The Concept of Varifold

COMMUNICATION

Ulrich Menne

of grain boundaries in an annealing pure metal. This was the starting point of the rapid development of mean curvature ﬂow, even for smooth surfaces.

What is a Varifold? Constructing nonparametric models of nonsmooth surfaces usually requires entering the realm of geometric measure theory. In the simplest case, we associate to each smooth surface 푀 the measure over the ambient space whose value at a set 퐴 equals the surface measure of the intersection 퐴 ∩ 푀. For varifolds, it is in fact expedient to similarly record information on the tangent planes Introduction of the surfaces. This yields weak continuity of area, ba- Motivation sic compactness theorems, and the possibility to retain Apart from generalisation, there are two main reasons to tangential information in the limit. include nonsmooth surfaces in geometric variational prob- lems in Euclidean space or Riemannian manifolds: firstly, Main Topics Covered the separation of existence proofs from regularity con- We will introduce the notational infrastructure together siderations and, secondly, the modelling of nonsmooth with a variety of examples. physical objects. Following the first principle, varifolds This will then allow us were introduced by F. Almgren in 1965 to prove, for every the attempt to to formulate the compact- intermediate dimension, the existence of a generalised ness theorem for integral minimal surface (i.e., a surface with vanishing first vari- comprise all varifolds, Theorem 1, and ation of area) in a given compact smooth Riemannian two key regularity theo- manifold. Then, in 1972, an important partial regularity objects that rems for integral varifolds result for such varifolds was established by W. Allard in his with mean curvature, The- foundational paper “On the first variation of a varifold” [2]. should be orems 2 and 3. The accom- These pioneering works still have a strong influence in considered panying drawings gener- geometric analysis as well as related fields. Particularly ally stress certain aspects prominent examples are the proof of the Willmore con- 푚-dimensional deemed important at the jecture by F. Marques and A. Neves and the development expense of accuracy. A ver- of a regularity theory for stable generalised minimal sur- surfaces sion of this article with faces in one codimension by N. Wickramasekera, both published in 2014. Following the second principle, vari- additional references is folds were employed by Almgren’s PhD student K. Brakke available from https://arxiv.org/abs/1705.05253. in 1978 to create a mathematical model of the motion Notation Ulrich Menne currently holds an interim professorship at the Uni- Suppose throughout this note that 푚 and 푛 are integers versity of Leipzig and a postdoctoral position at the Max Planck and 1 ≤ 푚 ≤ 푛. We will make use of the 푚-dimension- Institute for Mathematics in the Sciences. His e-mail address is al Hausdorff measure ℋ푚 over R푛. This measure is [email protected]. one of several natural outer measures over R푛 which For permission to reprint this article, please contact: associate the usual values with subsets of 푚-dimensional [email protected]. continuously differentiable submanifolds of R푛. DOI: http://dx.doi.org/10.1090/noti1589

