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Existence of Critical Points for the GinzburgLandau Functional on Riemannian Manifolds
by
Jeff Mesaric
A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto
Copyright c 2009 by Jeff Mesaric
Abstract
Existence of Critical Points for the GinzburgLandau 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 GinzburgLandau type equation on a Riemannian manifold. We prove that if N is a compact, orientable 3dimensional Riemannian manifold without boundary and is a simple, smooth, connected, closed geodesic in N satisfying a natural nondegeneracy condition, then for every > 0 sufficiently small, a critical point u H 1 (N ; C) of the GinzburgLandau functional E (u) := 1 2 ln   u2 +
N
(u2  1)2 22
and 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 exists
0
> 0 such that for every
(0, 0 ), E possesses a critical point u and
lim 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 GinzburgLandau 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 almostminimal currents that is a saddle point of E in an appropriate sense. ii
Dedication
For my fianc´e, Melissa. e
Acknowledgements
I would like to thank my supervisor Robert Jerrard for his guidance and patience over the past few years.
iii
Contents
1 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 73
3 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 Bibliography
iv
Chapter 1 Introduction
Over 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   u2 +
N
1
2
(u2  1)u = 0,
(1.1)
(u2  1)2 dX. 22
(1.3)
When k = 1, (1.3) is usually referred to as the AllenCahn functional and the GinzburgLandau functional when k = 2. Many problems regarding these functionals involve making a connection between critical points of (1.3) and mdimensional 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. Introduction
2
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 vectorvalued 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 3dimensional 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 mdimensional 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   u2 +
N
(u2  1)2 , 22
u : 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)
3
existence 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 GinzburgLandau equation in 3 dimensions. This is accomplished by finding a reasonable candidate for a critical point of E (the limit of the GinzburgLandau 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. Introduction
4
a 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) < }. We say that EU converges to EV as 0 if 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 , PV U (QU V (v))  v
V V
0 as 0,
0 and
lim 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 GinzburgLandau functional EU (u) = 1 2 ln   u2 +
(u2  1)2 dX, 22
where 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 that
can 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 2d 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 1currents 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. Introduction
5
2.1), 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 1current associated with an oriented line segment M joining points x0 , y0 . To be consistent with our notation below, we will label this 1current 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 nondegenerate 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 critical
The 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 > 0 is found to satisfy PW V (vs ) = 0, QV W (0) = vs , PW V QV W (w) = w for all w W , sup
wW,wa
(1.4)
EV (QV W (w)) < EV (vs ) for all a (0, r1 )
(1.5)
and EV (vs ) < EV (v) for v {v V : 0 < v  vs
V
0 , PW V (v) = 0}
(1.6)
where W is the open ldimensional ball of radius r1 centered at the origin for some r1 > 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 > 0 to be chosen
6
(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 < R about M , assuming 0 < cr2 for some absolute constant c > 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) =
1
1 2 ln 
 u2 +
N
(u2  1)2 , 22
where u U = H (N ; C), N is a 3dimensional 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 1current associated with
the 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 exists
0
> 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. Introduction
7
smooth, 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 1current 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 > 0 satisfying properties (1.4)(1.6), where, as in [11], we take W to be the open ldimensional ball of radius r1 centered at the origin for some appropriately chosen r1 > 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 u
W 1,
sufficiently small, then the length of can be computed in terms of
u through a Taylor expansion. Since is a geodesic, the firstorder term vanishes and, letting L := M(TM ) = lengthM , 1 length() = L + (Ju, u)L2 + u 2 SturmLiouville theory allows us to express u =
W 1, O(
u
2 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 2
n
2 H1
+
c p 4
2 H1
for some small positive constant c. In this situation, T will be close to TM in the flat norm, where T is the 1current 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=1
wi z i over M . Since uw
2 H1
Chapter 1. Introduction
8
goes 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 > 0 so that (1.6) holds. As in [11], we assume (1.7) and show T = TM for some 0 > 0. First, we find a number r0 > 0 so that the exponential map is welldefined on a thin tube Kr0 of radius r0 about
3 M and write Kr0 in coordinates through a map . Then, assuming 0 r0 , we show
that 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 > 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. Introduction
9
out 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 ) < 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 
S
f (x)dx2 for suitable
f , C > 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 S
S
f (x)dx, w Mw respectively constructed by Brian White. The overall approach
taken 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 wellversed in GMT regularity theory to extract all of the details. Therefore, the arguments are reconstructed in section 4.2.
Chapter 2 Definitions
d d Throughout this paper, Ua (p) (Ba (p)) will denote the open (closed) ddimensional ball d d d d of radius a centered at p. We will make the convention Ua := Ua (0), Ba := Ba (0).
2.1
Basic Geometric Measure Theory
Let 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 kforms with compact support in . If T Dk (), then T is referred to as a kcurrent in . The boundary of a kcurrent T , denoted T , is the (k  1)current defined by T () := T (d), Dk1 ().
