GMT Seminar: Introduction to Integral Varifolds DANIEL WESER These notes are from two talks given in a GMT reading seminar at UT Austin on February 27th and March 6th, 2019. 1 Introduction and preliminary results Definition 1.1. [Sim84] n k Let G(n; k) be the set of k-dimensional linear subspaces of R . Let M be locally H -rectifiable, n 1 and let θ : R ! N be in Lloc. Then, an integral varifold V of dimension k in U is a Radon 0 measure on U × G(n; k) acting on functions ' 2 Cc (U × G(n; k) by Z k V (') = '(x; TxM) θ(x) dH : M By \projecting" U × G(n; k) onto the first factor, we arrive at the following definition: Definition 1.2. [Lel12] n Let U ⊂ R be an open set. An integral varifold V of dimension k in U is a pair V = (Γ; f), k 1 where (1) Γ ⊂ U is a H -rectifiable set, and (2) f :Γ ! N n f0g is an Lloc Borel function (called the multiplicity function of V ). We can naturally associate to V the following Radon measure: Z k µV (A) = f dH for any Borel set A: Γ\A We define the mass of V to be M(V ) := µV (U). We define the tangent space TxV to be the approximate tangent space of the measure µV , k whenever this exists. Thus, TxV = TxΓ H -a.e. Definition 1.3. [Lel12] If Φ : U ! W is a diffeomorphism and V = (Γ; f) an integral varifold in U, then the pushfor- −1 ward of V is Φ#V = (Φ(Γ); f ◦ Φ ); which is itself an integral varifold in W . Definition 1.4. [Lel12] 1 n If V is a varifold in U and X 2 Cc (U; R ), then the first variation of V along X is defined by d δV (X) = M (Φt)#V ; (1.1) dt t=0 where Φt is the one-parameter family generated by X. 1 Proposition 1.5. [Lel12] n Let V = (Γ; f) be an integral varifold in U ⊂ R . Then the right hand side of (1.1) is well- defined and Z 1 n δV (X) = divTxΓX dµV for all X 2 Cc (U; R ): (1.2) U k Proof. By the standard simplifying arguments, we may assume Γ = F (R ) for F Lipschitz. Then, Z −1 k M((Φt)#V ) = f(Φt (z)) dH (z) Φt(Γ) Z k = f(z) JΦtjz dH (z) Γ Z k = f(z) JΦtjz dH (z) F (Rk) Z k = f(F (x))) JΦtjF (x) JF jx dH (x): Rk Hence, Z d d k δV (X) = M((Φt)#V ) = f(F (x))) JΦtjF (x) JF jx dH (x): t=0 k t=0 dt R dt d From computations before, we know that dt t=0JΦtjy = divTxΓX(y), which concludes the proof. Definition 1.6. [Lel12] We say that V has bounded generalized mean curvature if there exists a C ≥ 0 such that Z 1 n jδV (X)j ≤ C jXj dµV for all X 2 Cc (U; R ): (1.3) U Proposition 1.7. [Lel12] If V is a varifold in U with bounded generalized mean curvature, then there is a bounded Borel n map H : U ! R such that Z ~ 1 n δV (X) = − X · H dµV for all X 2 Cc (U; R ): (1.4) U H~ is called the generalized mean curvature of V and is defined µV -a.e. 1 n Proof. First, (1.2) tells us that δV is continuous in Cc (U; R ). Since inequality (1.3) holds for 1 n 0 all X 2 Cc (U; R ) and doesn't involve derivatives, by density it extends to all X 2 Cc . For 0 n 0 n X 2 Cc (U; R ), let sptX ⊂ BR. Thus, δV is a bounded linear functional on Cc (U; R ); so by