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First Order Quantities Definition (Integral varifold). An 푚-dimensional vari- Amongst the initial reasons for developing the notion of 푚- fold 푉 in R푛 is called integral if and only if there exists 푛 dimensional varifold in R was the attempt to comprise all a sequence of Borel subsets 퐵푖 of closed 푚-dimensional 푛 objects that should be considered 푚-dimensional surfaces continuously differentiable submanifolds 푀푖 of R with 푛 ∞ of locally finite area in R . Its definition is tailored for 푉 = ∑푖=1 v푚(퐵푖). compactness. An integral varifold 푉 is determined by its weight ‖푉‖; Definition (Varifold and weight). If G(푚, 푛) is the space in fact, associated to ‖푉‖ there are a multiplicity func- of unoriented 푚-dimensional subspaces of R푛, endowed tion Θ and a tangent plane function 휏 with values in the with its natural topology, then by an 푚-dimensional var- nonnegative integers and G(푛, 푚), respectively, such that 푛 푛 ifold 푉 in R , we mean a Radon measure 푉 over R × 푉(푘) = ∫ 푘(푥, 휏(푥))Θ(푥) dℋ푚 푥 G(푛, 푚), and we denote by ‖푉‖ the weight of 푉, that is, 푛 its canonical projection into R푛. whenever 푘 ∶ R × G(푚, 푛) → R is a continuous function with compact support. The theory of Radon measures yields a metrisable topology on the space of 푚-dimensional varifolds in R푛 Example 4 (Sum of two submanifolds). Suppose 푀푖 are closed 푚-dimensional continuously differentiable sub- such that 푉푖 → 푉 as 푖 → ∞ if and only if 푛 manifolds 푀푖 of R , for 푖 ∈ {1, 2}, and 푉 = v푚(푀1) + ∫ 푘 d푉 → ∫ 푘 d푉 as 푖 → ∞ 푚 푖 v푚(푀2). Then, for ℋ almost all 푧 ∈ 푀1 ∩ 푀2, we have whenever 푘 ∶ R푛 × G(푚, 푛) → R is a continuous function Θ(푧) = 2 and 휏(푧) = Tan(푀1, 푧) = Tan(푀2, 푧). with compact support. The theory of varifolds is particularly successful in the Proposition (Basic compactness). Whenever 휅 is a real- variational study of the area integrand. To describe some 푛 valued function on the bounded open subsets of R , the set aspects of this, diffeomorphic deformations are needed 푛 of 푚-dimensional varifolds 푉 in R satisfying ‖푉‖(푍) ≤ in order to define and compute its first variation. 휅(푍) for each bounded open subset 푍 of R푛 is compact. Definition (Area of a varifold). If 퐵 is a Borel subset of R푛 In general, neither does the weight ‖푉‖ of 푉 need to be and 푉 is called an 푚-dimensional varifold 푉 in R푛, then 푚-dimensional, nor does the Grassmannian information the area of 푉 in 퐵 equals ‖푉‖(퐵). of 푉 need to be related to the geometry of ‖푉‖: Example 5 (Area via multiplicity). If 푉 is integral, then Example 1 (Region with associated plane). For each plane 푚 ‖푉‖(퐵) = ∫퐵 Θdℋ . 푇 ∈ G(푛, 푚) and each open subset 푍 of R푛, the product of the Lebesgue measure restricted to 푍, with the Dirac Definition (Push-forward). Suppose 푉 is an 푚-dimension- 푛 measure concentrated at 푇, is an 푚-dimensional varifold al varifold in R and 휙 is a continuously differentiable 푛 푛 in R푛. diffeomorphism of R onto R . Then, the push-forward of 푉 by 휙 is the 푚-dimensional varifold 휙#푉 defined by Example 2 (Point with associated plane). For 푎 ∈ R푛 and 푇 ∈ G(푛, 푚), the Dirac measure