The mass and flat norm of a kcurrent T Dk () are given by M(T ) := sup
Dk ():
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 kcurrents 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 krectifiable set (a union of countably many Lipschitz ksubmanifolds of and a set of Hk measure zero), T (x) is, at Hk a.e. x, a simple kvector of unit length orienting the approximate tangent space apTx and m is a nonnegative integervalued Hk integrable function called the multiplicity of the current. Set Rk () := {T Rk () : M(T ) < , T = S for some S Rk+1 ()}, Fk () := {T Dk () : T = S for some S Dk+1 (), F(T ) < }. The restriction of T = (, m, T ) Rk () to a set A , denoted T A, is the kcurrent 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 righthand 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 as
n
(x) =
i=1
i (x)dXi
Chapter 2. Definitions where X = 1 (x). If f : N R is Lipschitz, then for a.e. x,
n
12
df (x) =
i=1
d (f )1 (x) dXi . dXi
Let 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 2
n1 n2
f (~) = x
i=1 j=1
i (f (~)) x
d ~ ~ 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 1current T corresponds to integration over if for any D1 (), T () =
I
i ((t))i (t)dt
where (t) = 1 ((t)). Let Ik () = {T Rk () : M(T ) < }. If T Ik (), then we say that T is an integral kcurrent in . Any integral 1current can be written as a finite or countable sum T =
i
Ti ,
where each Ti corresponds to integration over a Lipschitz curve i , M(T ) =
i
M(Ti ) =
i
H1 (i )
Chapter 2. Definitions and M(T ) =
i
13
M(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 kcurrent in such that M(T ) + M(T ) < 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 1current 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)
sign
d 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 3dimensional. For u = u1 + iu2 H 1 (), let J(u) denote the 2form du1 du2 and j(u) denote the 1form
1 (udu 2i
 udu). We can identify J(u) with
a 1current, denoted J(u), defined through its action on 1forms D1 () by J(u)() := J(u).
Similarly, j(u) can be associated with a 2current, denoted j(u), that acts on 2forms 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 2
L1 () .
(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. Definitions
14
2.2
Regularity
In 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 realvalued 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 nonnegative. For T = (, m, T ) Rk () and X , define (T ) :=
for c > 0.
(X, T (X))m(X)dX
and 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 . We
write 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 , ..., pn1 ), let
n1
§ (~, cn , p) := (X, c
i=1 §
pi i +
n ).
is called the nonparametric 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 .
15
Suppose K is a compact subset of , Q is a rectifiable 1current 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 Rn1 , 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))) , a
1 where C(t, a) := Rn1 ×Ba (t), a > 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 Point
Suppose U, V are Banach spaces, EU , EV : U, V (, ], (0, 1] and V0 = {v V : EV (v) < }. 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. Definitions
16
We will only be interested in limits for which the following compactness condition is satisfied: if sup (0,1] EU (u ) < , 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 PalaisSmale j=1 sequence if E(uj )
U
0 as j
and
{E(uj )} is bounded. j=1
The functional E is said to satisfy the PalaisSmale condition if every PalaisSmale 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 > 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 ) < EV (v) for v {v V : 0 < v  vs QV W (0) = vs , PW V QV W (w) = w for all w W and for every a > 0, sup
wW,wa V
0 , PW V (v) = 0},
(2.9) (2.10) (2.11)
EV (QV W (w)) < EV (vs ).
(2.12)
Chapter 3 Some Known Results
Recall 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 SturmLiouville 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]; Rn1 ), 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 dt
where y T = (y1 , ..., yn1 ), A : [0, L] Mn1,n1 (R) is a smooth function such that (A(t))T = A(t) for all t [0, L] and B Mn1,n1 (R) with BB T = B T B = I. There exists a nondecreasing, unbounded sequence {i } R and {Y i } C 1 ([0, L]; Rn1 ) such that {Y i } forms an orthornomal basis for L2 ([0, L]; Rn1 ) 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 dt
A 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 ) < and f : R is Lipschitz. Then M( T, f, s )ds sup  f (x)M(T ).
x
17
Chapter 3. Some Known Results
18
The 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 kcurrent supported in K and T minimizes X among all rectifiable kcurrents 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 > 0 such that if M(T ) < cN , then we can write T = S for some rectifiable 2current S with M(S) CN (M(T ))2 . Lemma 3.4 can be deduced as follows: first, cover N by finitely many open sets
n {j }m where each j is diffeomorphic to a ball URj . Let cN be minimum of diam(j k ) j=1
over all j, k such that j k = . Writing T as a sum of indecomposables T =
i
Ti ,
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 =
i
Si gives the desired result.
Next, we state a cone type inequality for integral 1currents 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 Results
19
The following is the important homotopy formula for 1currents; see [18], p.139 for a proof. Given 2 nonselfintersecting Lipschitz curves f1 , f2 supported in an open cylinder
n1 n1 n1 = UR × (a, b) that run from UR × {t = a} to UR × {t = b}, we use this to write
the 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 sptT
Here, E 1 is the standard 1current obtained by integration of 1forms 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  vs
V
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.
V
20 0 uniformly in w W as 0, (3.3)
(3.4)
Then given > 0, there exists ~ > 0 and a PalaisSmale sequence {uj } for every j=1 (0, ~) such that sup EU (uj )  EV (vs ) .
j
In particular, if EU satisfies the PalaisSmale condition for every , then there exists
0
> 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 Mass
Throughout this section, (N, g) will be a compact, orientable, ndimensional Riemannian manifold with N = that is isometrically embedded in Rn+m , n 2, m 1. For x N , we identify Tx N as an ndimensional subspace of Rn+m in the natural way. M will denote a 1dimensional, 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 shall
take 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), ..., un1 (t)} forms an orthonormal basis for
T(t) M for all t R/LZ. Therefore, for any t R/LZ,
{ (t), u1 (t), ..., un1 (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) < r}, r (0, r0 ), 0 < r0 < 1.
