12.09.2018

Geodesic flows: notes 5

Geodesian flows, notes 5: symplectic isoperimetric inequalities

If \(\Omega \subset \mathbb{R}^n\) is a bounded open set with smooth boundary, the isoperimetric inequality states that \[ \mathrm{Vol}_n(\Omega)^{\frac{n-1}{n}} \leq \frac{1}{n \omega_n^{1/n}} \mathrm{Vol}_{n-1}(\partial \Omega) \] with equality iff \(\Omega\) is a ball (here \(\omega_n = \mathrm{Vol}_n(B_1)\)). This inequality is equivalent with the (Gagliardo-Nirenberg-)Sobolev inequality \(W^{1,1}(\mathbb{R}^n) \subset L^{\frac{n}{n-1}}(\mathbb{R}^n)\) with sharp constant, \[ \lVert u \rVert_{L^{\frac{n}{n-1}}} \leq \frac{1}{n \omega_n^{1/n}} \lVert \nabla u \rVert_{L^1}. \] (One direction follows by applying the Sobolev inequality to a smoothed version of \(\chi_{\Omega}\), and the other direction involves the coarea formula and writing the \(L^{\frac{n}{n-1}}\) norm of a function in terms of super-level sets, see e.g. this link.)

Isoperimetric inequalities have been studied in many different geometries besides \(\mathbb{R}^n\) (sphere, Gaussian measure, Riemannian, ...). We intend to discuss a rather different looking variant, called a symplectic isoperimetric inequality, based on Viterbo (the notions that appear will not be explained at this time):

Theorem. One has \[ d(L)^{n/2} \leq c_n \mathrm{Vol}_n(L)\] where \(L\) is a (\(n\)-dimensional) Lagrangian manifold in \(\mathbb{R}^{2n}\) and \(d(L)\) is the so called displacement energy of \(L\).

(The displacement energy \(d\) is a "special capacity" on open subsets of a symplectic manifold and it satisfies \(d(A) > 0\) for nonempty open \(A \subset \mathbb{R}^{2n}\), \(d(A) \leq d(B)\) if \(A \subset B\), and \(d(F(A)) = d(A)\) if \(F\) is a symplectic diffeomorphism. See Hofer-Zehnder, Symplectic invariants and Hamiltonian dynamics.)

Why is such an inequality called isoperimetric? This is related to celebrated work of Gromov on symplectic geometry in the 1980s. In particular, Gromov proved that if \(L\) is any closed Lagrangian manifold in \(\mathbb{R}^{2n} = \mathbb{C}^n\), there is a holomorphic map \(f: \overline{\mathbb{D}} \to \mathbb{C}^n\) (where \(\mathbb{D}\) is the unit disc in \(\mathbb{C}\)) which maps \(\partial \mathbb{D}\) into \(L\). A subsequent result of Chekanov shows (presumably, I did not have access to the paper) that such a map can be chosen so that \[ \mathrm{Area}(f(\overline{\mathbb{D}})) \leq d_n \mathrm{Vol}_n(L)^{2/n}. \] If \(n=1\), any \(1\)-manifold \(L\) in \(\mathbb{R}^2\) is Lagrangian, and if \(L\) is a smooth curve bounding a simply connected domain, presumably this is a version of the standard isoperimetric inequality in \(\mathbb{R}^2\). This is related to the (vague) idea of symplectization introduced by Arnold, which means in particular that for a statement related to submanifolds of a smooth manifold, there should be an analogous symplectic statement phrased in terms of Lagrangian manifolds.

Let us work out a consequence of the symplectic isoperimetric inequality, which implies a version of the Aleksandrov-Bakelman-Pucci maximum principle. This is an important tool in the Krylov-Safonov estimates (De Giorgi-Nash-Moser for nondivergence form equations), and these are fundamental for the regularity theory of fully nonlinear elliptic PDE.

Theorem. (ABP) If \(\Omega \subset \mathbb{R}^n\) is a bounded smooth domain, and if \(u \in C^2(\Omega) \cap C(\overline{\Omega})\), then \[ \sup_{\Omega} u \leq \sup_{\partial \Omega} u + C \left[ \int_{\Omega} |\det(D^2 u)| \,dx \right]^{1/n} \] where \(D^2 u = (\partial_{jk} u)_{j,k=1}^n\) is the Hessian. (It is important for applications that the integral can actually be taken over the subset of \(\Omega\) where \(u\) touches its concave envelope, but we will not worry about this.)

We derive this from the following variant of the symplectic isoperimetric inequality, also proved by Viterbo:

Theorem. If \(K \subset \mathbb{R}^n\) is compact, and if \(L\) is a Lagrangian manifold contained in \(T^* K\), then \[ \gamma(L)^n \leq c_{n,K} \mathrm{Vol}_n(L) \] where \(\gamma(L)\) is another symplectic invariant.

Let \(u \in C^2_c(\Omega)\), and define \[ L = \{ (x, \nabla u(x)) \,;\, x \in \Omega \}. \] This turns out to be a Lagrangian manifold in \(\mathbb{R}^{2n}\). To compute its (usual Riemannian) volume, we note that the tangent vectors of \(L\) are of the form \(v_j = (e_j, \partial_j \nabla u)\) and the metric on \(L\) (induced by the Euclidean metric on \(\mathbb{R}^{2n}\) is given by \[ g_{jk} = v_j \cdot v_k = \delta_{jk} + \partial_j \nabla u \cdot \partial_k \nabla u = \delta_{jk} + \sum_{l=1}^n (D^2 u)_{jl} (D^2 u)_{lk}. \] Thus \(g = \mathrm{I} + (D^2 u)^2\), and \[ \mathrm{Vol}_n(L) = \int_{\Omega} \,dV_L(x) = \int_{\Omega} |\det(g_{jk}(x))|^{1/2} \,dx = \int_{\Omega} \sqrt{\det(\mathrm{I} + (D^2 u)^2)} \,dx. \] It also turns out that for this \(L\), one has \(\gamma(L) = \max_{\Omega} u - \min_{\Omega} u\). Thus the symplectic isoperimetric inequality gives \[ (\max_{\Omega} u - \min_{\Omega} u)^n \leq c_{n,\Omega} \int_{\Omega} \sqrt{\det(\mathrm{I} + (D^2 u)^2)} \,dx. \] Since \(u|_{\partial \Omega} = 0\) we have \(\max u \geq 0\) and \(\min u \leq 0\), and thus replacing \(u\) by \(\lambda u\) and letting \(\lambda \to \infty\) gives that \[ \lVert u \rVert_{L^{\infty}} \leq C_{n,\Omega} \lVert \det(D^2 u) \rVert_{L^1}^{1/n}, \qquad u \in C^2_c(\Omega). \]

(A related paper on symplectic Brunn-Minkowski. Some references for ABP: Caffarelli-Cabre, Taylor PDE 3 (section 14.13), Gilbarg-Trudinger (section 9.1), slides of Silvestre.)