2 =0 L2 Ur0 \U 2(u0  w )(X)G-1 (X)( (u0  w )(X)) |Jw |(X)dydt w w w (u0  w )(X)(I + O(|y|))( (u0  w )(X)) (|w (t)| + O(|y|))dydt w w 1 dydt + |y|2L=0 L2 Ur0 \U 2=0|w (t)|LO(|y|-1 )dydt02 Ur0 \U 22 Ur0 \U 2= 2| ln |0|w (t)|dt + O(1).· Estimate of E3 E3 =N \Kr0| y 0 |2 &lt; since N is compact.· Estimate of E4LE4 =0 U2 LC90(|uw (w (X))|2 - 1)2 |Jw |(X)dydt 22 (r2 / 2 - 1)2 rdr 22LC9 2 (r / = 6 LC9 = . 6- 1)3 |0Combining these estimates we haveLEU (uw ) =0|w (t)|dt + O(| ln |-1 )Chapter 5. Applications with | ln ||O(| ln |-1 )| bounded above by a constant independent of w. Therefore, EU (uw ) -0 0 L66|w (t)|dt = M(QV W (w))uniformly on w.5.2Existence of critical points for EU on NBefore we state our main result, we need to show the Ginzburg-Landau energy EU satisfies the Palais-Smale condition in the H 1 topology for every  (0, 1]. Proposition 5.1. If {uj }  U is a Palais-Smale sequence associated with the functional EU , then {uj } is precompact in U . Proof. By assumption, {uj } satisfies sup EU (uj ) &lt; , EU (uj )j U 0 as j  .  C. Hence there is a subsequenceThe energy bound immediately implies uj {jk } and a function u  H 1 (N ) such that ujk andH 1 (N )u in H 1 (N )(5.21)ujk  u in Lp (N ), 1  p &lt; 6.(5.22)This follows from Rellich's Compactness Theorem and the fact that every bounded sequence in a Hilbert space has a weakly convergent subsequence. Since EU (ujk )(v) = 1 | ln | ujk ·N kv+12(|ujk |2 - 1)ujk · v, (5.23)EU (ujk )(u) - 0 and |EU (ujk )(ujk )|  C EU (ujk )k U- 0,(5.24)Chapter 5. Applications we havek67lim| ujk |2N(5.24)=1 12 k 2 klim lim(1 - |ujk |2 )|ujk |2N(5.22)=(1 - |ujk |2 )ujk · uN(5.23)=klimujk ·Nu=(5.21),(5.22)klim ((ujk , u)H 1 (N ) - (ujk , u)L2 (N ) ) | u|2 .N=Thus, ujk  u in H and this completes the proof.1Theorem 5.2. Suppose N is a 3-dimensional compact, orientable Riemannian manifold with N = , M = {(t)} is a 1-dimensional, connected, smooth submanifold of N without boundary,  is a geodesic and the Jacobi operator associated with M has finite index and 0 nullity. Let TM  R1 (N ) denote the multiplicity 1 current corresponding to integration over . Then for EU , EV defined by (5.1),(4.5) respectively, there exists0&gt; 0 such that for all  (0, 0 ), EU possesses a critical point u and EU (u )  EV (TM )  0.asProof. Recalling Theorems 3.1,4.1,5.1 and Proposition 5.1, this will follow if we can verify-1 (3.1)-(3.4). Since Ow is a smooth function in w  W , (3.2) holds and (3.1),(3.3),(3.4)hold due to Lemmas 4.2,5.4,5.5 respectively.Chapter 6 AppendixThroughout this section,  will be an open subset of Rn , K will be a compact subset of  and T  I1 (Rn ) with M(T ) &lt; . Also, X = (y, t)  Rn-1 × R will denote a point of  and p(X) := t. Assumptions on T : (T1) T is (, , )-minimal in  with() decreasing in (T2) sptT is compact in K, sptT CR is compact  CR and spt(T CR )  CR (note that this implies p T = E 1 for some integer , where E 1 is the standard 1-current obtained by integration of 1-forms over R; we will assume that  = ±1) (T3) diam(sptT CR )  /4 (T4) ( T , X)  1 T -a.e. (T5) E(T, R) &lt; 1 = 2 Assumptions on : (P1)  is a non-negative, -elliptic parametric integrand of degree 1 on  × Rn that is C 1 on  × S n-1 (P2) -1 ||  (X, )  || for all X  ,   Rn (P3) |(X, ) - (Y, )|  |X - Y | for all X, Y  K,   S n-1 (P4) for every (X0 , 0 )  K × S n-1 , there is a certain 1-covector 0 = 0 (X0 , 0 ) such 68Chapter 6. Appendix that |0 |   and |(X, ) - (X0 , 0 ) - 0 ,  - 0 |  (|X - X0 | + | - 0 |2 ) for all (X, )  K × S n-1 For (P5) and (P6), X0  K and the reference frame ( 1 , ..., given.n)69centered at X0 are(P5) for all (~, cn )  K and |p|  1,  is a common bound for § , i and their first c p derivatives (P6) for all (~, cn )  K, all |p|  1 and every p0 with |p0 |  1, we have c § § (~, cn , p) - c (0, 0, p0 ) - pi pi  2 § (0, 0, p0 )(pj - p0 ) j pj pi§j (|~| + |cn | + (|~| + |cn | + |p - p0 |)|p - p0 |) c c for some positive function  defined on R+ with() is decreasing in 1 For  = (t)dt  D1 (UR ), define Sj () := T (yj ), j = 1, ..., n - 1. We can writeSj () =(t)fj (t)dt1 for some fj  BV (UR ). To see this, first note that the map   Sj () defines a bounded, 1 linear functional on Cc (UR ) and thus Sj can be represented by a Radon measure j . Since (t)dj= = = =(T 2)T (yj  dt) T (d(yj ) - dyj ) T (d(yj )) - T (dyj ) T (yj ) - T (dyj ) -T (dyj ) M(T ) 1 C(UR )=1 1 for any   Cc (UR ), we can mollify j to obtain measures j which we can identify withBV functions fj satisfying fjBV CM(T ). General compactness results guaranteeChapter 6. Appendix that there exists fj  BV such that fj  fj in L1 as70  0. Therefore, j can beidentified with the BV function fj since j converges to j weakly as measures. The following results can be found in [4]. Lemma 6.1 is Lemma 2 in [4] and Lemma 6.2 is a combination of Lemmas 17,18 and estimates in the proof of Lemma 18 in [4]. Lemma 6.1. If f = (f1 , ..., fn-1 ), where fj is the BV function above, then1 |Df |(UR )  3 E(T, R)R(6.1)Lemma 6.2. There exists a positive constant E0 = E0 (, (·), , (·)) such that ifRR + E(T, R)  E0and0() d  E0 , (6.2)1 1 1 then f  C 1 (BR/2 ) and T CR/2 = G (E 1 BR/2 ) where G(t) = (f (t), t), t  BR/2 . 1 Moreover, for any t  BR/2 , R|f (t) - f (0)|  C (R) +0() d (6.3)where  -1 () = c E(T, R) + E0 ( + (12)) R1/2.Assume () =  ,   (0, 1), 0 &lt;    &lt; 1. Combining (6.1) and (6.3), we have for1 any t  BR/2 ,|f (t)|  |f (t) - f (0)| +1 R|f (0)|dsUR/2 2 sup |f (s) - f (0)| +sBR/2 2  2C(1 +  )( cE(T,R) R1 1 |Df |(UR ) R-1 (1 + 12 )E0 )R/2 + 3 E(T, R)(6.4)+Chapter 6. Appendix Justification of existence of y 0 : First represent TM by a measure-valued 2-form µM via TM ( ) = µM ,  = (µM , ),   D2 ().71Assume r0 is sufficiently small so that  = arg(y(x)) is defined in K3r0 . Note that d(x) =-y2 (x) -1 d1 |y(x)|2+y1 (x) -1 d2 |y(x)|2and for any   D2 (K3r0 ), d   d = d   d.d2 ,  = Letting u(x) =y(x) |y(x)|(d, d ) =for x  K3r0 , we have j(u) = d, and, recalling (5.20) with w = 0, d2 ,  = d   j(u) = j(u)(d )=  j(u)( ) = 2 J(u)( ) = 2TM ( ) = Thus, d2  = 2µM in K3r0 . Recall that we have assumed that TM  R1 (N ). In particular, this means TM = SM for some SM  D2 (N ). If  is any co-closed 2-form, then µM ,  = TM ( ) = SM ( ) = SM (d ) = SM ( d ) = 0. Therefore, there exists a unique 2-form  with coefficients in W 1,q (N ) for q &lt; 3/2 that is orthogonal to all harmonic 2-forms and satisfies - = 2µM . This is classical if the right-hand side is in L2 (see [8], Chapter 5.2.5, Theorem 3). The extension to the case where the right-hand side is a measure is proved using the duality argument in [3]; see [2] for a more detailed account of a related result. Moreover, dµM = 0 since TM = 0, which implies d is harmonic. As d,  = 0 for every harmonic 3-form , it follows that d = 0. 