Second order rectifiability of varifolds of bounded mean curvature

We prove that the support of an $ m $ dimensional rectifiable varifold with a uniform lower bound on the density and bounded generalized mean curvature can be covered $ \mathscr{H}^{m} $ almost everywhere by a countable union of $m$ dimensional submanifolds of class $ \mathcal{C}^{2} $. We obtain this result using the notion of curvature of arbitrary closed sets originally developed in stochastic geometry and extending to our geometric setting techniques developed by Trudinger in the theory of viscosity solutions of PDE's.


Introduction
The concept of varifold goes back to the work of Almgren in the 60's and, since then, has played a central role in Geometric Measure Theory and in its applications. The definition is simple: an m-dimensional varifold V in an open subset Ω of R n is a Radon measure over Ω × G(n, m), where G(n, m) is the Grassmann manifold of all m dimensional subspaces of R n . Given such a V , we define (1) the weight measure V of V 1 , (2) the vector-valued distribution δV : C ∞ c (Ω, R n ) → R called (isotropic) first variation 2 and (3) the total variation 3 δV of δV . If the m dimensional upper density Θ * m ( V , x) is positive at V a.e. x ∈ Ω and if δV is a Radon measure over Ω, then the celebrated rectifiability theorem of Allard [All72, 5.5, 2.8(5)] asserts that the set {x : 0 < Θ * m ( V , x) < ∞} can be H m almost covered by the union of a countable collection of m dimensional submanifolds of class C 1 of R n and V = H m Θ m ( V , ·). See also [DPDRG18] for a recent extension of Allard's rectifiability result to the anisotropic case.
1 V is the Radon measure over Ω such that V (U ) = V (U × G(n, m)) for each open subset U of Ω.
2 δV (g) = D g • S dV (x, S) for every g ∈ C ∞ c (Ω), that is the initial rate of change of the total mass of the smooth deformation of V with initial velocity given by g.
The regularity theorems of Allard and Duggan, [All72,8] and [Dug86, Theorem 2.1], allows to conclude that if 2 ≤ m < p < ∞, θ > 0, α < ∞ and V is an m dimensional varifold such that Θ m ( V , x) ≥ θ for V almost every x ∈ Ω and such that δV (g) ≤ α( |g| p/(p−1) d V ) (p−1)/p for every g ∈ C ∞ c (Ω, R n ) 4 , then a dense open subset of spt V is an m dimensional submanifold M of class W 2,p . If we additionally assume that there exists β < 2 such that Θ m ( V , x) ≤ β for V almost every x ∈ Ω, then the conclusion can be strengthened to V (Ω ∼ M ) = 0. However, one may construct integral varifolds V of higher multiplicity such that δV ≤ α V (i.e. h(V, ·) ∈ L ∞ ( V , R n )) and V cannot be locally represented as a graph of multiple-valued function around each point of a set of positive V measure, see [All72,8.1(2)] and [Bra78,6.1]. It follows that, in the case of higher multiplicity, the structure around almost every point of a varifold cannot be studied using classical regularity theory, even under the rather strong assumption δV ≤ α V (however, in the case δV = 0, it is an open question if classical regularity holds almost everywhere).
On the other hand it is reasonable to presume that an integrable mean curvature should entail a certain amount of regularity around almost every point and this regularity has been effectively discovered in recent years in the case of integral 5 varifolds. In particular, the following results are currently known: (1) rectifiability of class C 2 has been completely solved in [Men13, Theorem 1] (see also [Sch04,5.1]-[Sch09, 3.1] for the first positive result ever obtained in this direction), (2) tilt excess decay rates has been systematically clarified in most of the cases in [Bra78], [Sch04], [Men12], [Men13] and [KM17], and (3) the equivalence of quadratic decay rates and rectifiability of class C 2 has been proved in [Sch09,3.1]. In contrast, for general rectifiable varifolds (i.e. the density function is real-valued), up to now, none of the aforementioned results is known (not even in the stationary case δV = 0). One problem to extend them to this more setting is that in the integral case they rely on the theory of Q-valued functions developed by Almgren and on a blow-up procedure, which has been originally developed by Brakke in [Bra78,5.6]. How to extend these techniques to non-integral varifolds is currently unclear.