22
n1 For r0 sufficiently small, define : Br0 × R/LZ Kr0 by n1
(y, t) := exp (t),
i=1
yi ui (t) .
Here, exp: {(x, v) : x M, v Tx M, v < r0 } Kr0 is a smooth map given by
exp(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 conjugacy
n1 class of t R. From now on, when thinking of as a map defined on Br0 × R, we will
write (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=1
n1 n1 if necessary so that Br0 × R/LZ m ti , we conclude that : Br0 × R/LZ Kr0 i=1
is 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 n1 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) and
D u(L) dt
=
D u(0)}. dt
For u A, there exists a positive number su such that su(t) < 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, secondorder, 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), ..., un1 (t)} be an orthonormal basis for T(t) M such that, for each i = p p
= (Ju, u)L2 (0,L) .
s=0
1, ..., 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), ..., un1 (0)} and {u1 (L), ..., un1 (L)} are orthonormal bases for T(0) M = T(L) M , p p p p
there exists a matrix B = (bij ) satisfying
n1
ui p
=
j=1
bij uj p
and
BB T = B T B = I.
Chapter 4. Saddle Points of Mass If u A and u =
n1 i=1
24
yi ui , it follows that y(L) = B T y(0), y (L) = B T y (0) and p
n1 n1
Ju =
i=1
yi +
j=1
aij yj
ui , p
where y T = (y1 , ..., yn1 ) 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 nondecreasing, unbounded sequence {i } R and {Y i } C 1 ([0, L]; Rn1 ) such that {Y i } forms an orthonormal basis for L2 ([0, L]; Rn1 ) and for each i, Y i satisfies S(Y i ) = i Y i , Setting z i =
n1 j=1
Y i (L) = B T Y i (0),
d i d Y (L) = B T Y i (0). dt dt
Yji 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 dt
and {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 z
Then (Jn, n)L2 (0,L) 1 n
2 H 1 (0,L) .
Moreover, there exists a positive constant c, independent of u, such that n and (Jp, p)L2 (0,L) c p
2 H 1 (0,L) . 2 L2 (0,L)
c n
2 H 1 (0,L)
Proof. The first claim is vacuous unless 1 < 0. If 1 < 0, we compute
l l
(Jn, n)L2 (0,L) =
i,j=1
i ci cj ij =
i=1
i c 2 1 n i
2 L2 (0,L)
1 n
2 H 1 (0,L) .
Chapter 4. Saddle Points of Mass For the second claim, note
l l
25
n
2 L2 (0,L)
C
i,j=1
ci cj  C/2
i,j=1
(c2 + c2 ) = lC n i j
2 L2 (0,L) ,
where C > 0 depends on z 1 , ..., z l . Thus, n
2 L2 (0,L)
(1 + lC)1 n
2 H 1 (0,L) .
Finally, to prove the third claim, first note
L
(p (t) + (a(t), p(t)))dt = (Jp, p)L2 (0,L) =
0 i,j=l+1
2
i ci cj ij l+1 p
2 L2 (0,L)
where a(t) = R( (t), p(t)) (t). Now, for (0, 1), p
2 L2 (0,L)
p
2 L2 (0,L)
+ (1  )[(Jp, p)L2 (0,L)  l+1 p
L 0
2 L2 (0,L) ]
= (Jp, p)L2 (0,L) 
((a(t), p(t)) + (1  )l+1 p(t)2 )dt.
~ ~ Since a Cp 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 p2 (l+1  (C + l+1 ))p2 0. Therefore, for this value of , p and (Jp, p)L2 (0,L) l+1 p + l+1
2 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 Mass
26
Theorem 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 < 0 < l+1 · · · if l := index of J > 0 and 0 < 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 > 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 reason
for 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 = u
W 1, l i i=1 ci z , p
u W 1, O( u 2 1 ) H 1 (Jp, p)L2 + u W 1, O( u 2
2 H1 )
=
i i=l+1 ci z .
If c > 0 is the constant from Lemma 4.1 and u
2 H 1 )
is small enough so that u M(T ) L 
W 1, O(
c 8
u
2 H1 ,
we have
1 c  1 2 2
n
2 H1
+
c p 4
2 H1 .
Chapter 4. Saddle Points of Mass
27
Thus, 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 lvector (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 d
L
C/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 Fnorm. 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 ,
28
we 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))
L
By definition of p, v and , we have p((t)) = (t), v((t)) = u(t) and ((t)) = t. Therefore, since u(t) < r0 /2 for every t [0, L) and {z i } are orthonormal in L2 (0, L), i=1
L
PW V (T ) =
0
(z 1 (t) · u(t), ..., z l (t) · u(t))dt
L L
=
0
z 1 (t) · u(t)dt, ...,
0
z 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 > 0 such that if T R1 (N ), F(T  TM ) < 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 .
29
First 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 1dimensional. 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 ndimensional surface, n 2, stretches between two sets, the ndimensional 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 taken
sufficiently small, then there exists a 1current 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 2current S with sptS K2r0 /3 and M(S ) < . Proof. 1. Assumption (4.6) implies there exists S1 D2 (N ) such that S1 = T  TM in N and M(S1 ) < 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 Mass
30
Let's compute  d(x) for x Kr0 . Set G to be the 3 × 3 matrix with entries gij = (Xi , Xj ) and G1 = (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 velocity
n1 i=1
yi 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 n1
d((y, t)) =
0
y,t (s)ds = y,t (0) =
i=1
yi ui (t) = y.
