The spherical image of singular varieties of bounded mean curvature

In this paper we study singular varieties of arbitrary codimension and bounded mean curvature in a viscosity sense. These varieties, introduced by White, naturally arise in the study of the area-blow up of sequences of smooth submanifolds with uniformly bounded mean curvature and provide a non-variational analogous for the class of varifolds of bounded mean curvature. We introduce notions of mean curvature and second fundamental form and we establish a Coarea-type formula which allows to compute the area of the spherical image of the singular variety in terms of its curvature. Such a formula is new even for integral varifolds of bounded (or zero) mean curvature. Then we extend the celebrated Almgren sphere theorem to these class of singular varieties. Our proof, which is based on Almgren's ideas, combines the aforementioned Coarea-type formula, the barrier principle of White and the Alexandrov's technique of moving planes to derive crucial quantitative informations without assuming a-priori regularity for the variety (no analogous of Allard's regularity theorem is available in our non-variational setting).


Introduction
The general setting. In this paper we study some geometric-analytic aspects of certain singular varieties introduced by Brian White in [Whi16] to analyse the area-blow up of sequences of smooth submanifolds of arbitrary codimension and mean curvature uniformly bounded by a non negative number h.
1.1 Definition. (see [Whi16,2.1] 1 ) Suppose 1 ≤ m < n are integers, Ω is an open subset of R n , Γ is relatively closed in Ω and h ≥ 0. We say that Γ is an 1 Problem. Is it possible to introduce meaningful notions of mean curvature and second fundamental form for (m, h) sets? This question is of interest also in case of those varifolds that are (m, h) sets, for which it is a difficult problem to understand how to introduce a notion of second fundamental form.
2 Problem. A basic fact in differential geometry asserts that the area of the spherical image of a smooth variety can be computed in terms of the curvature (see (1)). Is it possible to make a similar assertion for (m, h) sets? This question is open even for those varifolds that are (m, h) sets.
3 Problem. Almgren in [Alm86] proved that if V is an m dimensional rectifiable varifold with a uniform lower bound on the density, δV ≤ h 0 V for some 0 ≤ h 0 < ∞ and the normal component of h(V, ·) is bounded exactly by m, then H m (spt V ) ≥ H m (S m ) and, in case H m (spt V ) = H m (S m ), then V is the standard m dimensional unit sphere. This statement can be naturally formulated in the larger class of (m, h) sets: if Γ is an (m, m) set in R n , is it true that H m (Γ) ≥ H m (S m ) with equality if and only if Γ is the round sphere?
Second fundamental form and Coarea formula for the spherical image. If M is an m dimensional C 2 submanifold of R n without boundary, N (M ) is the unit normal bundle and Q M is the second fundamental form then the area 2 In this paper we adopt the terminology in [Alm86, Appendix C] for varifolds; in particular note that the variation function h(V, ·) differs from the one adopted in Allard's paper [All72,4.2] by a sign.
3 Problems 1, 2 and 3 can be analogously formulated for rectifiable varifolds of locally bounded first variation and possibly unbounded mean curvature. As we point out later in the introduction only for Problem 1, and only for integral varifolds, a partial answer follows from deep results of Schätzle [Sch04], [Sch09] and Menne [Men13]. (1) Smoothness of M readily reduces the proof of this result to an application of Federer's Coarea formula. On the other hand it is well known that integral varifolds V with δV ≤ h V (in particular (m, h) sets) can fail to be C 1 submanifolds almost everywhere and the singular set can be extremely complicated, as the example in [Bra78,6.1] shows. Therefore it is a non-trivial question to understand how to introduce a second fundamental form for these varifolds. Up to now, the best result in this direction 4 follows from the C 2 rectifiability results proved by Schätzle [Sch04,5.1]-[Sch09, 3.1] and Menne [Men16a,4.8]. In fact, these results imply the existence of an approximate second fundamental form, see 2.1, whose associated approximate mean curvature coincides with the variational mean curvature of the varifold. However, a Coarea-type formula as in (1) does not directly follow from C 2 rectifiability and a deeper analysis seems to be necessary. In this paper we provide an approach to solve Problems 1 and 2 for all (m, h) sets (which are unions of countably many sets of finite H m measure) which is essentially independent of C 2 rectifiability results. Our starting point is the notion of curvature intruduced in the series of papers [Sta79], [HLW04] and [San17b], where for an arbitrary closed set Γ a notion of unit normal bundle N (Γ), which is a countably (H n−1 , n − 1) rectifiable subset of R n × S n−1 , and second fundamental form which is a symmetric bilinear form associated to H n−1 a.e. (z, η) ∈ N (Γ), are defined. A summary of the relevant definitions for the present work is provided in 2.2-2.5. This concept has already found several applications in stochastic geometry; however this is the first time that it is applied in variational problems. However, even if it is possible for arbitrary closed sets to prove Coarea-type formulas integrating suitable curvature functions on the normal bundle N (Γ), see [San17b,4.11(3), 5.6], a Coarea formula as in (1) cannot be proved for arbitrary closed sets. In fact, what turns out to be true is that such a formula holds whenever Γ is a closed subset of R n whose normal bundle N (Γ) satisfies the following property: It is worth to mention that a second fundamental form is introduced in a variational way by Hutchinson in [Hut86], defining a special class of integral varifolds known as curvature varifolds (see also [Man96] for further extensions and [Men16b, 15.6] for a characterization in terms of weakly differentiable functions). However not every integral varifolds such that δV ≤ κ V (or even δV = 0) is a curvature varifolds.
where Γ (m) = Γ ∩ {a : 0 < H n−m−1 (N (Γ, a)) < ∞}; see 3.3. We call such a property m dimensional Lusin (N) condition, in analogy with the terminology used in classical works on the Area formula for mappings. It follows from a recent result of Schneider [Sch15] that a typical (in the sense of Baire category) compact convex hypersurface in R n (which is known to be C 1 ) does not possess the n − 1 dimensional Lusin (N) condition. It requires some technical effort to prove that all (m, h) sets which are countable unions of sets of finite H m measure satisfy this property. The main result of section 3 can be summarized in the following theorem.