In this paper, following a completely different approach, we prove rectifiability of class C 2 for varifolds with a uniform lower bound on the density and bounded generalized mean curvature, thus providing the first positive regularity results valid for almost every point of varifolds with real-valued densities with possible higher multiplicity. Our main result reads as follows: 1.1 Theorem. Suppose 1 ≤ m < n are integers, Ω ⊆ R n is an open set, V is an m dimensional varifold in Ω, S = spt V and the following two conditions hold: Then S can be H m almost covered by a countable collection of m dimensional submanifolds of class 2 in R n .
We explain now the strategy of the proof. The basic tools of our proof are taken from the theory of curvature for arbitrary closed sets, developed in [Sta79], [HLW04] and [San17]. This theory is based on the definition for a closed subset A ⊆ R n of the generalized unit normal bundle of A: (here δ A is the distance function from A), whose fiber at a is denoted by N (A, a). Since N (A) is a countably n − 1 rectifiable subset of R n × R n (in the sense of [Fed69, 3.2.14]), one may use Coarea formula [Fed69,3.2.22] with the projectionmaps p and q (see section 2 for notation) to generalize several integral formulas from smooth varieties to general closed sets (see [HLW04,Theorem 2.1] and [San17,4.11(3), 5.4]). These formulas are expressed in terms of the generalized principal curvatures of A and the second fundamental form Q A of A; see section 2 for more details. Of course, this theory alone is too general to produce useful results for our purpose. Therefore, in order to proceed, we need to understand how it specializes for the class of closed subsets that are supports of those varifolds considered in 1.1. First, given an arbitrary closed set A ⊆ R n , we introduce the following stratification of A: The m-th stratum A (m) is the set of points where A can be touched by balls from n − m linearly independent directions. A crucial step for our result has been done in [MS17], where it is proved that, for an arbitrary closed set A, the m-th stratum A (m) can be covered by countably many m dimensional submanifolds of class 2. Therefore the main point of the present paper is to show that if S is the support of a varifold as in 1.1 then H m (S ∼ S (m) ) = 0. To prove it, we first introduce the following key definition. This is essentially everything we need to known from varifold's theory and most of the results of this paper can actually be obtained for arbitrary closed sets whose normal bundle satisfies the Lusin (N) condition. The first important consequence of this assumption is the Coarea-type formula in 3.6. We use such a formula in the main result the paper (which is Lemma 3.9) to extend one of the key results of the theory of viscosity solutions of elliptic PDE's, the Alexandrov-Bakelmann-Pucci (ABP) estimate (see [CC95,Theorem 3.2]), to our geometric setting. We do not explicitly write such a formula in the statement of our results, since the study of the ABP inequality in the context of varifolds (or, more generally, in the abstract setting of closed sets) would be beyond the scope of the present paper; however, the reader might recognize the resemblance in inequality (20) of Lemma 3.9. The validity of the ABP inequality is the central point to obtain the criterion for rectifiability of class C 2 in 3.10, whence, as one can easily see from what has been pointed out above, Theorem 1.1 follows as a special case. The proof of Lemma 9 and its main consequence Theorem 3.10 are built upon a careful generalization of the argument employed by Trudinger in [Tru89, Theorem 1] to prove twice super-differentiability almost everywhere of a viscosity subsolution of an elliptc operator. A moment of reflection reveals that the conclusion of our Theorem 3.10, H m (A ∼ A (m) ) = 0, effectively corresponds to twice super-differentiability almost everywhere for A in an higher-codimensional and non-graphical setting.
We conclude noting that in this paper we do not use the full strength of Theorem 3.10; in fact to prove Theorem 1.1 it would have been enough to have f constant in 3.10. However, we decide to state 3.10 with a much less restrictive hypothesis (and this hypothesis is maybe the optimal one) because it seems natural to think that this approach could also be useful to treat classes of varifolds with possibly unbounded mean curvature. However, verifying the Lusin (N) condition in these more general cases presents several additional nontrivial complications. It is our plan to investigate them in future works.