~1 ~1 Therefore, letting Y (x) = (1 (x), ..., n1 (x), 0), we have d(x) = d ~ ~ g ij ( 1 (x)) (d )1 (x) Xi ( 1 (x)) ~ dXj i,j=1 n1 ~1 j (x) ij ~1 ~ = g ( (x)) X ( 1 (x)) Y (x) i i,j=1 ~ Y (x)G1 ( 1 (x))(Y (x))T . Y (x)2
n
and  d(x)2 = Using Lemma 3.2, it follows that
r0
M( S1 , d, r )dr
r0 /2
sup
xKr0 \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 , and
2 ~ M(T ) L + 2C1 r0 .
~ M(S1 ) M(S1 ) < 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). Since
n1 ((y, t)) = t for all (y, t) Ur0 × R/LZ,
31
D (t), ui (t) dt
=0
we have
(x) =
n i=1
~ ~ g in ( 1 (x))Xi ( 1 (x)) and g nn (X)
2 1 nn (g nn (0, t), ..., gyn1 (0, t)) g nn (0,t) y1
 ((X)) =
= g nn (0, t) +
· y + O(y2 )
= 1  1 ( y1 gnn (0, t), ..., yn1 gnn (0, t)) · y + O(y2 ) 2
= 1 + O(y2 ). 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 )
xKr0
2 2 (1 + C2 r0 )(L + 2C1 r0 ) ()
(4.15)
4L, ~ which implies M( T , , t ) < 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 Mass
32
level 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 C1
0 2 C1 (1 + C2 r0 )0 ()
~ M( S1 , , t )dt
0 L
~ M( S1 , , t )dt
2C1 0 , which gives
2 0  16C1 0 /r0 16C1 r0 .
(4.17)
5. Similar to (4.15), we compute ~ M(T ) 1 2 1 + C2 r0
L
~ M( T , , t )dt
0
(1 
2 C2 r0 )(1 
+ 22 ).
Chapter 4. Saddle Points of Mass
33
2 Combining this with (4.13),(4.14) and (4.17), we see that 2  3(6C1 + LC2 )r0 , which
implies
2 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 )dt
1
(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 that
2 1,1  L  10(7C1 + LC2 )r0 .
t 1 :
i
H0 (~i 1 (t)) = j
.
(4.20)
The explicit formula (2.3) for the slice of an indecomposable 1current implies ~ M( T , , t )
i
~ M( Ti , , t ) =
i
H0 (~i 1 (t))
Chapter 4. Saddle Points of Mass for a.e. t. It follows that
i
34
H0 (~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 have
2 L + 2C1 r0 (4.13)
=
~ M(T ) ~ M(Ti )
i
=
i
H1 (~i ) 1 2 1 + C2 r0 1 (1  (1  H0 (~i 1 (t))dt
i
=
(4.21)
1 2 C2 r0
2 1 + C2 r0
j1,j 
j=1
2 C2 r0 )
1,1  + 2
j=2
1,j 
=
2 C2 r0 )(21 
 1,1 ).
This together with (4.18) implies (4.20). 8. Taking r0 sufficiently small, we have 1,1  > L/2 > 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 Mass
35
9. 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 ) = M
i=k
Ti
=
i=k
M(Ti ) =
i
M(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 ~
n1 (3.6) with the convex set = Ur0 /2 × (t , t + L) and f1 (t) = (t), f2 (t) = (0, t) for n1 t (t , t + L), we can find a 2current 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) < .
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) < , 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 .
n1 For X Ur0 × R and Rn , let
F (X, ) := D(X),
Chapter 4. Saddle Points of Mass
36
which is a parametric integrand of degree 1 (see Section 2.2). Here, we take D(X) to represent
n i=1 i Xi (X)
T(X) N .
n1 From this, we can define a functional on R1 (Ur0 × R) by
F (T ) :=
F (X, T (X))m(X)dH1 (X),
n1 where T = (, m, T ). Also, for X Ur0 × R and T = (, m, T ) R1 (Rn ), define
the `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 n1 and the existence of a constant C3 = C3 () > 0 such that if sptT B (X) Ur0 × R,
then FX (T )  F (T ) C3 M(T ). (4.23)
n1 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), ..., un1 (t)} is an orthonormal basis for T(t) N , we have D(0, t) =  Therefore, letting X = (y, t), F (X, )   =
MVT
for 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, ) 3/2
n1 for all X Ur0 × R and Rn .
37
(4.25)
n1 Lemma 4.6. If T = (, m, T ) R1 (Ur0 × R), then
(1  C3 r0 )M(T ) F (T ) (1 + C3 r0 )M(T ).
n1 In addition, for every X Ur0 × R,
(4.26)
(1  C3 r0 )M(T ) FX (T ) (1 + C3 r0 )M(T ).
n1 ~ ~ 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)
n1 ~ 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 implies
1 (4.22) F (Q ) = 1  C3 r0 C3 Lr0 . 1  C3 r0
1+
C3 r0 1  C3 r0
M(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 2current 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) < }. 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 > 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 > 0 is to be chosen. We claim that there exists a C > 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)
39
C 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 .
4.2
Regularity
This 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 < .
