$C^{1,1}$-rectifiability and Heintze-Karcher inequality on $\mathbf{S}^{n+1}$

In this paper, by isometrically embedding $(\mathbf{S}^{n+1},g_{\mathbf{S}^{n+1}})$ into $\mathbf{R}^{n+2}$, and using nonlinear analysis on the codimension-2 graphs, we will show that the level-sets of the distance function from the boundary of any open set in sphere, are $C^{1,1}$-rectifiable. As a by-product, we establish a Heintze-Karcher inequality on sphere.


Introduction
The isoperimetric theorem, a fundamental but important topic in the calculus of variations, has attached well attention of many mathematicians. From the perspective of the modern calculus of variations, sets of finite perimeter are believed to be the natural competition class in which the isoperimetric theorem shall be formulated.
Starting from De Giorgi [2,3], who managed to show by using Steiner symmetrization and the compactness theorem of sets of finite perimeter that Euclidean balls are the only isoperimetric sets (global minimizers) among sets of finite perimeter, mathematicians have been working in various context of minimizers to study the isoperimetric problem for decades. Such problem is already found to be very subtle in the context of local minimizers, due to the lack of regularity in the higher dimensional situation (and hence the classical moving plane method fails to be applicable), see [21] for an example of local volume-constrained perimeter minimizer admitting singularities. Despite these obstacles, very recently, Delgadino-Maggi [6] solved the very important open problem: the characterization of critical points of the Euclidean isoperimetric problem among sets of finite perimeter. In the weakest assumption (critical points of Euclidean isoperimetric problem), they obtained the following (see also [5] for the anisotropic version, which is solved by using a completely different method with [6], and also [12] for recent progress on a class of warped-products).
Their advanced techniques to approach this problem are two-fold: 1. By using subtle nonlinear analysis and geometric measure theory, they established the following C 1,1 -rectifiability result of the level sets of the distance function from the boundary of any bounded sets of finite perimeter in R n+1 .
The C 1,1 -rectifiability of Γ t s allows them to define the principal curvatures a.e. on Γ + s , which are bounded from above and by below due to the definition of Γ t s . As a by-product, they followed the proof of [1] and derived a Heintze-Karcher inequality for sets of finite perimeter ( [6], Thm. 8). Note that although Proposition B is stated in the context of sets of finite perimeter, the statement remains valid when Ω is only a bounded open set, we refer to the details of the proof in [6], Steps 1-3.
2. They exploited the Schättzle's strong maximum principle for the codimension-1 integer rectifiable varifolds [19], Theorem 6.2 and showed that the flow method used by Montiel-Ros in [14], Theorem 3 to prove the Heintze-Karcher inequality for C 2 -closed hypersurfaces can be modified to apply in the context of sets of finite perimeter, thus proved their main result Theorem A.
It is worth mentioning that aforementioned works on the isoperimetric problems are contextualized in the Euclidean space. In view of [14,16,17], a natural question arises: is there any characterization of geodesic balls as the only critical points in the isoperimetric problem that is brought up in space forms?
Motivated by this natural question and the celebrated work [6], in this note, we follow the subtle nonlinear analysis carried out by Delgadino-Maggi and prove a C 1,1 -rectifiability result in (S n+1 , g S n+1 ). On the other hand, we manage to prove a Heintze-Karcher-type inequality for any open set that is contained in a hemisphere, enlightened by Brendle's monotonicity approach [1]. To introduce our main results, let us first fix some notations.
Let (S n+1 , g S n+1 ) be the space form with sectional curvature which is identically 1, for simplicity, we abbreviate it by S n+1 in the rest of this paper. Let dist g denote the distance function of S n+1 .
For any nonempty open set Ω ⊂ S n+1 , we define u(y) = dist g (y, ∂Ω) to be the distance function with respect to ∂Ω on S n+1 . For s > 0, we define the super level-set and the level-set of u in Ω by Ω s = {y ∈ Ω : u(y) > s}, ∂Ω s = {y ∈ Ω : u(y) = s}. (1.2) As a parallel version of the Euclidean one in [6], we introduce the following definitions.

Main results
The main purpose of this paper is to establish the following C 1,1 -rectifiability result, which extends Proposition B to S n+1 .