Notation and preliminary results
The open and closed balls of radius r and center a are respectively denoted by U(a, r) and B(a, r). The closure and the boundary in R n of a set A are denoted by Clos A and ∂A. The symbol • denotes the standard inner product of R n . If T is a linear subspace of R n , then T ♮ : R n → R n is the orthogonal projection onto T and T ⊥ = R n ∩ {v : v • u = 0 for u ∈ T }. If X and Y are sets and We adopt the language of symmetric algebra to write in a compact form our formulas: if f : V → W is a linear map between vector spaces, then there exists a unique linear map 2 f : 2 V → 2 W , which is the restriction of the unique preserving algebra homeomorphism * f :

Curvatures of arbitrary closed sets
The reference for this section is [San17].
Suppose A is a closed subset of R n . The distance function to A is denoted by δ A . If U is the set of all x ∈ R n such that there exists a unique a ∈ A with |x − a| = δ A (x), we define the nearest point projection onto A as the map ξ A characterised by the requirement and we say that x ∈ U (A) is a regular point of ξ A if and only if ξ A is approximately differentiable 7 at x with symmetric approximate differential and Combining these two facts, we now briefly describe how a general notion of second fundamental form for arbitrary closed sets has been introduced in [San17, section 4]. This notion will be repeatedly used in the rest of this paper. First of all, we define the generalized unit normal bundle of A as one uses the rectifiability properties of the distance sets {x : δ A (x) = r} (see [San17,2.13]) to conclude that N (A) is a countably n − 1 rectifiable subset of R n × S n−1 in the sense of [Fed69, 3.2.14]. Then we introduce the following definition: if x ∈ R(A) then we say that ψ A (x) is a regular point of N (A), and we denote the set of all regular points of N (A) by R(N (A)). One may check (see [San17,4.5 and we define a symmetric bilinear form Q(a, u) : here σ 1 ∈ R n is any vector such that ap D ξ A (x)(σ 1 ) = τ 1 . This is a wellposed definition, see [San17, 4.6, 4.8]. We call Q A (a, u) second fundamental form of A at a in the direction u. It is not difficult to check that if A is smooth submanifold, then Q A agrees with the classical notion of differential geometry. Moreover, if (a, u) ∈ R(N (A)) we define the principal curvatures of A at (a, u) to be the numbers κ A,1 (a, u) ≤ . . . ≤ κ A,n−1 (a, u), 1 (a, u)

The second-order rectifiable stratification
The reference for this section is [MS17]. Suppose A ⊆ R n is a closed subset of R n . For each a ∈ A we define (see This stratification and its rectifiability properties will play a crucial role in our results. In fact, we achieve rectifiability of class C 2 for a varifold V as in 1.1 proving that H m (spt V ∼ (spt V ) (m) ) = 0.

Curvature under diffeomorphic deformations
In this section we prove an explicit formula for the second fundamental form Q F [A] of a diffeomorphic deformation F [A] of an arbitrary closed set A, in terms of Q A . This formula appears to be new even in the smooth setting.
2.1 Lemma. Suppose A ⊆ R n is a closed set, F : R n → R n is a diffeomorphism of class 2 onto R n and ν F : R n × S n−1 → R n × S n−1 is given by Then ν F is a diffeomorphism of class 1 onto R n × S n−1 , (ν F ) −1 = ν F −1 and In particular, F A (m) = F (A) (m) for m = 0, . . . , n.
Proof. A direct computation shows that ν F is a diffeomorphism of class 1 onto If (a, u) ∈ N (A) and r > 0 such that U(a + ru, r) ∩ A = ∅, we let  Noting that for each a ∈ R n the function mapping u ∈ S n−1 onto q(ν F (a, u)) is a diffeomorphism onto S n−1 the postscript follows from (1) and (2) 2.2 Theorem. Suppose A is a closed subset of R n and F : R n → R n is a diffeomorphism of class 2 onto R n . Proof. We define g : R n × R n → R n to be g(a, u) = (D F (a) −1 ) * (u) for (a, u) ∈ R n × R n .