T Hr
Select 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 FedererFleming 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 ) < 1. Since Tij Hr , Tij  TM = Sj for some Sj D2 (N ) with sptSj K3r0 /2 and M(Sj ) < . For this j, set Sr = Sr,j + Sj D2 (N ). Then Tr  TM = Sr ,sptSr K3r0 /2 and M(Sr ) < . 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 .
j
It follows that M (Tr ) = mr = min M (T ).
T Hr
Lemma 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 2current S with M(S) < . 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.
41
We 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)
i
Tr,i ,
where each Tr,i corresponds to integration over a closed, Lipschitz curve r,i , M(Tr ) =
i
M(Tr,i ) and M(Tr ) =
i
M(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  + 2r  1 2
r 1 L
M( Tr , , t )dt +
r 2
M( Tr , , t )dt
=
0
M( Tr , , t )dt
2 (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 2
r := 1,j
t r : 1
i
H0 (r,i 1 (t)) = j
and using an argument similar to the one in Step 7 in the proof of Lemma 4.4, we have
2 r  2r   L  2C2 Lr0 , 1,1 1
Chapter 4. Saddle Points of Mass
()
42
2 which implies r  L  4C2 Lr0 > L/2 > 0. From now on, t will denote a point 1,1 r
in 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), set
n1 Qr to be the 1current compactly supported in Br × [t  2L, t + 2L] r r n1 defined on cylinders of height L by 1 Tr R1 (Ur0 × R/LZ).
Note that M(Qr ) = 2 for every r (0, 2r0 /3). Let
n1 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 =
i
Qr,i , where each Qr,i corresponds to integration over a closed (in
n1 n1 Ur0 × (t  2L, t + 2L)) Lipschitz curve r,i : [0, Lr,i ] Br × [t  2L, t + 2L]. Note r r r r
that 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 curves
n1 {r,i : i = j} correspond to closed loops in Br × [t  2L, t + 2L]. Thus, recalling that r r
p(X) = t for X = (y, t) Rn1 × R, sptQr p1 (t) = Also, M(Qr C(t, L/2)) = M(Qr C(t + L/2, L/2)) r
n1 H1 (r,j Br ×[t ,t +L] ) r r
for 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 r
43
Our goal now is to show that Qr is an almost minimizer for F ; this means that small perturbations of Qr by closed, rectifiable 1currents 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.
n1 Lemma 4.9. There exists > 0 such that F is elliptic in Ur0 × R. n1 ~ Proof. Set FX,c () := FX ()  c, X Br0 × R, Rn , c > 0. 0
~ First, we will show that if there exists a c > 0 such that FX,c is convex for all
n1 n1 X Ur0 × R, then F is c1 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 > 0 such
n that T2  T1 = S for some 2current S with sptS Ua .
Note that
T2 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 all
n 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 Mass
n1 which means F is c1 elliptic in Ur0 × R.
44
n1 ~ We claim that there exists c > 0 such that FX,c is convex for all X Ur0 × R. Let
lX () = 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) . 3
n i=1
1 Here, lX is a map that takes the vector
Xi T(X) N to Rn . Let
3 1 lX 4
c=
1 min lX n1 2 XBr0 ×R/LZ
. = 1 for every t R and
Note c > 0 for r0 sufficiently small since l(0,t)
1 = l(0,t)
1 n1 ~ lX , lX are continuous functions of X. Then FX,c is convex for all X Ur0 × R and n1 F is elliptic in Ur0 × R with = c1 .
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 almostminimality, his argument does not exacly apply here. Lemma 4.10. Let () = . There exists a positive constant , independent of r, such
that Qr is (F, , )minimal for all r (0, 2r0 /3) and r0 sufficiently small. Proof. Set := r0
1/4 ()
< L/2.
n1 For X Ur0 × R and (0, ], let
S := {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 show
n n ~ F (Qr B (X)) (1 + C)FX (Qr B (X) + T )
45
(4.39)
n ~ for some absolute constant C > 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 ) +
()
1/4
C 1  C3 r0
n 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 that
n ~ ~ H1 (sptT sptQr ) > 0 otherwise (4.39) again follows easily since FX (Qr B (X) + T ) = n ~ FX (Qr B (X)) + FX (T ).
Since
n 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.
46
n1 n ^ ~ Using Lemma 3.3, we conclude sptT Br × [2L, 3L] B (X) as spt(Qr sptT ) n1 n is contained in the convex set Br × [2L, 3L] B (X). Thus,
n1 n ~ sptT Br × [2L, 3L] B (X).
(4.41)
~ ~ Let S minimize M among all 2currents with boundary equal to T . Using (4.41),
n1 n ~ Lemma 3.3 implies sptS Br × [2L, 3L] B (X). Now we would like to show n ~ ~ M(S) < . First, as sptT B (X), Lemma 3.5 implies that there exists a 2current 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 ) < 96L.
47
(4.45)
n1 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 Mass
48
n ~ Since < L/2, sptT B (X) C(s, L/2) for some s [t  3L/2, t + 3L/2], r r
and thus
n 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 implies
n 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 dt
and ur
W 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 nonnegative, ~ E(Qr , t, r0 )
I
49
i=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 p1 (t) (after a translation), it suffices to show the assumptions made in the Appendix hold with T = Qr , = F
n1 n1 = Ur0 × R, K = B3r0 /4 × [2L, 3L] () = , 0 < () R = r0 < 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
() 1/4
and thus
13r0 /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 have
1 ~ ~ spt(Qr C(t, R/2)) = {(yr (s), s), s BR/2 (t)}
for some C 1 function yr with yr
L
C5 r0 .
