Proof of the Michael–Simon–Sobolev inequality using optimal transport

. We give an alternative proof of the Michael–Simon–Sobolev inequality using techniques from optimal transport. The inequality is sharp for submanifolds of codimension 2 .


Introduction
In this paper, we use techniques from optimal transport to prove the following result.  ρ(|z| 2 + |y| 2 ) dy.
Let Σ be a compact n-dimensional submanifold of R n+m , possibly with boundary ∂Σ. Then where H denotes the mean curvature vector of Σ.
The proof of Theorem 1 is based on an optimal mass transport problem between the submanifold Σ and the unit ball in R n+m , the latter equipped with a rotationally invariant measure. A notable feature is that this transport problem is between spaces of different dimensions.
In Theorem 1, we are free to choose the density ρ. For m ≥ 2, it is convenient to choose the density ρ so that nearly all of the mass of the measure ρ(|ξ| 2 ) dξ onB n+m is concentrated near the boundary. This recovers the main result of [2]. Corollary 2. Let n ≥ 2 and m ≥ 2 be integers. Let Σ be a compact ndimensional submanifold of R n+m , possibly with boundary ∂Σ. Then The first-named author was supported by the National Science Foundation under grant DMS-2103573 and by the Simons Foundation. The second-named author was supported by the START-Project Y963 of the Austrian Science Fund. 1 where H denotes the mean curvature vector of Σ.
Note that the constant in (3) is sharp for m = 2. Earlier proofs of the non-sharp version of the inequality were obtained by Allard [1], Michael and Simon [8], and Castillon [4]. In particular, the Michael-Simon-Sobolev inequality implies an isoperimetric inequality for minimal surfaces. We refer to [3] for a recent survey on geometric inequalities for minimal surfaces.
Finally, we refer to [5], [6], [7] for some of the earlier work on optimal transport and its applications to geometric inequalities.

Proof of Theorem 1
Let Σ be a compact n-dimensional submanifold of R n+m , possibly with boundary ∂Σ. We denote by g the Riemannian metric on Σ and by d(·, ·) the Riemannian distance. For each point x ∈ Σ, we denote by II(x) : x Σ the second fundamental form of Σ. As usual, the mean curvature vector H(x) ∈ T ⊥ x Σ is defined as the trace of the second fundamental form.
We first consider the special case when |Σ| = 1. Let µ denote the Riemannian measure on Σ. We define a Borel measure ν on the unit ballB n+m by for every Borel set G ⊂B n+m . With this understood, µ is a probability measure on Σ and ν is a probability measure onB n+m . Let J denote the set of all pairs (u, h) such that u is an integrable function on Σ, h is an integrable function onB n+m , and for all x ∈ Σ and all ξ ∈B n+m . By Theorem 5.10 (iii) in [11], we can find a pair (u, h) ∈ J which maximizes the functional In fact, the result in [11] shows that the maximizer (u, h) may be chosen in such a way that h is Lipschitz continuous and for all x ∈ Σ. Note that our notation differs from the one in [11]. In our setting, the space X is the unit ballB n+m equipped with the measure ν; the space Y is the submanifold Σ equipped with the Riemannian measure µ; the cost function is given by c(x, ξ) = − x, ξ for x ∈ Σ and ξ ∈B n+m ; the function ψ in [11] corresponds to the function −h; and the function φ in [11] corresponds to the function −u in this paper. The fact that ψ can be chosen to be a c-convex function implies that h is Lipschitz continuous (see [11], Definition 5.2). The fact that φ can be taken as the c-transform of ψ corresponds to the statement (6) above (see [11], Definition 5.2).
It follows from (6) that u is the restriction to Σ of a convex function on R n+m which is Lipschitz continuous with Lipschitz constant at most 1. In particular, u is Lipschitz continuous with Lipschitz constant at most 1. Moreover, u is semiconvex with a quadratic modulus of semiconvexity (see [11], Definition 10.10 and Example 10.11).
Proof. For every positive integer j, we define a compact set G j ⊂B n+m by for all x ∈ Σ and all ξ ∈B n+m . Therefore, (u j , h j ) ∈ J for each j. Since the pair (u, h) maximizes the functional (5), we obtain Finally, we pass to the limit as j → ∞. Note that G j+1 ⊂ G j for each j. Since E is compact and u is continuous, we obtain Putting these facts together, we conclude that This completes the proof of Lemma 3.