A sufficient condition for C 2 rectifiability for closed sets
This section is the main technical part of the paper. We work in the abstract setting of closed subsets whose generalized unit normal bundle satisfies the Lusin (N) condition. The main point here is to provide a general criterion for rectifiability of class C 2 (see Theorem 3.10). Then, in the next section we verify that the support of a varifold as in Theorem 1.1 satisfies the hypothesis of this criterion, thus obtaining the announced result for varifolds. In case Ω = R n , we say that N (A) satisfies the m dimensional Lusin (N) condition.
We have introduced this terminology in analogy with the theory of functions: f : R n → R n is said to satisfy the Lusin (N) condition if L n (f (A)) = 0 whenever L n (A) = 0, see [MZ92]. Actually, we can think N (A) to be a setvalued function associating at each point a the set N (A, a). Therefore we can interpret the Lusin (N) condition given in 3.1 as a property of the graph of N (A). The preservation of the Lusin (N) condition under diffeomorphisms is a subtle point. In fact, the following example shows that if we had define the Lusin condition in 3.1 replacing H n−1 (N (A)|S) = 0 with the weaker property H n−1 (q(N (A)|S)) = 0, then the resulting condition would not be preserved under diffeomorphisms, as the following example shows for n = 3 and m = 2.
3.6 Theorem. Suppose 1 ≤ m < n is an integer, Ω ⊆ R n is open, A ⊆ R n is closed and N (A) satisfies the m dimensional Lusin (N) condition in Ω.
Then for every H n−1 measurable set B ⊆ N (A)|Ω, We need the following simple fact from linear algebra in the proof of the next result.
3.7 Lemma. Suppose V and W are finite dimensional vector spaces with inner products such that dim V = m and dim W = n, f ∈ Hom(V, W ), 0 < t < ∞ and b ∈ 2 W such that b(w, w) ≤ t|w| 2 whenever w ∈ W . Then Proof. By [Fed69, 1.7.3] we can choose an orthonormal basis v 1 , . . . , v m of V and an orthonormal basis w 1 , . . . , w n of W such that Combining the two equations we get the left side. The right side is trivial.
3.8 Definition. If 0 < t < ∞, a ∈ R n an T ∈ G(n, n − 1), we define The criterion for second-order-differentiability in 3.10, that is the central result of this section, can be deduced by standard arguments from the somewhat more subtle result in 3.9.
Proof. Firstly we notice that C 4r (T, a) ⊆ B(a, 4 √ 2r) for every a ∈ R n and T ∈ G(n, n − 1). If δ is given as in 3.9, with the help of [Fed69,2.4.11], for H m a.e. a ∈ A ∩ Ω we can select s > 0 and v ∈ S n−1 such that B(a, 4 √ 2s) ⊆ Ω, 4 Proof of theorem 1.1 Here we prove Theorem 1.1. The main point will be to check that the closure 8 in R n of the support S of V satisfies the hypothesis of the general criterion for C 2 rectifiability in 3.10. These hypothesis have been already checked for V in several different papers, so we just need to collect them here.
(1) H m (S ∩ K) < ∞ for every compact set K ⊆ Ω. This follows combining the upper-semicontinuity of the density function Θ m ( V , ·), see [All72,8.6], with the fact that V = H m Θ m ( V , ·). In fact, we obtain the stronger conclusion H m S ≤ θ −1 V .
(3) For V a.e. a ∈ Ω there exists an m dimensional plane T such that This follows from [Sim83,17.11].