1.2 Theorem (Coarea formula for the spherical image map of (m, h) sets).
The main tool to prove the Lusin (N) condition is the Barrier principle [Whi16, 7.1], which allows to prove that for L 1 a.e. r ∈ (0, r 0 ) the sum of the first m approximate principal curvatures of the level set S(Γ, r) of the distance function δ Γ from Γ, is bounded from above by h at H n−1 almost every point of suitable subset of S(A, r). In turn, this estimate implies that the approximate jacobian of the nearest point projection ξ Γ onto Γ must be positive on this subset. Since there exists a bi-lipschitzian correspondence between the level sets S(Γ, r) and the unit normal bundle N (Γ), see [San17b,3.18,4.3], the positivity of the approximate jacobian of ξ Γ readily implies the Lusin (N) condition by 3.5. Moreover, the relation between the approximate principal curvatures of S(Γ, r) and the principal curvatures of Γ (i.e. the eigenvalues of Q Γ ) given in (2) allows to establish that trace Q Γ (z, η) ≤ h at H n−1 almost every (z, η) ∈ N (Γ). We remark that for arbitrary closed sets both the aforementioned bilipschitzian correspondence and the relation between principal curvatures in (2) are new facts, established in [San17b].
Theorem 1.2 clearly shows that Q Γ and trace Q Γ naturally describe key geometric properties of general (m, h) sets, thus providing natural notions of second fundamental form and mean curvature for this class of varieties. If we consider those varifolds V that are also (m, h) sets, it is important to understand if the first variation function h(V, ·) of V agrees with trace Q spt V in order to regard Q spt V as a suitable notion of second fundamental form for V . In this paper, on the basis of the locality theorem of Schätzle [Sch09], we provide an answer for integral varifolds.
1.3 Theorem (Second fundamental for integral varifolds of bounded mean curvature). Suppose 1 ≤ m ≤ n − 1, V ∈ V m (R n ) is an integral varifold such that the total variation δV of the first variation δV is absolutely continuous with respect to the variation measure V and the first variation function h(V, ·) ∈ L ∞ ( V , R n ). Then We finally remark that a similar Coarea-type formula has been announced in [Men12b] for m dimensional integral varifolds V in R m+1 with δV absolutely continuous with respect to V , h(V, ·) ∈ L m ( V , R m+1 ) and m ≥ 2.
Almgren sphere theorem for (m, m) sets. Our extension of Almgren result reads as follows. The proof of this theorem, as well as the content of section 3, owes much to Almgren's ideas developed in [Alm86] and to some new insights contained in a proof of Almgren's result due to Ulrich Menne (see [Men12a]). On the other hand, critical difficulties in extending Almgren's result to the non-variational setting of (m, m) sets are both the lack of those variational structures that are naturally associated with varifolds (e.g. first variation and first variation function), and the lack of a powerful regularity theory (Allard's theorem). Therefore the proof of 1.4 presents aspects that are new even for the varifold's case treated by Almgren. We describe now the main steps of this proof. For the inequality case we use compactness of Γ to see that for each η ∈ S n−1 there exists an (n − 1) dimensional plane π perpendicular to η such that Γ lies on one side of π and touches π at least in one point. This is the Alexandrov technique of moving planes and can be precisely stated saying that the projection onto S n−1 of the contact set equals S n−1 . Then the estimate trace Q Γ ≤ m in 1.2 and the more elementary fact that Q Γ has a sign when restricted on C allows to obtain the estimate H m (Γ) ≥ H m (S m ). This crucial quantitative estimate is obtained working directly on the projection of the contact set of C on Γ, combining the Coarea formula 1.2 and the Barrier principle of White [Whi16, 7.1], and with no structural or smoothness assumptions at the touching points (see the introduction of [DM19] for further comments about the role of regularity in arguments based on Alexandrov moving plane technique and Maximum principles). This argument originates from the approach to Almgren's theorem developed in [Men12a]. We discuss now the proof of the equality case. In the varifold case Almgren, after have proved that V must be a unit density varifold with mean curvature of constant length equal to m and spt V must be contained in the boundary of its convex hull, he reduces the proof to the codimension 1 case and makes use of the weak* convergence of the blow up of a varifold to a stationary cone, in combination with a strong barrier principle based on first variation computations (see [Alm86, Appendix C (12)]), to show that at each point of spt V there exists a unique unit-density varifold-tangent, which corresponds to the unique supporting hyperplane of the convex hull of spt V . Then he concludes the proof employing Allard regularity theorem to prove that spt V coincides with the boundary of a smooth convex body, which must be the sphere because of the constant length of the mean curvature. In following this strategy, we need first to replace weak* convergence with the Kuratowski converge of the blow up of an (m, h) to an (m, 0) set (see [Whi16,3.2]) and using the strong barrier principle in [Whi16,7.3] to conclude that at each point of Γ the tangent cone of Γ is the unique supporting hyperplane of the convex hull of Γ. This implies that Γ actually coincides with the boundary of its convex hull and it is a C 1 hypersurface. At this point we cannot conclude using Allard's regularity theorem, since such a theorem is not available in our context. Therefore to conclude the proof we use an idea that we have learned from [Men12a]. We apply the barrier principle [Whi16, 7.1] in combination with a result of Federer [Fed69, 3.1.23] to gain some further regularity for Γ, namely it is a C 1,1 hypersurface. At this point the conclusion can be easily deduced from a straightforward computation.
Acknowledgements. Most of the work in section 3 was carried out when the author was a Phd student in the Geometric Measure Theory group led by Prof. Ulrich Menne at Max Planck Institute for Gravitational Physics.
The author thanks Prof. Ulrich Menne for many conversations on the subject of the present paper and to have kindly made available his unpublished lecture notes [Men12a], where some of the key ideas of the present work originate from.