Riesz we can find a Radon measure jjδV jj, which is the total variation measure of δV , and a jjδV jj-measurable function ~ν with j~νj = 1 jjδV jj-a.e. such that Z δV (X) = X · ~νdjjδV jj: U Furthermore, for A open, 0 n jjδV jj(A) = supfδV (X): X 2 Cc (A; R ); jjXjjC0 ≤ 1g: (1.5) 2 0 n Looking back at (1.3), for each X 2 Cc (A; R ), we know Z jδV (X)j ≤ C jXj dµV ≤ µV (A \ U) jjXjjC0 : (1.6) U Hence, we combine (1.5) and (1.6) to get jjδV jj(A) = supfδV (X): :::g ≤ supfCµV (U \ A)jjXjjC0 : :::g ≤ CµV (U \ A); so jjδV jj is absolutely continuous with respect to µV . Hence, if we label the Radon-Nikodym derivative of jjδV jj with resepct to µV as −H, which exists µV -a.e., and we label H~ = H~ν then ~νdjjδV jj = −H~ dµV and Z δV (X) = − X · H~ dµV U as desired. Remark 1.8. If we didn't have the inequality jδV (X)j ≤ µV (A \ U) jjXjjC0 0 n for X 2 Cc (A; R ), then we couldn't conclude that jjδV jj is absolutely continuous with respect to µV . However, if we knew that V had locally bounded first variation given by jδV (X)j ≤ cjjXjjC0 ; we still could still use Riesz and apply a Lebesgue decomposition to jjδV jj to write ~νdjjδV jj = −H~ dµV + dµsing: Then, we use a polar decomposition on µsing to write dµsing = ~νco dσ; where ~νco is the generalized co-normal and σ is the generalized boundary measure. In this way, we recover the full tangential divergence theorem Z δV (X) = divTxΓX dµV U Z Z = − X · H~ dµV + X · ~νco dσ: U U See [Sim84] Ch.8 for more details. Definition 1.9. [Lel12] 1 n V is stationary if δV (X) = 0 for all X 2 Cc (U; R ). Remark 1.10. If V = (Γ; f) is stationary, then the proposition is telling us that H~ ≡ 0, so V has k zero generalized mean curvature. If we suppose f ≡ 1, we can see that µV = H Γ\U : Then, for all 1 n X 2 Cc (U; R ), we have Z Z k δV (X) = − X · H~ dµV = − X · H~ dH = 0: Rn Γ Hence, Γ has zero mean curvature in U, so Γ is a minimal surface in U. 3 2 Compactness Theorem 2.1. [Sim84] Suppose fVjg is a sequence of integral varifolds in U which are of locally bounded first variation in U, sup µVj (W ) + jjδVjjj(W ) < 1 8 W ⊂⊂ U: j Then, there exists a subsequence fVj0 g ⊂ fVjg and an integral varifold V of locally bounded first variation in U such that Vj0 ! V in the sense of Radon measures on U × G(n; k), and jjδV jj(W ) ≤ lim infj jjδVj0 jj(W ) for all W ⊂⊂ U. Corollary 2.2. If in addition the fVjg are stationary, then, by the LSC property of the total variation measure jjδV jj, the limit varifold V is also stationary. Remark 2.3. First, we note that convergence in the sense of Radon measures on U × G(n; k) is 1 n called varifold convergence. Second, for fixed X 2 Cc (U; R ), we note that the first variation functional is continuous with respect to varifold convergence. By definition, Vj ! V as varifolds if 0 n 1 n Vj(') ! V (') for all ' 2 Cc (U × G(n; k); R ). For X 2 Cc (U; R ), Z Z δV (X) = divS X(x) dV (x; S) = '(x; S) dV (x; S); U×G(n;k) U×G(n;k) 0 n