Let us fix a large positive constant
We refer to ∂u(x) as the subdifferential of u at the pointx.
Lemma 4. Fix a pointx ∈ Σ and let ξ ∈B n+m . Let ξ tan denote the orthogonal projection of ξ to the tangent space TxΣ.
By Rademacher's theorem, u is differentiable almost everywhere. At each point where u is differentiable, the norm of its gradient is at most 1. By Alexandrov's theorem (see Theorem 14.1 and Theorem 14.25 in [11]), u admits a Hessian in the sense of Alexandrov at almost every point.
In the following, we fix a pointx ∈ Σ\∂Σ with the property that u admits a Hessian in the sense of Alexandrov atx. Letû be a smooth function on Σ such that |u(x) −û(x)| ≤ o(d(x,x) 2 ) as x →x.
Let us fix a small positive real numberr so that √ n 2r < d(x, ∂Σ) and √ n 2r is smaller than the injectivity radius atx. For each r ∈ (0,r), we denote byω(r) the smallest nonnegative real number ω with the property that |z − ∇ Σû (x)| ≤ ω whenever x ∈ Σ, z ∈ ∂u(x), and d(x,x) ≤ Proof. The first statement follows immediately from the definition. The second property follows from the basic properties of the Alexandrov Hessian; see [11], Theorem 14.25 (i'). This completes the proof of Lemma 5.
For each r ∈ (0,r), we denote byδ(r) the smallest nonnegative real number δ with the property that D 2 Proof. The first statement follows immediately from the definition. To prove the second statement, we argue by contradiction. Suppose that lim sup r→0δ (r) > 0. Then we can find a positive real number δ 0 , a sequence of points x j ∈ Σ, and a sequence ξ j ∈B n+m with the following properties: • For each j, the first eigenvalue of D 2 Σû (x j ) − II(x j ), ξ j is less than −δ 0 . After passing to a subsequence, we may assume that the sequence ξ j converges toξ ∈B n+m . Sinceû is a smooth function, it follows that the first eigenvalue of D 2 Σû (x) − II(x),ξ is strictly negative. Moreover, Σû (x) − II(x),ξ ≥ 0. This is a contradiction. This completes the proof of Lemma 6.
Let {e 1 , . . . , e n } be an orthonormal basis of TxΣ. For each r ∈ (0,r), we consider the cube We denote by the image of the cube W r under the exponential map. We further define Clearly, E r is a compact subset of Σ and A r is a compact subset of the normal bundle of Σ. We define a smooth map Φ : T ⊥ Σ → R n+m by Φ(x, y) = ∇ Σû (x) + y for x ∈ Σ and y ∈ T ⊥ x Σ. Moreover, we denote by G r = {ξ ∈B n+m : ∃ (x, y) ∈ A r with |ξ − Φ(x, y)| ≤ω(r)} the intersection ofB n+m with the tubular neighborhood of Φ(A r ) of radiuŝ ω(r). Clearly, G r is a compact subset ofB n+m . Lemma 7. Let r ∈ (0,r). Then u(x) − h(ξ) − x, ξ > 0 for all x ∈ E r and all ξ ∈B n+m \ G r .
Proof. We argue by contradiction. Suppose that there is a point x ∈ E r and a point ξ ∈B n+m \ G r such that u(x) − h(ξ) − x, ξ = 0. Let ξ tan denote the orthogonal projection of ξ to the tangent space T x Σ. By Lemma 4, ξ tan ∈ ∂u(x).
Proof. This follows by combining Lemma 3 and Lemma 7.
Proposition 9. Fix a pointx ∈ Σ \ ∂Σ with the property that u admits a Hessian in the sense of Alexandrov atx. Letû be a smooth function on Σ such that |u( Proof. In the following, we fix an arbitrary positive integer j. We define , y ≥ −j −1 g}. For each r ∈ (0,r), we decompose the normal space T ⊥ x Σ into compact cubes of size r. Let Q r denote the collection of all the cubes in this decomposition. Moreover, we denote by Q r,j ⊂ Q r the set of all cubes in Q r that are contained in the set S j . We define a smooth map where P z : T ⊥ x Σ → T ⊥ expx(z) Σ denotes the parallel transport along the geodesic t → expx(tz) (see [9], pp. 114-115). Since lim r→0ω (r) = 0 and lim r→0δ (r) = 0, we obtain provided that r is sufficiently small (depending on j). This implies provided that r is sufficiently small (depending on j).
We next observe that Hence, if r is sufficiently small (depending on j), then we obtain for each cube Q ∈ Q r,j . To justify (10), we argue as in the proof of the classical change-of-variables formula (see [10], pp. 150-156). We also use the fact that lim r→0ω (r) r = 0. Summation over all cubes Q ∈ Q r,j gives provided that r is sufficiently small (depending on j). On the other hand, Lemma 8 implies that µ(E r ) ≤ ν(G r ) for each r ∈ (0,r). Thus, we conclude that Σû (x) − II(x), y )| ρ(|∇ Σû (x)| 2 + |y| 2 ) + j −1 dy.
Finally, we pass to the limit as j → ∞. Note that S j+1 ⊂ S j for each j. Moreover, ∞ j=1 S j = S. This gives Since D 2 Σû (x) − II(x), y ≥ 0 for all y ∈ S, the assertion follows. This completes the proof of Proposition 9.
Corollary 10. Fix a pointx ∈ Σ \ ∂Σ with the property that u admits a Hessian in the sense of Alexandrov atx. Letû be a smooth function on Σ such that |u( where α is defined by (1).
Proof. We argue by contradiction. If the assertion is false, then there exists a real numberα > α such that for all y ∈ S. Using Proposition 9, we obtain In the last step, we have used the definition of α; see (1). Thusα ≤ α, contrary to our assumption. This completes the proof of Corollary 10.
After these preparations, we may now complete the proof of Theorem 1. Corollary 10 implies that (11) n α − 1 n ≤ ∆ Σ u + |H| almost everywhere, where ∆ Σ u denotes the trace of the Alexandrov Hessian of u. The distributional Laplacian of u may be decomposed into its singular and absolutely continuous part. By Alexandrov's theorem (see Theorem 14.1 in [11]), the density of the absolutely continuous part is given by the trace of the Alexandrov Hessian of u. The singular part of the distributional Laplacian of u is nonnegative since u is semiconvex. This implies for every nonnegative smooth function η : Σ → R that vanishes in a neighborhood of ∂Σ. Combining (11) and (12), we obtain for every nonnegative smooth function η : Σ → R that vanishes in a neighborhood of ∂Σ. By a straightforward limiting procedure, this implies This completes the proof of Theorem 1 in the special case when |Σ| = 1. The general case follows by scaling.