For any nonempty open set Ω ⊂ S n+1 and for every 0 < s < t < π, there exists a countable collection {U j } j≥1 of compact subsets of Γ t s such that H n (Γ t s \ ∞ j=1 U j ) = 0, with each U j contained in a C 1,1hypersurface. Moreover, denoting by N the gradient of the distance function with respect to u(·) = dist g (·, ∂Ω), then N | Uj is Lipschitz for every j ≥ 1.
With the C 1,1 -rectifiability in force, the principal curvatures (κ t s ) i of Γ t s , the viscosity boundary ∂ ν Ω of Ω, and the viscosity mean curvature H ν Ω of Ω are thus naturally defined in Proposition 5.1, Definitions 5.3 and 5.5, see also [6], Lemma 7 for the Euclidean version.
Consequently, following Brendle's monotonicity approach [1], Section 3, we prove the following Heintze-Karcher-type inequality on S n+1 . Theorem 1.3 (A Heintze-Karcher-type inequality on sphere). If Ω ⊂ S n+1 is an open set lying completely in a hemisphere, which is mean convex in the viscosity sense as in Definition 5.5, then for every 0 < s < π 2 , Moreover, as s → 0 + , the limit of the right-hand side of (1.3) always exists in (0, ∞]. Our strategy for proving the main rectifiability theorem follows largely from [6] and is as follows: by isometrically embedding S n+1 into R n+2 , our goal becomes: to show that the aforementioned Γ t s is a n-rectifiable set in R n+2 . Using the Hinge version of Topogonov theorem, we obtain an estimation of |N (y) · (y − y)|, where y , y ∈ Γ t s and N (y) is the derivative of the distance function at y, which will be proved to be well-defined everywhere on Γ t s in Proposition 3.5, both y, y , and N (y) are considered as vectors in R n+2 , '· denotes the standard Euclidean inner product. By virtue of this basic estimation, we can use the C 1 -Whitney extension theorem and the C 1 -implicit function theorem to show that Γ t s is C 1 -rectifiable. To prove the C 1,1 -rectifiability, in the Euclidean case, Delgadino-Maggi's proof of C 1,1 -rectifiability is built on the fact that Γ t s can be written as a codimension-1, C 1 -graph in R n+1 . In our situation, the main obstacle to prove the C 1,1 -rectifiability is that, the analysis of codimension-2 graph in R n+2 seems more subtle. Regarding this, our approach can be viewed as a codimension-2 counterpart of the one presented in [6], Theorem 1, Step 1].
In view of the classical rigidity result of C 2 , CMC hypersurfaces in S n+1 [14], part C, and the Alexandrov theorem for critical points of isoperimetric problem among sets of finite perimeter Theorem A (see in particular [6], Thm. 1, step 4), it is interesting to see whether one can establish an Alexandrov theorem on S n+1 among sets of finite perimeter, we hope that our rectifiability result can serve as a fundamental step for solving this interesting open problem. On the other hand, we believe our codimension-2 analysis can be used in a wider range of problems that deal with graphs on S n+1 .
As pointed out to us by a referee, our rectifiability theorem can be deduced quite easily from the corollary of [13], Theorem 4.12, which asserts that if S is a closed subset of R n+1 and S * is the set of x ∈ S such that there exists an open ball B x with B x ∩ S = ∅ and x ∈ ∂B x , then S * can be H n -almost covered by the union of countable collection of C 2 -hypersurfaces. This theorem readily implies that (using local charts) all the level-sets of the distance function from an arbitrary closed set in a complete (n + 1)-dimensional Riemannian manifold can be H n -almost covered by the union of embedded C 2 -hypersurfaces. This fact, together with the applications of a few standard tools in Geometric Measure Theory, yields Theorem 1.2 even when the standard sphere (S n+1 , g S n+1 ) is replaced by an arbitrary complete Riemannian manifold and for all 0 < s < t < ∞. In this regard, our proof of Theorem 1.2 serves as an alternative approach by using the nonlinear analysis arguments.

Organization of the paper
In Section 2 we collect some background material from geometric measure theory. In Section 3 we study the fine properties of Γ t s . In Section 4 we prove the main rectifiability result Theorem 1.2. In Section 5, we define the viscosity mean curvature and boundary of any open set in S n+1 , and we establish a Heintze-Karcher-type inequality (1.3).