1/8
Chapter 4. Saddle Points of Mass
50
~ Since t (0, L) was chosen arbitrarily, it follows that spt(Qr C(L/2, L/2)) coincides
n1 with the graph of a function yr C 1 ([0, L]; Br ) satisfying yr (0) = yr (L), yr (0) = yr (L)
and yr
W 1,
(1 + C5 )r0 .
n1
1/8
Setting ur (t) :=
i=1
yr,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 r
and
W 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 . l
Lemma (4.3) implies c2 = nr r,i
2 L2 .
P (Tr ) =
i=1
2
(4.47)
Assume sr = ur
C([0,L])
= yr
C([0,L])
> 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) = (ss1 yr (t), t, ss1 yr (t), 1), r r we have d3 M(Tr,s ) = ds3 which implies Rr,2 (s) C7 r0 s3 ur r
1/8 2 3 H 1 s L 3 2 H 1 )s .
51
=0
s=0
Indeed, viewing F as a function of Y =
YsT
0
d 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 2
2 L2 2 L2
=
=
(4.48),()
L + 1 (Jnr , nr ) + 1 (Jpr , pr ) + C nr 2 2 L + ( 1 + C c) nr 2
2 H1
+ Rr,2 (sr )
+
c 2
pr +
2 H1 c 4
+ Rr,2 (sr ) pr
2 H1 .
L + ( 1 + c(C  1 )) nr 2 4
2 H1
If we choose C >
c21 , 4c
then nr and pr are identically 0, which implies Tr = TM .
Chapter 4. Saddle Points of Mass
52
4.3
Construction of QV W
l i=1
l l Let W = Ur1 where r1 > 0 is to be chosen. For w Br1 , define zw (t) :=
wi z i (t).
Assume r1 is small enough so that
l zw (t) < 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) := (1s)X1 +sX2 .
l Fix w1 , w2 Ur1 . Using Lemma 3.6 with fi (t) = 1 (wi (t)) for i = 1, 2, t (0, L), n1 we can find Sw1 ,w2 D2 (Ur0 × (0, L)) such that
Sw1 ,w2 = 1 QV W (w2 )  1 QV W (w1 ) + Tf1 (0),f2 (0)  Tf1 (L),f2 (L) and M(Sw1 ,w2 ) L f1  f2
L
f1 + f2
L
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 = zw
C([0,L])
53
> 0. Consider the variation
hw : (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(w2 ). Note that
l
(Jzw , zw ) =
i=1
2 i wi l w2 .
Taking r1 small enough so that
o(w2 ) w2
l /2, we have
M(QV W (w)) L + l w2 /2. Thus, sup
l wUr1 ,wa
M(QV W (w)) L + l a2 /2 < L.
Chapter 5 Applications
Throughout this chapter, (N, g) will be a 3dimensional compact, orientable Riemannian manifold without boundary, U = H 1 (N ; C), V = F1 (N ) and EV will be given by (4.5). Define the GinzburgLandau energy EU : U R by EU (u) := 1 2 ln   u2 +
N
(u2  1)2 . 22
(5.1)
5.1
convergence of EU on N
Theorem 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 ) < , 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. Applications
55
( Xi x , Xj x ), we have G1 (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 +
URi
1 (v i (X)2  1)2 dX 22 1 (v i (X)2  1)2 dX 22
v i (X)G1 (X)( v i (X)) + i
URi
=
i
v i 1 (x) G1 (1 (x))( v i 1 (x) ) i i i i 1 (u (x)2  1)2 J1 (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 (UR
i
;C) (v
i
) and Theorem 3.2 in [11], there exists
Q1 F1 (UR1 ) and a subsequence { 1 } (0, 1] such that j FUR1 ( Jv 11  Q1 ) 0
j
as j .
Since sup EH 1 (UR
i
;C) (v
i
) < 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 } { j
i1 j }
such that as j .
FURi ( Jv ii  Qi ) 0
j
Setting
j
=
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 i
Let {i } be a partition of unity subordinate to the cover {i }, T = and Tj =
m i=1
Tji i = Ju j  T . It follows from (5.3) that ~ FN (Tj ) ~ C 0
m i=1
~ FN (Tji i ) ~ Fi (Tji ) (5.4)
m i=1
as j .
~ for some constant C > 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 > 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. Applications
57
We 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 1dimensional curve and apTX0  exists, then
3 (Br (X0 )) = 0. 3 r0 (Br (X0 ))
lim
By definition, the above hypotheses mean there exists a line l R3 passing through X0 and > 0 such that r H1 l = 0 as r 0 where r (A) := 1 (X0 + rA). r
3 3 Proof. Without loss of generality, assume X0 = 0. Since 0 ((B1 )) = 0 (B1 ) = 0
and r
0 , we have
3 3 lim r (B1 ) = 0 (B1 ) = 0
r0
and
r0
3 3 lim r (B1 ) = 0 (B1 ) > 0.
Therefore,
3 3 3 (Br ) r (B1 ) 0 (B1 ) = lim = = 0. 3 3 3 r0 r (B1 ) r0 (Br ) 0 (B1 )
lim
With 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 straight
line l for r sufficiently small. By slicing Q Br (X) orthogonal to l and using known 2d 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 ) < .
