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Currents and Flat Chains Associated to Varifolds, with an Application to Mean Curvature Flow

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Currents and Flat Chains Associated to Varifolds, with an Application to Mean Curvature Flow CURRENTS AND FLAT CHAINS ASSOCIATED TO VARIFOLDS, WITH AN APPLICATION TO MEAN CURVATURE FLOW BRIAN WHITE Abstract We prove under suitable hypotheses that convergence of integral varifolds implies con- vergence of associated mod 2 flat chains and subsequential convergence of associated integer-multiplicity rectifiable currents. The convergence results imply restrictions on the kinds of singularities that can occur in mean curvature flow. 1. Introduction N + Let U be an open subset of R .LetLm-rec(U,Z ) denote the space of functions on U that take values in nonnegative integers, that are locally L1 with respect to Hausdorff m-dimensional measure on U, and that vanish except on a countable disjoint union of m-dimensional C1-submanifolds of U. We identify functions that agree except on a set of Hausdorff m-dimensional measure zero. Let Lm-rec(U,Z2) be the + corresponding space with the nonnegative integers Z replaced by Z2, the integers mod 2. The space of m-dimensional integral varifolds in U is naturally isomorphic to + Lm-rec(U,Z ): given any such varifold V , the corresponding function is the density function (V,·) given by μ (B(x,r)) (V,x) = lim V , → m r 0 ωmr where μV is the radon measure on U determined by V and ωn is the volume of the unit ball in Rm. In particular, this limit exists and is a nonnegative integer for Hm-almost every x ∈ U. Similarly, the space of m-dimensional rectifiable mod 2 flat chains in U is naturally isomorphic to Lm-rec(U,Z2): given any such flat chain A, the corresponding function DUKE MATHEMATICAL JOURNAL Vol. 148, No. 1, c 2009 DOI 10.1215/00127094-2009-019 Received 15 May 2008. Revision received 8 November 2008. 2000 Mathematics Subject Classification. Primary 49Q15; Secondary 53C44. Author’s research supported by National Science Foundation grants DMS-0406209 and DMS-0707126. 41 42 BRIAN WHITE is the density function (A, ·) given by μ (B(x,r)) M(A ∩ B(x,r)) (A, x) = lim A = lim , → m → m r 0 ωmr r 0 ωmr where μA is the radon measure on U determined by A. In particular, this limit exists and is 0 or 1 for Hm-almost every x ∈ U. The surjective homomorphism + [·]: Z → Z2, k → [k] + determines a homomorphism from Lm-rec(U,Z ) to Lm-rec(U,Z2) and thus also a homomorphism from the additive semigroup of integral varifolds in U to the additive group of rectifiable mod 2 flat chains in U.IfV is such a varifold, we let [V ] denote the corresponding rectifiable mod 2 flat chain. Thus [V ] is the unique rectifiable mod 2 flat chain in U such that ([V ],x) = [(V,x)] for Hm-almost every x ∈ U. Although in some ways integral varifolds and rectifiable mod 2 flat chains are similar, the notions of convergence are quite different. Typically (and throughout this article), convergence of varifolds means weak convergence as radon measures on N N N U × Gm(R ) (where Gm(R ) is the set of m-dimensional linear subspaces of R ), and convergence of flat chains means convergence with respect to the flat topology (see Section 4). A sequence V (i) of integral varifolds may converge even though the associated flat chains [V (i)] do not converge. Similarly, the flat chains [V (i)] may converge even though the varifolds V (i) do not. Furthermore, the V (i) and [V (i)] may converge to limits V and A, respectively, with A = [V ] (see Section 2.4 for examples). This article identifies an important situation in which convergence of integral varifolds implies convergence of the corresponding mod 2 flat chains to the expected limit. In practice, one often proves existence of convergent sequences of integral varifolds by appealing to Allard’s compactness theorem (described in Section 3). Here we prove that if a sequence of integral varifolds with limit V satisfies the hypotheses of Allard’s compactness theorem plus one additional hypothesis, then the corresponding mod 2 flat chains converge to [V ]. THEOREM 1.1 Let V (i) be a sequence of m-dimensional integral varifolds in an open set U of RN that converges to a limit V . Suppose that CURRENTS, FLAT CHAINS, AND VARIFOLDS 43 (1) the V (i) satisfy the hypotheses of Allard’s compactness theorem for integral varifolds, and (2) the boundaries ∂[V (i)] of the mod 2 flat chains [V (i)] converge in the flat topology. Then the chains [V (i)] converge in the flat topology to [V ]. I do not know whether hypothesis (2) is really necessary. There is an analogous theorem with rectifiable currents in place