Rectifiable sets
The main purpose of this paper is to establish the rectifiability result, here we list some fundamental concepts and tools that are needed in the sequel, and we refer to [4,11,20] for more details. We must point out that, by virtue of the embedding S n+1 → R n+2 , in most of this paper, we will be working in R n+2 , and we use H k to denote the k-dimensional Hausdorff measure in R n+2 .

Rectifiable sets
Definition 2.1 (n-rectifiable set, [6], Sect. 2.1). A Borel set N ⊂ R n+2 is a locally H n -rectifiable set if N can be covered, up to a H n -negligible set, by countably many Lipschitz images of R n to R n+2 , and if H n N is locally finite on R n+2 . N is called H n -rectifiable if in addition, H n (N ) < ∞.
is the Jacobian of f with respect to A at x(see for example [20], (12.3) and [11], (11.1)), which exists for H k -a.e. x ∈ A.
Proposition 2.3 (coarea formula for k-rectifiable set, [20], (12.6)). For k ≥ m, if A ⊂ R n is a H k -rectifiable set and f : R n → R m is a Lipschitz map, then The following proposition is also needed in our codimension-2 argument, which amounts to be a simple modification of [6], Section 2.1(iv). The proof follows exactly from [6] and hence is omitted here. In particular, if ψ : R n → R n+2 is a Lipschitz map and E ⊂ R n is a Borel set, then Note that ∇ M F (x) denotes the tangential differential of F with respect to M at x, which exists for H n -a.e. x ∈ M by virtue of the Rademacher-type theorem [11], Theorem 11.4.
To close this section, we list some well-known facts about the space form S n+1 , which will be needed in our proof. Note that part of them are already mentioned in the introduction.
2.3. Geometry of (S n+1 , g S n+1 ) In this paper, we consider the unit sphere S n+1 ⊂ R n+2 , endowed with the Riemannian metric g S n+1 that is induced from the isometrically embedding S n+1 → (R n+2 , g euc ), and we have the following basic facts: 1. S n+1 is a smooth, complete, compact Riemannian manifold without boundary, having constant sectional curvature which is identically 1. 2. The injective radius of S n+1 is π, i.e., inj(S n+1 ) = π.
3. The only geodesics on S n+1 are great circles. 4. For x = z ∈ S n+1 with dist g (x, z) < π, there exists a unique minimizing unit-speed geodesic joining x and z. 5. The geodesic ball of radius r centered at some x ∈ S n+1 , denoted by B r (x), is umbilical in S n+1 with constant principal curvatures cot r.

Fine properties of Γ t s
In this section, we explore the fine properties of Γ t s . For any nonempty open set Ω ⊂ S n+1 , we define Γ t s and Γ + s as in Definition 1.1. Recall that for any y ∈ Ω, we define u(y) = dist g (y, ∂Ω) to be the distance function from ∂Ω. Following [8], we define the unique point projection mapping on S n+1 , see [8], Definition 4.1 for the Euclidean version. associates with y ∈ Unp(∂Ω) the unique x ∈ ∂Ω such that u(y) = dist g (x, y).

Lemma 3.2.
For any nonempty open set Ω ⊂ S n+1 and for every 0 < s < t < π, let Γ t s be as in Definition 1.1. Then, for any y ∈ Γ t s , it has a unique point projection onto ∂Ω, which reads as ξ(y) = x; in other words, Γ t s ⊂ Unp(∂Ω). Similarly, it has a unique point projection onto ∂Ω t .
Proof. By definition, for any y ∈ Γ t s , there exists x = γ y (s) ∈ ∂Ω and z = γ y (s − t) ∈ ∂Ω t . Since y ∈ ∂Ω s , if there exists x = x ∈ ∂Ω such that dist g (x , y) = s, then by the triangle inequality we have contradicts to the fact that z ∈ ∂Ω t . Therefore, we have showed that x is the unique point of ∂Ω nearest to y; that is, ξ(y) = x.