58
Let x0 sptT and be an open subset of N such that x0 and is dif3 feomorphic to BR for some R > 0 and diffeomorphism . Assume (0) = x0 and { X1 (0), X2 (0), X3 (0)} forms an orthonormal basis for Tx0 N . Let G = (gij ) be the
3 × 3 matrix with entries gij = (Xi , Xj ), G1 = (g ij ) and { j } be a subsequence such that
j
lim Ej (u j ) = m := lim inf E (u ) < .
(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, we
have 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)d1 ((X)), T ((X)) m((X)) detG(X)dX. i
Letting T = i Xi , = D1 (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 )) > 0 for all r > 0.
3 Also, since T has finite mass and (5.7) holds, L1 (UR ; d), which implies
59 (5.10)
(5.11)
r0
lim
1 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 by
µG (A) :=
1 2 ln 
Xv A
(X)G1 (X)(
Xv
(X)) + detG(X)dX,
1 (v (X)2  1)2 22
where 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 zaxis onto the line through X0 in the direction of (X0 )). We claim that
3 d (Br (X0 )) (X0 ) = lim for µa.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 and
G(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 

A
Xv
(X)2 +
1 ((v (X)2  1)2 dX. 22

S
Xv
(X1 , X2 , t)2 +
1 (v (X1 , X2 , t)2  1)2 dX1 dX2 . 22
Chapter 5. Applications
3 where S BR p1 (t).
60
Since we are assuming (5.5), Theorem 5.2 in [10] implies 3 can be represented by
t 3 slices 3 . This means in particular that for any open ball U BR , t 3 (U p1 (t))dt
3 (U ) =
(5.15) di (t)ai (t) for
t 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 p1 (t). The theorem also implies
t lim inf µ j ,t (U p1 (t)) 3 (U p1 (t))
(5.16)
3 for any open ball U BR . Therefore, 3 µ(Br (X0 )) (5.14)
=
Fatou
3 (1  Cr) lim inf µ j (Ur (X0 ))
(1  Cr) lim inf (1  Cr) (1  Cr)
3 µ j ,t (Ur (X0 ) p1 (t))dt
3 lim inf µ j ,t (Ur (X0 ) p1 (t))dt t 3 3 (Ur (X0 ) p1 (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 conclude
3 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 0
61 exists µa.e..
µ. Thus,
1 3 ({X UR : X is a good point}) 1 d (X)dµ 3 {XUR :Xis a good point} dµ
3 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 .
j
j
Finally, we need to show lim inf EU (u ) M(T ). Fix > 0. We can cover sptT
3 by disjoint open sets {i = i (URi )}, where i is a diffeomorphism as above, and
M(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 > 0 is the number chosen in Section 4.3 and
QV W (w) is the multiplicity 1 current corresponding to integration over w , where w (t) = exp((t), zw (t)), zw (t) =
l i=1
wi 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 w
t 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 Let
2
62
Kw,r := for r (0, r0 /2] and
exp w (t),
i=1
yi ui (t) w
2 : 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=1
yi ui (t)). Set yi (x) :=
Define Ow : N N by exp exp(( (x)), (v(x))zw ( (x))), Ow (x) := x where
2 i yi (x)vw (x) i=1
for x Kr0 , for x N \Kr0
i 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 small
enough, 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 \Kr0
where 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 w
Ju0 = 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). w
It is wellknown 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 \K
1
Lemma 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 ), G1 = (gw ) and w
aw,i (x) = uw (x) ×
d (u w ) dXi w
1 w (x)
 u0 (x) × w
d (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 V
Juw  Ju0 ) w
L1 (N )
=
j(uw )  j(u0 ) w
Kw, L 0 U2
1 2 1 = 2
ij 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. Applications
64
2 l Note that since w is a smooth function in both X Br0 × R/LZ and w Br1 , there
exists an absolute constant C9 > 0 such that G1 (X) + Jw (X) C9 w
2 l for all X Br0 × R/LZ and w Br1 . Also, aw,3 (w (X)) = 0 and aw,i (w (X)) 3 y
for
all 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 U2
1 dydt y
2 = 12C9 L
which implies PV U (uw )  QV W (w)
V
 0 uniformly on W .
0
Lemma 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 22
1 = (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) G1 (w (x))( (uw w )w (x) ) w
=
0 U2 L
(uw w )(X)G1 (X)( (uw w )(X)) Jw (X)dydt w  (uw w )(X)2 dydt (1, i, 0)T / 2 dydt
C9
0 L U2
= C9
0 U2
= 2LC9 .
Chapter 5. Applications
65
d · 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,
L
1 2 2 (y  iy1 y2 , y1 y2 + iy1 , 0)T . y3 2
E2 =
0 L
2 Ur0 \U 2
(u0 w )(X)G1 (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 + y2
L
=
0 L
2 Ur0 \U 2
=
0
w (t)
L
O(y1 )dydt
0
2 Ur0 \U 2
2 Ur0 \U 2
= 2 ln 
0
w (t)dt + O(1).
· Estimate of E3 E3 =
N \Kr0
 y 0 2 <
since N is compact.