concentrated at (푎, 푇) is ∫ 푘 d휙#푉 = ∫ 푘(휙(푧), image(D 휙(푧)|푆))푗푚휙(푧, 푆)d푉(푧, 푆) an 푚-dimensional varifold in R푛. whenever 푘 ∶ R푛 × G(푚, 푛) → R is a continuous function with compact support. Here, 푗 휙(푧, 푆) denotes the This generality is in fact expedient to describe higher- 푚 푚-dimensional Jacobian of the restriction to 푆 of the dif- (or lower-) dimensional approximations of more regular 푚- ferential D 휙(푧) ∶ R푛 → R푛. dimensional surfaces, so as to include information on the 푚-dimensional tangent planes of the limit. Such situations Example 6 (Push-forward). If 퐵 and 푀 are as in Example 3, may be realised by elaborating on Examples 1 and 2. then, by the transformation formula (or area formula), However, we will mainly consider integral varifolds, that 휙#(v푚(퐵)) = v푚(image(휙|퐵)). is, varifolds in the spirit of the next basic example, which Definition (First variation). The first variation of a vari- avoid the peculiarities of the preceding two examples. fold 푉 in R푛 is defined by Example 3 (Part of a submanifold). If 퐵 is a Borel sub- (δ푉)(휃) = ∫ trace(D 휃(푧) ∘ 푆 )d푉(푧, 푆) set of a closed 푚-dimensional continuously differentiable ♮ submanifold 푀 of R푛, then, by Riesz’s representation the- whenever 휃 ∶ R푛 → R푛 is a smooth vector field with com- 푛 orem, the associated varifold v푚(퐵) in R may be defined pact support, where 푆♮ denotes the canonical orthogonal by projection of R푛 onto 푆. 푚 ∫ 푘 dv푚(퐵) = ∫퐵 푘(푥, Tan(푀, 푥)) dℋ 푥 Example 7 (Meaning of the first variation). If the maps 푛 푛 푛 whenever 푘 ∶ R × G(푚, 푛) → R is a continuous function 휙푡 ∶ R → R satisfy 휙푡(푧) = 푧 + 푡휃(푧) whenever 푡 ∈ with compact support, where Tan(푀, 푥) is the tangent R, |푡| Lip 휃 < 1, and 푧 ∈ R푛, then one may verify that, 푛 푛 space of 푀 at 푥. Then, the weight ‖v푚(퐵)‖ over R equals for such 푡, 휙푡 is a smooth diffeomorphism of R onto 푚 푛 the restriction ℋ ⌞ 퐵 of 푚-dimensional Hausdorff mea- R and ‖(휙푡)#푉‖(spt 휃) is a smooth function of 푡 whose sure to 퐵. differential at 0 equals (δ푉)(휃).

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Example 8 (First variation of a smooth submanifold I). Suppose that 푀 is a properly embedded, twice continuously differentiable 푚-dimensional submanifold-with- 푛 boundary in R and 푉 = v푚(푀). Then, integrating by parts, one obtains 푚 (δ푉)(휃)= − ∫푀 h(푀, 푧) • 휃(푧)dℋ 푧 푚−1 + ∫휕푀 휈(푀, 푧) • 휃(푧)dℋ 푧, where h(푀, 푧) ∈ R푛 is the mean curvature vector of 푀 at 푧 ∈ 푀, 휈(푀, 푧) is the outer normal of 푀 at 푧 ∈ 휕푀, and • denotes the inner product in R푛. In fact, h(푀, 푧) belongs to the normal space, Nor(푀, 푧), of 푀 at 푧. Figure 1. One-dimensional submanifolds (by the spheres drawn) of R2 converging to a limit nonzero, Second Order Quantities nonintegral varifold with all one-dimensional tangent 2 In general, even integral varifolds—which already are planes equally weighted at every point of R . significantly more regular than general varifolds—give meaning only to first order properties such as tangent planes but do not possess any second order The preceding theorem is fundamental both for var- properties such as curva- ifolds and the consideration of limits of submanifolds. tures. Yet, for varifolds of The summand ‖δ푉‖(푍) therein may not be omitted: For varifolds of locally bounded first varia- Example 12 (Lattices of spheres; see Figure 1). If 푉 is the tion, one may define mean 푖 varifold associated to locally bounded curvature. 