Preliminaries
The notation and the terminology used without comments agree with [Fed69,. For varifolds our terminology is based on [Alm86, Appendix C]. The closure and the boundary in R n of a set A are denoted by A and ∂A and, if λ > 0 and x ∈ R n then λ(A − x) = {λ(y − x) : y ∈ A}. The gradient of a function f is denoted by ∇f . 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 The maps p, q : If A ⊆ R n and m ≥ 1 is an integer, we say that A is countably (H m , m) rectifiable of class 2 if A can be H m almost covered by the union of countably many m dimensional submanifolds of class 2 of R n ; we omit the prefix "countably" when H m (A) < ∞. If X and Y are metric spaces and f : X → Y is a function such that f and f −1 are Lipschitzian functions, then we say that f is a bi-Lipschitzian homeomorphism.

Approximate second fundamental form
In this paper we employ weak notions of second fundamental form and mean curvature that can be naturally associated to each set A ⊆ R n at those points a ∈ R n where A is approximately differentiable of order 2 in the sense of [San17]. In order to keep this preliminary section relatively short we directly refer to [San17b, 2.8-2.12], where relevant definitions and remarks about the theory developed in [San17] are summarized. On the basis of [San17b, 2.8-2.9] we can introduce the following definitions. If A is an m dimensional submanifold of class 2 then these notions agree with the classical notions from differential geometry, see [San17b, 2.10].