On the other hand, we note that by using the triangle inequality again, it is easy to see that z is the unique point in ∂Ω t such that dist g (y, z) = t − s. Now that we have showed that Γ t s ⊂ Unp(∂Ω), we can explore the unique point projection mapping ξ on Γ t s . Indeed, we have the following. 1. The function u(·) = dist g (·, ∂Ω) is a Lipschitz function on S n+1 with Lipschitz constant at most 1, i.e., for any x, y ∈ S n+1 , 2. For 0 < s < t < π, ξ is continuous on Γ t s . Proof. For any x, y ∈ S n+1 . Since ∂Ω ⊂ S n+1 is compact, we may let a ∈ ∂Ω such that u(x) = dist g (a, x). Without loss of generality, assume that u(y) ≥ u(x). By the triangle inequality, we find For (2), suppose on the contrary that there exists some > 0 and a sequence of points y 1 , y 2 , y 3 , . . . ∈ Γ t s , converges to y ∈ Γ t s , such that dist g (ξ(y i ), ξ(y j )) ≥ for i = 1, 2, . . . By definition, for each i, we have, for i large, there holds Using the triangle inequality and the fact that y i converges to y, we find This means, all the points {ξ(y i )} i are lying in ∂Ω ∩ B s+ (y), which is a bounded subset of the compact set ∂Ω, and hence by passing to a subsequence, we can assume that {ξ(y i )} i converges to some point x ∈ ∂Ω. But then, since u is continuous on S n+1 , we have which implies that x = ξ(y) since we have proved that y ∈ Γ t s ⊂ Unp(∂Ω) in Lemma 3.2. However, this contradicts to the assumption that dist g (x, ξ(y)) = lim i→∞ dist g (ξ(y), ξ(y i )) ≥ , and hence completes the proof. With the help of Lemmas 3.2 and 3.3, we collect the fine properties of Γ t s as follows (see [6], Thm. 1 for the Euclidean version).

Proposition 3.5.
For any nonempty open set Ω ⊂ S n+1 and for every 0 < s < t < π, the following statements hold: s , Γ t s is bounded by two geodesic balls in S n+1 , mutually tangent at y, i.e., 4. The distance function u is differentiable at every y ∈ Γ t s . Proof.
(1) The first part of the statement follows easily from the definition of Γ t s , while by virtue of the inclusion, it is apparent that Γ + s = lim t→s + Γ t s .
(2) It suffice to prove that Γ t s is a closed set in S n+1 , i.e., if a sequence of points in Γ t s , say {y i } ∞ i=1 , converges to y, then it must be that y ∈ Γ t s . By definition of Γ t s , for each y i , there exists corresponding points x i ∈ ∂Ω, z i ∈ ∂Ω t . By Lemma 3.3, ξ is continuous on Γ t s , and hence we have: Similarly, we deduce that dist g (y, ∂Ω t ) = dist g (y, z) = t. By using the triangle inequality, we find it is easy to see that the geodesic joining x and z must pass through y, and hence y ∈ Γ t s by definition.
Remark 3.6. The lemmas and propositions in this section can be extended easily to the case when (S n+1 , g S n+1 ) is replaced by any complete Riemannian manifold.
In this section, we finish the proof of Theorem 1.2. As mentioned in the introduction, our proof is based on the isometrically embedding S n+1 → R n+2 . We point out that, in the rest of the paper, we will be working in R n+2 . For every 0 < s < t < π, and for any nonempty open set Ω ⊂ S n+1 , we define Γ t s , Γ + s , ∂Ω s as in Definition 1.1, (1.2), respectively. For any y ∈ Γ t s , by virtue of Proposition 3.5(4), u is differentiable at y, and we denote by N (y) its gradient, which belongs to T y S n+1 . In all follows, thanks to the embedding, N (y) will be considered as a vector in R n+2 .
The following well-known fact motivates our estimation.