푀 = R푚+1 ∩{푧 : |푖푧 − 푎|=3−1푖−1/푚 for some 푎 ∈ Z푚+1}, first variation, Example 9 (A particular 푖 continuously differentiable for every positive integer 푖, then the nonzero limit of one may define submanifold without—even this sequence is the product of the Lebesgue measure approximate—curvatures; over R푚+1 with an invariant Radon over G(푚 + 1, 푚). mean see R. Kohn [3]). There Definition (Mean curvature). Suppose 푉 is a varifold in R푛 exists a nonempty closed of locally bounded first variation and 휎 is the singular curvature. continuously differentiable part of ‖δ푉‖ with respect to ‖푉‖. Then, there exist a 휎 al- 푚-dimensional submani- most unique, 휎 measurable function 휂(푉, ⋅) with values fold 푀 of R푚+1 such that for any twice continuously in the unit sphere S푛−1 as well as a ‖푉‖ almost unique, differentiable 푚-dimensional submanifold 푁 of R푚+1, ‖푉‖ measurable, locally ‖푉‖ summable, R푛-valued func- there holds ℋ푚(푀 ∩ 푁) = 0. tion h(푉, ⋅)—called the mean curvature of 푉—satisfying Definition (Locally bounded first variation). A varifold 푉 (δ푉)(휃)= − ∫ h(푉, 푧) • 휃(푧) d‖푉‖ 푧 in R푛 is said to be of locally bounded first variation if and only if ‖δ푉‖(푍), defined as + ∫ 휂(푉, 푧) • 휃(푧) d휎 푧 sup{(δ푉)(휃) :휃 a smooth vector field compactly whenever 휃 ∶ R푛 → R푛 is a smooth vector field with com- supported in 푍 with |휃| ≤ 1} pact support. 푛 whenever 푍 is an open subset of R , is finite on bounded Example 13 (Varifold mean curvature of a smooth subsets. Then ‖δ푉‖ may be uniquely extended to a Radon manifold). If 푀, 푉, and 휈 are as in Example 8, then 휎 = 푛 measure over R , also denoted by ‖δ푉‖. ℋ푚−1 ⌞ 휕푀, 휂(푉, 푧) = 휈(푀, 푧) for 휎 almost all 푧, and Clearly, for any open subset 푍 of R푛, ‖δ푉‖(푍) is lower h(푉, 푧) = h(푀, 푧) for ‖푉‖ almost all 푧. semicontinuous in 푉. Yet, in general, the mean curvature vector may have Example 10 (Functions of locally bounded variation). If tangential parts, and the behaviour of 휎 and 휂(푉, ⋅) may 푚 푚 = 푛, then, 푚-dimensional varifolds in R of locally differ from that of boundary and outer normal: bounded first variation are in natural correspondence to real-valued functions of locally bounded first variation Example 14 (Weighted plane). If 푚 = 푛, Θ is a positive, 푚 on R푚. continuously differentiable function on R , and 푉(푘) = ∫ 푘(푧, R푚)Θ(푧)dℋ푚 푧 whenever 푘 ∶ R푚 × G(푚, 푚) → R is Example 11 (First variation of a smooth submanifold II). a continuous function with compact support, then If 푀 and 푉 are as in Example 8, then we have ‖δ푉‖ = |h(푀, ⋅)|ℋ푚 ⌞ 푀 + ℋ푚−1 ⌞ 휕푀. h(푉, 푧) = grad(log ∘ Θ)(푧) for ‖푉‖ almost all 푧. Theorem 1 (Compactness of integral varifolds; see Allard Example 15 (Primitive of the Cantor function). If 퐶 is [2, 6.4]). If 휅 is a real-valued function on the bounded the Cantor set in R, 푓 ∶ R → R is the associated func- open subsets of R푛, then the set of integral varifolds with tion (i.e., 푓(푥) = ℋ푑(퐶 ∩ {휒 : 휒 ≤ 푥}) for 푥 ∈ R, where ‖푉‖(푍) + ‖δ푉‖(푍) ≤ 휅(푍) for each bounded open sub- 푑 = log 2/ log 3), and 푉 is the varifold in R2 ≃ R × R as- set 푍 of R푛 is compact. sociated to the graph of a primitive function of 푓, then 푉