Curvature for arbitrary closed sets
Besides the concept of approximate second fundamental form, in this paper we make use of a more general notion of second fundamental form introduced in [San17b] that can be associated to arbitrary closed sets. The theory of curvature for arbitrary closed sets has been developed in [Sta79], [HLW04], [San17b] and here we summarize those concepts that are relevant for our purpose in the present paper.
Suppose A is a closed subset of R n .
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 at x with symmetric approximate differential and ap lim inf y→x ρ(A, y) ≥ ρ(A, x) > 1 5 . The set of regular points of ξ A is denoted by R(A). If x ∈ R(A) then we say that ψ A (x) is a regular point of N (A). We denote the set of all regular points of N (A) by R(N (A)). For every (a, u) ∈ R(N (A)) we define where x is a regular point of ξ A such that ψ A (x) = (a, u), τ ∈ T A (a, u), τ 1 ∈ T A (a, u) and v 1 ∈ R n such that ap D ξ A (x)(v 1 ) = τ 1 . We say that Q A (a, u) is the second fundamental form of A at a in the direction u.
If (a, u) ∈ R(N (A)) we define the principal curvatures of A at (a, u), 1 (a, u), . . . , κ A,m (a, u) are the eigenvalues of Q A (a, u) and m = dim T A (a, u). Moreover are the eigenvalues of ap D ν A (x)|{v : v • ν A (x) = 0} for x ∈ R(A).
It follows from [San17b,4.10] that if r > 0 and x ∈ S(A, r) ∩ R(A) then

Level sets of distance function
We conclude this preliminary section providing a structural result for the level sets of the distance function from an arbitrary closed set, which is sufficient for the purpose of the present work. Other structural results are available, in particular we refer to [RZ12] and references therein.

Theorem
Then there exists an open neighborhood V of x and a Lipschitzian function Proof. Since δ A is differentiable at L n a.e. x ∈ R n , it follows from [Fed59, 4.8(3)] and Coarea formula that ν A (x) = ∇δ A (x) for H n−1 a.e. x ∈ S(A, r) and for L 1 a.e. r > 0. Henceforth, it follows from [Men16a, 3.14] and 2.7 that for L 1 a.e. r > 0 the level set S(A, r) is pointwise differentiable of order 1 at H n−1 a.e. for H n−1 a.e. x ∈ S(A, r) and for L 1 a.e. r > 0. It follows that S(A, r) is pointwise differentiable of order (1, 1) at H n−1 a.e. x ∈ S(A, r) and for L 1 a.e. r > 0 (see [Men16a,3.3]) and we employ [Men16a, 5.7(3)] to conclude that S(A, r) is pointwise differentiable of order 2 at H n−1 a.e. x ∈ S(A, r) and for L 1 a.e. r > 0. Now the conclusion can be easily deduced with the help of 2.7, [Men16a, 3.14, footnote of 3.12] and [San17b, 3.13].

Area formula for the spherical image
We introduce now the key concept of Lusin (N) condition for the generalized unit normal bundle. The following coarea-type formula is a crucial consequence of the Lusin (N) condition.
3.3 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.
Finally the following immediate consequence of Federer's Coarea formula is needed.
3.5 Lemma. Suppose 0 ≤ µ ≤ m are integers, W is a (H m , m) rectifiable and H m measurable subset of R n , S ⊆ R ν is a countable union of sets with finite H µ measure and f : W → R ν is a Lipschitzian map such that Then In the proof of the next result it is convenient to introduce the following Borel sets (see [San17b,3.9]).
3.6 Definition. If A ⊆ R n is closed and λ ≥ 1 we define (see 2.3) of Ω that is a countable union of sets with finite H m measure and A = Γ.
3.9 Remark. Note that if V ∈ V m (Ω) is an integral varifold such that δV is absolutely continuous with respect to V and h(V, ·) ∈ L ∞ ( V , R n ) then it follows from [Men13, Theorem 1] and [Alm86, Appendix C(8)] that spt V is (H m , m) rectifiable of class 2. In view of 3.7(1), if Γ = spt V in 3.8 then we can replace A (m) with Γ in the conclusion of the theorem; i.e.
for H n−1 a.e. (z, η) ∈ N (A)|Ω. 3.10 Remark. As pointed out in 2.6 the second fundamental form Q Γ of an arbitrary closed set Γ when restricted over Γ (m) may not be fully described by ap b Γ (m) . In a certain sense 3.8 draws an interesting analogy with the theory of functions of bounded variation. In fact, it is well known that the total differential of a BV function is not equal to the approximate gradient, unless the function belongs to the Sobolev space. Following this analogy, (m, h) sets correspond to Sobolev functions.