for '(x; S) = divS X(x) 2 Cc (U × G(n; k); R ). Hence, δVj(X) ! δV (X): 3 Monotonicity formula ? For a differentiable function g : U ! R and a varifold V = (Γ; f) in U, we denote by r g(x) the ? orthogonal projection of rg onto (TxΓ) , e.g. the normal part of the gradient. For fixed ξ 2 U, define r(x) := jx − ξj. Theorem 3.1. [Lel12] Let V be an integral varifold of dimension k in U with bounded generalized mean curvature H~ . Fix ξ 2 U. For every 0 < σ < ρ < dist(ξ; U) we have the Monotonicity Formula Z Z ? 2 µV (Bρ(ξ)) µV (Bσ(ξ)) H~ 1 1 jr rj k − k = · (x − ξ) k − k dµV + k dµV ; ρ σ Bρ(ξ) k m(r) ρ Bρ(ξ)nBσ(ξ) r ρjjHjj1 −k where m(r) = maxfr; σg. Hence, the map ρ 7! e ρ µV (Bρ(ξ)) is monotone increasing. 1 Proof. Without loss of generality, we assume ξ = 0. We fix a function γ 2 Cc ([0; 1]) such that γ ≡ 1 jxj in some neighborhood of 0. For s 2 [0; dist(0; @U)], we define the vector field Xs(x) := γ s x. 1 Then, Xs 2 Cc (U), so we can combine (1.2) and (1.4) to conclude Z Z ~ divTxΓXs dµV = − Xs · H dµV : (3.1) 4 Our goal now is compute both sides of (3.1), rearrange it in a smart way, use a dominated conver- gence argument to replace γ by the indicator function 1[0;1], and conclude the identity. We fix a point x, and let π = TxΓ. Let e1; : : : ; ek be an orthonormal basis for π, and complete n it to an orthonormal basis for R . Now, recalling that r := jx − ξj = jxj, we compute k r X r x · ej div X = k γ + e · x γ0 π s s j s jxj s j=1 k 2 r r r X x · ej = k γ + γ0 s s s jxj j=1 2 3 n 2 r r r X x · ej = k γ + γ0 1 − s s s 4 jxj 5 j=k+1 r r r = k γ + γ0 1 − jr?rj2 : (3.2) s s s Now, we insert (3.2) into (3.1), divide both sides by sk+1, and integrate s between σ and ρ: Z ρ Z Z ρ Z k jxj r 0 jxj ? 2 k+1 γ dµV (x) ds + k+2 γ 1 − jr rj dµV (x) ds σ Rn s s σ Rn s s Z ρ Z H~ · x jxj = − k+1 γ dµV (x) ds: σ Rn s s Then, we use Fubini's theorem to change the order of integration, we distribute the integrand in the term (1 − jr?rj2), and we move the jr?rj2 term to the right hand side to find: Z Z ρ k jxj jxj 0 jxj k+1 γ + k+2 γ ds dµV (x) (3.3) Rn σ s s s s Z Z ρ Z Z ρ ? 2 jxj 0 jxj ~ 1 jxj = jr rj k+2 γ ds dµV (x) − H · x k+1 γ dµV (x) ds: (3.4) Rn σ s s Rn σ s s Looking at (3.3), we note that d 1 jxj k jxj 1 jxj −|xj γ = − γ + γ0 ; ds sk s sk+1 s sk s s2 so that Z ρ k jxj jxj 0 jxj − k+1 γ + k+2 γ ds σ s s s s Z ρ d 1 jxj = k γ ds σ ds s s 1 jxj 1 jxj = γ − γ : ρk ρ σk σ Now, we subsititute this indentity and the resulting identities Z ρ Z ρ k jxj jxj 0 jxj 1 jxj 1 jxj − k+1 γ ds = k+2 γ ds + k γ − k γ σ s s σ s s ρ ρ σ σ Z ρ Z ρ jxj 0 jxj k jxj 1 jxj 1 jxj − k+2 γ ds = k+1 γ ds + k γ − k γ σ s s σ s s ρ ρ σ σ 5 into (3.3) to find Z Z Z Z ρ −k jxj −k jxj ~ 1 jxj ρ γ dµV (x) − σ γ dµV (x) − H · x k+1 γ ds dµV (x) Rn ρ Rn σ Rn σ s s Z Z ρ ? 2 −k jxj −k jxj k jxj = jr rj ρ γ − σ γ + k+1 γ ds dµV (x): (3.5) Rn ρ σ σ s s 1 Our initial choice of γ 2 Cc ([0; 1]) was arbitrary, and we arrived at (3.6).