2µM ,  .Chapter 6. Appendix Fix x0  N \Kr0 and for x  N \Kr0 , set 0 (x) =72 (d )T , where x0 ,x is anyx0 ,xsmooth curve supported in N \Kr0 connecting x0 and x. We claim that 0 is well-defined modulo 2. To see this, it suffices to check that if  is a smooth closed loop in N \Kr0 , then(d )T = 2d for some d  Z. Using Stokes' Theorem and that fact that dd  = 0in N \Kr0 , we deduce that there exist t  R/LZ and d  Z such that (d )T = d C(d )Tfor some path C = C() = (r0 cos , r0 sin , t), 0   &lt; 2. Since dd  = d2  = 2µM in K3r0 , (d )T =C C(d)T =Cd.The change of variables y1 (x) = r0 cos , y2 (x) = r0 sin  gives2d =C 0r0 cos  -r0 sin  d(r0 cos ) + d(r0 sin ) = 2. 2 2 r0 r0Define ei in K3r0 \Kr0 by ei(x) = e-i0 (x)y(x) . |y(x)|Note that d0 = d  by construction and thus d = -d  + d. We can recover  by integrating over paths (up to an additive constant). Since d2  = 0, Stokes' Theorem implies that  is well-defined. Also, as  and  are smooth in K3r0 \Kr0 , so is . Now,   i(|y(x)|/3)(x)  e in K3r0 \Kr0 , 0 i0 (x) y (x) = e   1 in N \K3r0 satisfies the required properties.Bibliography[1] G. Alberti, S. Baldo, and G. Orlandi. Variational Convergence for Functionals of Ginzburg-Landau Type. Indiana Univ. Math. J., 54(5):1411­1472, 2005. [2] S. Baldo, R. Jerrard, G. Orlandi, and M. Soner. Convergence of Ginzburg-Landau functionals in 3-d condensed matter physics. In preparation. [3] S. Baldo and G. Orlandi. A note on the Hodge theory for functionals with linear growth. Manuscripta Math., 97:453­467, 1998. [4] E. Bombieri. Regularity Theory for Almost Minimal Currents. Arch. Rational Mech. Anal., 7:99­130, 1982. [5] M.P. do Carmo. Riemannian Geometry. Birkh¨user, 1992. a [6] L. Evans. Partial Differential Equations. AMS, 1998. [7] H. Federer. Geometric Measure Theory. Springer, 1969. [8] M. Giaquinta, G. Modica, and J. Souek. Cartesian Currents in the Calculus of c Variations I. Springer-Verlag, 1998. [9] R.L. Jerrard and H.M. Soner. Functions of Bounded Higher Variation. Indiana Univ. Math. J., 51(3):645­677, 2002. [10] R.L. Jerrard and H.M. Soner. The Jacobian and the Ginzburg-Landau Energy. Calc. Var. P.D.E., 14(2):151­191, 2002. 73Bibliography74[11] R.L. Jerrard and P. Sternberg. Critical Points Via Gamma-Convergence: General Theory and Applications. To appear. [12] R.V. Kohn and P. Sternberg. Local Minimizers and Singular Perturbations. Roy. Soc. of Edin. Sect. A, 111(1-2):69­84, 1989. [13] M. Kowalczyk. On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions. Annali di Matematica, 184:17­52, 2005. [14] L. Modica. The Gradient Theory of Phase Transitions and the Minimal Interface Criterion. Arch. Rat. Mech. Anal., 98(2):123­142, 1987. [15] L. Modica and S. Mortola. Un Essempio di -convergenza. Boll. U.M.I., 14-B:285­ 299, 1977. [16] J. Montero, P. Sternberg, and W. Ziemer. Local minimizers with vortices to the Ginzburg-Landau system in 3d. CPAM, 57(1):99­125, 2004. [17] F. Pacard and M. Ritor´. From constant mean curvature hypersurfaces to the grae dient theory of phase transitions. J. Diff. Geom., 64(3):359­423, 2003. [18] L. Simon. Lectures on Geometric Measure Theory. Centre for Math. Anal., Australian Nat. Univ., 1983. [19] S. Wenger. Flat Convergence for Metric Integral Currents. Calc. Var. P.D.E., 28(2):139­160, 2007. [20] B. White. A strong minimax property of nondegenerate minimal submanifolds. Journal f¨r die reine und angewandte Mathematik, 457:203­218, 1994. u`

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