Lemma 4.1. For any y ∈ Γ t s , let x = γ y (s) ∈ ∂Ω and z = γ y (s − t) ∈ ∂Ω t be the corresponding points that y admits. Then there holds Proof. These are well-known facts and one can check by a direct computation. Notice for example that y − tan sN (y) = 1 cos s x. See Figure 1 for an illustration. Proof of Theorem 1.2. Throughout the proof, | · | will denote the Euclidean norm in R n+2 , ∇ will denote the gradient in Euclidean space and " · " will denote the Euclidean inner product in R n+2 . To make a distinction, we use ·, · to denote the Euclidean inner product in R n . For any two points x = y ∈ S n+1 such that dist g (x, y) < π, we use xy to denote the unique image of the minimizing geodesic on S n+1 joining x and y.
cylinder at the origin with axis along N 1 , N 2 ∈ T S n+1 , radius ρ and height 2ρ in R n+2 . By the fact that at any y ∈ Γ t s , {y} = ∂B t−s (z) ∩ ∂B s (x), ν S n+1 (y) = y and Γ t s is locally H n -rectifiable, we have: Γ t s admits an approximate tangent plane at H n -a.e. of its points and this plane is then exactly span N (y), ν S n+1 (y) ⊥ , which is a n-dimensional affine plane in R n+2 , i.e., for H n -a.e. y ∈ Γ t s .
We want to show that N (y) is Lipschitz on each U j , namely, for some constants C j , it holds that To have a chance to prove this, let us first point out that it suffice to consider the case when y 1 , y 2 are close enough. Precisely, for r j < ρ j /3, we may assume that y 1 ∈ y 2 + C(N (y 2 ), y 2 , ρ j ) (4.14) or otherwise, |y 1 − y 2 | ≥ c(n)r j and it is trivial to see that |N (y 1 ) − N (y 2 )| ≤ 2 ≤ C j |y 1 − y 2 |. Next, with (4.14), we may further assume, up to a rigid motion as before, that In this way, (4.11) reads as with ψ i j ∈ C 1 (V j ) satisfying (4.10). By (4.14) again, y 1 = (z 1 , ψ 1 j (z 1 ), ψ 2 j (z 1 )) for some z 1 ∈ V j with |z 1 | < r j . Since ψ i j is C 1 on V j , the Taylor theorem gives In view of this and invoking (4.10), (A.1) thus gives a normal vector field in the form where |∇ z ψ 1 j | ≤ 1 on each V j due to (4.10). In particular, set N (y) =Ñ (y)/|Ñ (y)| and we readily see that once provided Let us verify the validity of (4.20) by Delgadino-Maggi's approach (see [6], proof of (3-25) for a detailed codimension-1 argument). First, for any y, y 0 ∈ y 2 + C(N (y 2 ), y 2 , ρ j ) as in (4.16), we set , y = (z, ψ 1 j (z)), y 0 = (z 0 , ψ 2 j (z 0 )). (4.21) , y = (z, ψ 2 j (z)), y 0 = (z 0 , ψ 2 j (z 0 )). (4.22) Note that similar with (A.2), we may write a normal vector field as Observe that a 1 (y) = − N 2 (y), y 1 + |∇ z ψ 1 j | 2 1 + |∇ z ψ 2 j | 2 , and from (4.10) we know that by continuity, we may assume that a 2 (y), (−∇ z ψ 1 j , 1, 0), y 0 are closed to 1, and we have for the second term, since Γ t s is trapped between two mutually tangent geodesic balls at y, we may project Γ t s and the two geodesic balls over the plane T y2 S n+1 to find that the projected graph is punctually second order differentiable at z and which is controlled by o(|y − y 0 | 2 ). On the other hand, by continuity we may assume that |Ñ (y)| ≤ 2 and hence (4.24) together with (4.7) yields that We exploit (4.25) in the manner of Delgadino-Maggi, with y = y 1 and y 0 = (z 0 , h 0 1 , h 0 2 ), defined by where e 0 = − ∇zψ 2 j (z1) |∇zψ 2 j (z1)| is a unit vector, determined as in [6], . (4.25) then gives (4.27) To proceed, let us note that for all |z| < ρ j such that (z, ψ 1 j (z), ψ 2 j (z)) ∈ Γ t s , it holds that this is a direct consequence of the following fact: near y 2 = ( 0, 1, 0), at every y ∈ Γ t s , Γ t s is trapped between two mutually tangent geodesic balls. Due to this, we note that we can only use the estimate (4.28) for those points that lies in Γ t s . By definition of z 0 , we have |z 0 | ≤ 2|z 1 | < 2r j < ρ j , if y 0 lies