Sphere theorem
The following lemma will be useful in the proof of the rigidity theorem.
4.1 Lemma. Let 1 ≤ m ≤ n be integers and let B be an m dimensional submanifold of class 1 in R n . If 0 < λ < 1 and ϕ : B → R n is a Lipschitzian map such that It follows that if U is a convex subset of f (U(b, tr) ∩ B) such that f (b) ∈ U and U is relatively open in A then Therefore one uses (8) and Lip f ≤ t −1 to conclude Proof. We assume H m (Γ) < ∞. We define and we notice that C is a closed subset of Γ × R n , C(a) is a closed convex cone 7 containing 0 for every a ∈ Γ and, since Γ is compact, for every η ∈ S n−1 there exists a ∈ Γ such that inf w∈Γ (w • η) = (a • η); in other words, q(C ∩ (Γ × S n−1 )) = S n−1 .
If T ∈ G(n, m) and v ∈ T ⊥ we define We readily infer that there exists 0 < γ(n, m) < ∞ such that  (15)), Let A be the convex hull of Γ and let B be the relative boundary of A. Note that A − a ⊆ {v : v • η ≥ 0} for every a ∈ Γ and η ∈ C(a). Then it follows from (17) and (18)  for H m a.e. a ∈ Γ, whence we deduce that dim A ≤ m + 1. If dim A = m then we could apply [San17b, 2.10] with M replaced by the relative interior of A to infer that ap h Γ (x) = 0 for H m a..e x ∈ Γ. Since this contradicts (19) we have proved that dim A = m + 1. Then we notice that H m (Γ ∼ B) = 0 and, since Γ = spt(H m Γ), it follows that Γ ⊆ B.
At this point it is not restrictive to assume m = n − 1 in the sequel. Now we prove that if x ∈ Γ then Tan(Γ, x) is the unique supporting hyperplane of A at x. We fix x ∈ Γ. By [Sch14, 1.3.2] there exists a closed halfspace H of R m+1 such that 0 ∈ ∂H and A − x ⊆ H. By [AF09, Theorem 1.1.7] we choose a sequence λ i converging to +∞ and a closed set Z in R m+1 such that (see [ Since Lip f < ∞ it follows that Tan(∂A, y)∩T ⊥ = {0} for y ∈ W ∩∂A and, since we have proved in the previous paragraph that Tan(Γ, y) is an m dimensional plane for every y ∈ Γ, we employ [Fed69, first paragraph p. 234] to conclude T = T ♮ (Tan(Γ, y)) = Tan(T ♮ (W ∩ Γ), T ♮ (y)) for every y ∈ W ∩ Γ.
Noting that T ♮ (W ∩ Γ) is relatively closed in U , we infer 9 that T ♮ (W ∩ Γ) = U and W ∩ ∂A = W ∩ Γ. Since x is arbitrarily chosen in Γ, it follows that ∂A = Γ. We combine the assertions of the previous two paragraphs with [Sch14,2.2.4] to conclude that ∂A is an m dimensional submanifold of class 1 in R m+1 . Moreover, it is well known that dmn ξ A = R m+1 , Lip ξ A ≤ 1 (see [Sch14, 1.2]) and {x : δ A (x) < r} is an open convex set whose boundary S(A, r) is an m dimensional submanifold of class C 1,1 for r > 0 (see [Fed59,4.8]). Let 0 < r < m −1 and ξ = ξ A |S(A, r). For H m a.e. x ∈ S(A, r) we apply the barrier principle 3.4, with T , η and f replaced by {v : v • ν A (x) = 0}, ν A (x) and a concave function whose graph corresponds to S(A, r) in a neighborhood of x, to infer (see 2.4) that and we combine these inequalities to conclude that χ A,i (x) ≤ m for i = 1, . . . , m. Therefore D(ξ − 1 S(A,r) )(x) ≤ mr < 1 for H m a.e. x ∈ S(A, r) and, noting that ξ is univalent by [Fed59,4.8(12)], we apply 4.1 to conclude that the function ξ −1 : ∂A → S(A, r) is a locally Lipschtzian map and the unit normal vector field on ∂A, η = ν A • ξ −1 , is locally Lipschitzian. Combining [San17,3.25] with (19) and (20), we infer for H m a.e. x ∈ Γ and for u, v ∈ Tan(Γ, x) that whence we conclude that D(η − 1 Γ )(x) = 0 for H m a.e. x ∈ Γ. Therefore there exists a ∈ R m+1 such that η(z) = z − a for every z ∈ Γ and, since |η(z)| = 1 for z ∈ Γ, we conclude that Γ = ∂B(a, 1).