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In contrast, if 푝 ≥ 푚, then ℋ푚 ⌞ spt ‖푉‖ ≤ ‖푉‖ whenever 푉 is an integral varifold in 퐻(푝); in particular, spt ‖푉‖ has locally finite 푚-dimensional measure. Definition (Singular). An 푚-dimensional varifold 푉 in R푛 is called singular at 푧 in spt ‖푉‖ if and only if there is no neighbourhood of 푧 in which 푉 corresponds to a positive multiple of an 푚-dimensional continuously differentiable submanifold. Example 18 (Zero sets of smooth functions). If 퐴 is a closed subset of R푚, then there exists a nonnegative Figure 2. Varifold (blue, on the right) with fractal smooth function 푓 ∶ R푚 → R with 퐴 = {푥 : 푓(푥) = 0}. “boundary.” Left: Cantor function. Right: A primitive Example 19 (Touching; see Figure 3). Suppose 퐴 is a of the Cantor function. closed subset of R푚 without interior points such that ℋ푚(퐴) > 0, 푓 is related to 퐴 as in Example 18, 푛 = 푚+1, 푛 푚 푛 R ≃ R × R, and 푀푖 ⊂ R , for 푖 ∈ {1, 2}, correspond is an integral varifold of locally bounded first variation, to the graph of 푓, and R푚 × {0}, respectively. Then, h(푉, 푧) = 0 for ‖푉‖ almost all 푧, and spt 휎 corresponds 푉 = v푚(푀1) + v푚(푀2) (see Examples 4 and 13) is an to 퐶 via the orthogonal projection onto the domain of 푓, integral varifold in 퐻(∞) which is singular at each point as in Figure 2. of 푀1 ∩ 푀2 ≃ 퐴 × {0}. Definition (Classes 퐻(푝) of summability of mean curva- Proposition (Structure of one-dimensional integral vari- ture). For 1 ≤ 푝 ≤ ∞, the class 퐻(푝) is defined to consist folds in 퐻(1); see [4, 12.5]). If 푚 = 1 and 푉 ∈ 퐻(1) 푛 of all 푚-dimensional varifolds 푉 in R of locally bounded is an integral varifold, then, near ‖푉‖ almost all points, first variation satisfying the following three conditions: if 푉 may be locally represented by finitely many Lipschitzian 푝 > 1, then ‖δ푉‖ is absolutely continuous with respect graphs. to ‖푉‖; if 1 < 푝 < ∞, then their mean curvature is locally 푝-th power ‖푉‖ summable; and if 푝 = ∞, it is locally Proposition (Regularity of one-dimensional integral vari- ‖푉‖ essentially bounded. folds with vanishing first variation; see Allard and Alm- gren [1, §3]). If 푚 = 1 and 푉 is an integral varifold with 푝 1/푝 푚 Observe that (∫푍 |h(푉, 푧)| d‖푉‖푧) is lower semicon- δ푉 = 0, then the set of singular points of 푉 has ℋ mea- tinuous in 푉 ∈ 퐻(푝) if 푍 is an open subset of R푛 and sure zero. 1 < 푝 < ∞; a similar statement holds for 푝 = ∞. By It is a key open prob- Theorem 1, this entails further compactness theorems for lem to determine whether integral varifolds. the hypothesis 푚 = 1 is es- Example 16 (Critical scaling for 푝 = 푚). If 푉 ∈ 퐻(푚), sential; the difficulty of the 0 < 푟 < ∞, and 휙 ∶ R푛 → R푛 satisfies 휙(푧) = 푟푧 for case 푚 ≥ 2 arises from the 푛 푧 ∈ R , then 휙#푉 ∈ 퐻(푚) and we have possible presence of holes: 푚 푚 ∫image(휙|퐵)|h(휙#푉, 푧)| d‖휙#푉‖푧 = ∫퐵|h(푉, 푧)| d‖푉‖푧, Example 20 (Swiss cheese; see Brakke or [4, 10.8]). If and, in case 푚 = 1, additionally ‖δ(휙#푉)‖(image(휙|퐵)) = ‖δ푉‖(퐵) whenever 퐵 is a Borel subset of R푛. 2 ≤ 푚 < 푛, then there exist 푉 ∈ 퐻(∞) integral, Regularity by First Variation Bounds 푇 ∈ G(푛, 푚), and a Borel subset 퐵 of 푇 ∩ spt ‖푉‖ For twice continuously differentiable submanifolds 푀, Figure 3. A simple integral with ℋ푚(퐵) > 0 such the mean curvature vector h(푀, 푧) at 푧 ∈ 푀 is defined as varifold with bounded that no 푏 ∈ 퐵 belongs to trace of the second fundamental form mean curvature whose the interior of the orthog- set of singular points has b(푀, 푧) ∶ Tan(푀, 푧) × Tan(푀, 푧) → Nor(푀, 푧). onal projection of spt ‖푉‖ positive weight measure. For varifolds, bounds on the mean curvature (or, more gen- onto 푇 relative to 푇. So, for ℋ푚 almost all 푏 ∈ 퐵, 휏(푏) = 푇 and 푉 is singular at 푏 (cf. erally, on the first variation) are defined without reference Example 4); see Figure 4. to a second order structure and entail more regularity, similar to the case of weak solutions of the Poisson Yet, the preceding varifold needs to have nonsingular equation. The key additional challenges are nongraphical points in spt ‖푉‖. To treat ‖푉‖ almost all points, however, behaviour and higher multiplicity: weaker concepts of regularity are needed. Example 17 (Cloud of spheres). If 1 ≤ 푝 < 푚 < 푛 and Theorem 2 (Allard’s regularity theorem—qualitative ver- 푍 is an open subset of R푛, then there exists a countable sion; see Allard [2, 8.1 (1)]). If 푝 > 푚 ≥ 2 and 푉 ∈ 퐻(푝) collection 퐶 of 푚-dimensional spheres in R푛 such that is integral, then the set of singular points of 푉 contains no 푉 = ∑푀∈퐶 v푚(퐶) ∈ 퐻(푝) and spt ‖푉‖ = Closure 푍. interior points relative to spt ‖푉‖.

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References [1] W. K. Allard and F. J. Almgren Jr., The structure of station- ary one dimensional varifolds with positive density, Invent. Math. 34 (1976), no. 2, pages 83–97. MR 0425741 [2] William K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), pages 417–491. MR 0307015 [3] Robert V. Kohn, An example concerning approximate dif- ferentiation, Indiana Univ. Math. J. 26 (1977), no. 2, pages 393–397. MR 0427559 [4] Sławomir Kolasiński and Ulrich Menne, Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds, NoDEA Nonlinear Differential Equations Appl. 24 Figure 4. Constructing an integral varifold with (2017), Art. 17, pages 1–56. MR 3625810 bounded mean curvature with holes near a set of [5] Ulrich Menne, Second order rectifiability of integral vari- positive weight measure. Left: Building blocks (from folds of locally bounded first variation, J. Geom. Anal. 23 side). Right: Result (from top). (2013), no. 2, pages 709–763. MR 3023856

Image Credits Theorem 3 (Existence of a second order structure; see [5, All figures courtesy of Ulrich Menne. 4.8]). If 푉 is an integral varifold in 퐻(1), then there exists a Author photo courtesy of Max Planck Institute for Gravitational countable collection 퐶 of twice continuously differentiable Physics (Albert Einstein Institute). 푚-dimensional submanifolds such that 퐶 covers ‖푉‖ almost all of R푛. Moreover, for 푀 ∈ 퐶, there holds h(푉, 푧) = h(푀, 푧) for ‖푉‖ almost all 푧. Hence, one may define a concept of second fundamental form of 푉 such that, for ‖푉‖ almost all 푧, its trace is h(푉, 푧). Moreover, Theorem 3 is the base for studying the fine behaviour of integral varifolds in 퐻(푝); see [5] and [4].

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