of mod 2 flat chains. Suppose that A is an m-dimensional integer-multiplicity rectifiable current in U and that V is an m-dimensional integral varifold in U. Recall that A determines an integral varifold v(A) by forgetting orientations (see [Si, Section 27]). We say that A and V are compatible provided that V = v(A) + 2W for some integral varifold W in U. Thus A and V are compatible if and only if they determine the same mod 2 rectifiable chain. Equivalently, A and V are compatible provided that (V,x) − (A, x) is a nonnegative, even integer for Hm-almost every x ∈ U. The analog of Theorem 1.1 for integer-multiplicity currents is the following theorem. THEOREM 1.2 Let V (i) and A(i) be sequences of m-dimensional integral varifolds and integer- multiplicity currents, respectively, in U, such that V (i) and A(i) are compatible for each i. Suppose that the V (i) satisfy the hypotheses of Allard’s compactness theorem for integral varifolds. Suppose also that the boundaries ∂A(i) converge (in the integral flat topology) to a limit current. Then there is a subsequence i(k) such that the V (i(k)) converge to an integral varifold V ,theA(i(k)) converge to a limit integer-multiplicity current A, and A and V are compatible. The existence of a subsequence for which the limits V and A exist follows im- mediately from Allard’s compactness theorem for integral varifolds and from the Federer-Fleming compactness theorem for integer-multiplicity currents. What is new here is the compatibility of the limits A and V . 44 BRIAN WHITE 2. Preliminaries 2.1. Terminology For mod 2 flat chains, see Fleming’s original paper [Fl] or, for a different approach, Federer’s book [Fe, Section 4.2.26]. Unfortunately (for the purposes of this article), a multiplicity [1] plane does not qualify as a mod 2 flat chain under either definition.∗ By contrast, a multiplicity 1 plane does qualify as an integral varifold. Thus, in order for the map V → [V ] (as described in Section 1) to be a homomorphism from integral varifolds to mod 2 flat chains, one must either restrict the class of varifolds or enlarge the class of flat chains. If one prefers to restrict, then one should (throughout this particle) replace “vari- fold” with “compactly supported varifold” and “flat chain” with “compactly supported flat chain.” (Federer’s flat chains are automatically compactly supported, but Fleming’s + need not be.) Likewise, Lm-rec(U,Z ) and Lm-rec(U,Z2) should be replaced by the subsets consisting of compactly supported functions. In particular, the main theorem, Theorem 3.3, remains true with those replacements. However, in this article we have chosen instead to enlarge the class of flat chains. Fortunately, only a slight modification in Fleming’s definition (or Federer’s) is required to produce the enlarged class of flat chains. (Flat chains so defined would, in the terminology of [Fe], be called “locally flat chains” However, although locally flat chains over the integers are briefly mentioned in [Fe, Section 4.1.24], the mod 2 versions are not.) See Section 4 for the required modification. When the coefficient group is the integers (with the standard metric), the “correct” class of flat chains is defined in [Si], and the rectifiability and compactness theorems are proved there. 2.2. Notation Suppose that M is a Borel subset of a properly embedded m-dimensional C1- submanifold of U, or of a countable union of such manifolds. If M has locally finite Hm measure, we let [M] denote the mod 2 flat chain associated to M,andwelet v(M) denote the integral varifold associated to M. More generally, if f : M → Z+ is a function such that the extension F : U → Z+, f (x) if x ∈ M, F (x) = 0 if x ∈ U \ M ∗Federer’s definition requires that a flat chain have compact support, and Fleming’s definition requires that a flat chain have finite flat norm. CURRENTS, FLAT CHAINS, AND VARIFOLDS 45 + is in Lm-rec(U,Z ),thenweletv(M,f) be the integral varifold in U corresponding to F . 2.3. Pushforwards Suppose that V is an integral varifold in U and that φ : U → W is a C1-map that is ∩ proper on U spt(μV ). Then the pushforward φ#V is also an integral varifold in W, and it satisfies = (φ#V,y) (V,x) (1) φ(x)=y for Hm-almost every y ∈ W. Similarly, if A is a rectifiable mod 2 flat chain in U,andifφ : U → W is locally ∩ Lipschitz and proper on U spt μA, then the image chain φ#A satisfies = [(φ#A, y)] [(A, x)] (2) φ(x)=y for Hm-almost every y ∈ W. Hm Note that (2) determines (φ#A, y) for -almost every y since its value is 0 or 1 almost everywhere. In other words, for Hm-almost every y ∈ W, 1 if (A, y) is odd, = φ(x)=y (φ#A, y) (3) 0 if the sum is even.
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