· Estimate of E4
L
E4 =
0 U2
LC9
0
(uw (w (X))2  1)2 Jw (X)dydt 22 (r2 / 2  1)2 rdr 2
2
LC9 2 (r / = 6 LC9 = . 6
 1)3 0
Combining these estimates we have
L
EU (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 L
66
w (t)dt = M(QV W (w))
uniformly on w.
5.2
Existence of critical points for EU on N
Before we state our main result, we need to show the GinzburgLandau energy EU satisfies the PalaisSmale condition in the H 1 topology for every (0, 1]. Proposition 5.1. If {uj } U is a PalaisSmale sequence associated with the functional EU , then {uj } is precompact in U . Proof. By assumption, {uj } satisfies sup EU (uj ) < , EU (uj )
j U
0 as j . C. Hence there is a subsequence
The energy bound immediately implies uj {jk } and a function u H 1 (N ) such that ujk and
H 1 (N )
u in H 1 (N )
(5.21)
ujk u in Lp (N ), 1 p < 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 k
v+
1
2
(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 have
k
67
lim
 ujk 2
N
(5.24)
=
1 1
2 k 2 k
lim lim
(1  ujk 2 )ujk 2
N
(5.22)
=
(1  ujk 2 )ujk · u
N
(5.23)
=
k
lim
ujk ·
N
u
=
(5.21),(5.22)
k
lim ((ujk , u)H 1 (N )  (ujk , u)L2 (N ) )  u2 .
N
=
Thus, ujk u in H and this completes the proof.
1
Theorem 5.2. Suppose N is a 3dimensional compact, orientable Riemannian manifold with N = , M = {(t)} is a 1dimensional, 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 exists
0
> 0 such that for all (0, 0 ), EU possesses a critical point u and EU (u ) EV (TM ) 0.
as
Proof. 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 Appendix
Throughout this section, will be an open subset of Rn , K will be a compact subset of and T I1 (Rn ) with M(T ) < . Also, X = (y, t) Rn1 × 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 1current obtained by integration of 1forms over R; we will assume that = ±1) (T3) diam(sptT CR ) /4 (T4) ( T , X) 1 T a.e. (T5) E(T, R) < 1 = 2 Assumptions on : (P1) is a nonnegative, elliptic parametric integrand of degree 1 on × Rn that is C 1 on × S n1 (P2) 1  (X, )  for all X , Rn (P3) (X, )  (Y, ) X  Y  for all X, Y K, S n1 (P4) for every (X0 , 0 ) K × S n1 , there is a certain 1covector 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 n1 For (P5) and (P6), X0 K and the reference frame ( 1 , ..., given.
n)
69
centered 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 write
Sj () =
(t)fj (t)dt
1 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 with
BV functions fj satisfying fj
BV
CM(T ). General compactness results guarantee
Chapter 6. Appendix that there exists fj BV such that fj fj in L1 as
70 0. Therefore, j can be
identified 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 , ..., fn1 ), where fj is the BV function above, then
1 Df (UR ) 3 E(T, R)R
(6.1)
Lemma 6.2. There exists a positive constant E0 = E0 (, (·), , (·)) such that if
R
R + E(T, R) E0
and
0
() 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)) R
1/2
.
Assume () = , (0, 1), 0 < < 1. Combining (6.1) and (6.3), we have for
1 any t BR/2 ,
f (t) f (t)  f (0) +
1 R
f (0)ds
UR/2
2 sup f (s)  f (0) +
sBR/2 2 2C(1 + )( cE(T,R) R
1 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 measurevalued 2form µM via TM ( ) = µM , = (µM , ), D2 ().
71
Assume 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)2
and 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 coclosed 2form, then µM , = TM ( ) = SM ( ) = SM (d ) = SM ( d ) = 0. Therefore, there exists a unique 2form with coefficients in W 1,q (N ) for q < 3/2 that is orthogonal to all harmonic 2forms and satisfies  = 2µM . This is classical if the righthand side is in L2 (see [8], Chapter 5.2.5, Theorem 3). The extension to the case where the righthand 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 3form , 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 any
x0 ,x
smooth curve supported in N \Kr0 connecting x0 and x. We claim that 0 is welldefined 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 = 0
in N \Kr0 , we deduce that there exist t R/LZ and d Z such that (d )T = d
C
(d )T
for some path C = C() = (r0 cos , r0 sin , t), 0 < 2. Since dd = d2 = 2µM in K3r0 , (d )T =
C C
(d)T =
C
d.
The change of variables y1 (x) = r0 cos , y2 (x) = r0 sin gives
2
d =
C 0
r0 cos r0 sin d(r0 cos ) + d(r0 sin ) = 2. 2 2 r0 r0
Define ei in K3r0 \Kr0 by ei(x) = ei
0 (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 welldefined. 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 GinzburgLandau Type. Indiana Univ. Math. J., 54(5):14111472, 2005. [2] S. Baldo, R. Jerrard, G. Orlandi, and M. Soner. Convergence of GinzburgLandau functionals in 3d 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. SpringerVerlag, 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 GinzburgLandau Energy. Calc. Var. P.D.E., 14(2):151191, 2002. 73
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[11] R.L. Jerrard and P. Sternberg. Critical Points Via GammaConvergence: 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(12):6984, 1989. [13] M. Kowalczyk. On the existence and Morse index of solutions to the AllenCahn 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., 14B:285 299, 1977. [16] J. Montero, P. Sternberg, and W. Ziemer. Local minimizers with vortices to the GinzburgLandau 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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