exactly in Γ t s ∈ U j , by (4.28) and the definition of y 0 , we find and hence, recalling the definition of e 0 , (4.27) gives this shows (4.20) when y 0 ∈ Γ t s . On the other hand, if y 0 / ∈ Γ t s , we let 0 be the largest > 0 such that Our definition of 0 then assures the existence of z * ∈ V j with |z * − z 0 | = 0 so that On the other hand, the definition of µ j in (4.12) shows that ω n (3|z 1 |) n (1 − µ j (3|z 1 |)) ≤ H n (Γ t s ∩ C(N (y 2 ), y 2 , 3|z 1 |)) ≤ ω n (3|z 1 |) n (1 + µ j (3|z 1 |)). Moreover, recall that U j is the graph of ψ j = (ψ 1 j , ψ 2 j ) over V j , and the Jacobian of the where D i Ψ j (z) = (0, . . . , 1, 0 . . . , 0, ∂ i ψ 1 j (z), ∂ i ψ 2 j (z)) is the directional derivative of Ψ j along e i , with {e 1 , . . . , e n } is the standard Euclidean coordinate of R n . We can use the Laplace expansion for the n × n matrix (< D i Ψ j , D k Ψ j >) 1≤i,k≤n to see that J Ψj (z) = 1 + terms involving (∂ i ψ 1 j (z), ∂ k ψ 2 j (z)). (4.36) In particular, by virtue of (4.10), (4.12) and the continuity of ∇ z ψ 1 j , ∇ z ψ 2 j , we find: J Ψj (0) = 1, J Ψj (z) is close to 1 near z = 0, and hence non-vanishing on V j . Again, by definition of µ j in (4.12), we find where we have used (4.12) in the Laplace expansion of J Ψj (z) for the first inequality, the area formula for Ψ j in the second equality, and (4.34) for the last inequality.

A Heintze-Karcher-type inequality on sphere
With the C 1,1 -rectifiability in force, we are going to derive the definitions of the principal curvature, viscosity mean curvature and boundary in the spirit of Delgadino-Maggi, thus extends [6], Lemma 7 from Euclidean space to S n+1 . In this section, we continue to use the notations in Definition 1.1, (1.2) and in the proof of Theorem 1.2.
Proposition 5.1. For any nonempty open set Ω ⊂ S n+1 , there holds 1. N is tangentially differentiable along Γ t s at H n -a.e. y ∈ Γ t s , with denote the principal curvatures of N along Γ t s at y when they exist, which are indexed in the increasing order. 2. For a.e. 0 < s < π, H n (Γ + s ) = H n (∂Ω s ). 3. For every r < s < t, the map g r : Γ t s → Γ t s−r , given by g r (y) = cos ry − sin rN (y) for y ∈ Γ t s , is a bijection from Γ t s to Γ t s−r and is Lipschitz when restricted to U j , with for H n -a.e. y ∈ Γ t s . Proof. We begin by recalling that in Theorem 1.2, we have constructed a sequence of compact sets U j , on which N is Lipschitz (see (4.13)) such that H n (Γ t s \ ∞ j=1 U j ) = 0.
(1) By virtue of Lemma 2.4, to study the tangential gradient of N on Γ t s , it suffice to work on each U j , see (4.10) and (4.11) for the construction of U j , where U j is contained in the graph of the C 1,1 -map (ψ 1 j (·), ψ 2 j (·)). Now, for a fixed y ∈ U j , we consider a natural Lipschitz extension of N , from U j ∩ (y + C (N (y) , y, ρ j )) to y + C(N (y), y, ρ j ), denoted by N * and defined as where N (y + z) is just the normal of the graph (x , ψ 1 j (x ), ψ 2 j (x )) at y + z ∈ U j . Set Ψ j (z) := y + z + ψ 1 j (z)y + ψ 2 j (z)N (y) for |z| < ρ j , by (2.4) we have: for H n -a.e. y ∈ U j and for any τ ∈ T y U j , where e = (∇Ψ j ) −1 , then for any z ∈ span {N (y), y} ⊥ , by definition, this shows that (5.4) where S j (Ψ j ) denotes the classical shape operator in differential geometry, with respect to the graph of ψ j = (ψ 1 j , ψ 2 j ). Notice that Γ t s is trapped between two mutually tangent geodesic balls on S n+1 with radius s and t − s, and hence the principal curvatures of the graph of ψ j is bounded from below by − cot s and from above by cot (t − s) , i.e., By the chain rule for Lipschitz functions and the fact that ∇ 2 ψ j is a H n -a.e. classical differential, the above argument holds for H n -a.e. y ∈ U j , which completes the proof of (1).
(2) In view of (4.2), given r ∈ [−s, t − s], we consider the map f r : Γ t s → ∂Ω s+r , defined by f r (y) = cos ry + sin rN (y). By definition of Γ t s , it is clear that f r is surjective, thus we have using the area formula, we find a direct computation then gives that, for H n -a.e. y ∈ Γ t s , For 0 < r < t − s, we have, cot r > cot(t − s) ≥ (κ t s ) i (y) by virtue of (5.5), it follows that cos r − sin r(κ t s ) i > 0 for each i, and hence we can use the Cauchy-Schwarz inequality to find which proves (2).
(3) For r ∈ (0, s), consider the bijection g r : Γ t s → Γ t s−r , defined by g r (y) = cos ry − sin rN (y), which is Lipschitz on each U j . We claim that, if N is tangential differentiable at y along Γ t s , then N is also tangential differentiable at g r (y) along Γ t s−r . Indeed, by a simple geometric relation on sphere (as illustrated in Fig. 2), we have g r (y) + tan r 2 N (g r (y)) = y − tan r 2 N (y), which implies N (g r (y)) = − (cos r − 1) y − sin r − tan r 2 N (y) · 1 tan r 2 = sin ry + cos rN (y).
That is, if y is a point of tangential differentiability of N along Γ t s and τ ∈ T y Γ t s , then τ ∈ T gr(y) Γ t s and take τ = τ i (y) to be the eigenvectors of the shape operators S j in (5.4), we obtain from this we have .
Hence {τ i (y)} n i=1 is an orthonormal basis for T gr(y) Γ t s−r , and the eigenvalues of ∇ Γ t s−r N (g r (y)) are given by: which completes the proof of (3).
Remark 5.2. In view of the rectifiability result Theorem 1.2 and Proposition 5.1(2), we have proved the following: for any nonempty open set Ω ⊂ S n+1 , the level-set ∂Ω s of the distance function u is C 1,1 -rectifiable for a.e. 0 < s < π.
Now we are in the position to generalize the viscosity mean curvature defined in [6] from Euclidean space to S n+1 . Definition 5.3 (Principal curvature and second fundamental form on Γ + s ). For every 0 < s < π, the principal curvatures (κ s ) i of Γ + s are defined H n -a.e. on Γ + s by setting where (κ t s ) i is well-defined on H n -a.e. y ∈ Γ t s by virtue of Proposition 5.1(1). In particular, we can define the mean curvature and the length of the second fundamental form of ∂Ω s with respect to ν Ωs at H n -a.e. points of Γ + s as follows: Lemma 5.4. For every 0 < s = π/2 < π and for H n -a.e. y ∈ Γ + s where the principal curvatures are well-defined, let x = g s (y), then the limit exists by monotonicity.
Proof. Let y ∈ Γ + s be the point where the principal curvatures are well-defined, and let x = g s (y). For 0 < r 1 < r 2 < s, by (5.8) we have .
Notice also that (κ t s ) i (y) is a bounded number as in (5.5), and hence (κ t s−r ) i (x) is monotone decreasing on r, it follows that (5.9) exists.
Definition 5.5. For any nonempty open set Ω in S n+1 , the viscosity boundary of Ω is defined as and the corresponding viscosity mean curvature of Ω is defined at those x where the limit (5.9) exists for some s > 0, defined by We would like to mention that the viscosity principal curvatures are strictly related to the principal curvatures defined and investigated in [9,18].
Finally, we can prove a Heintze-Karcher-type inequality in the spirit of Brendle's monotonicity approach [1], see also [6], Theorem 8 for the Euclidean version.
Thanks to the C 1 -differentiability of ψ i , we can take directional derivative near z = 0 along e 1 , . . . , e n to obtain where z i denotes the i-th component of z ∈ R n . In particular, this, together with (A.3), shows that ∇ z ψ 1 ( 0) = 0, which completes the proof.