We propose to study the Hessian metric of given functional in the space of
probability space embedded with L^2–Wasserstein (optimal transport) metric. We name it
transport Hessian metric, which contains and extends the classical L^2–Wasserstein metric.
We formulate several dynamical systems associated with transport Hessian metrics.
Several connections between transport Hessian metrics and math physics equations are
discovered. E.g., the transport Hessian gradient flow, including Newton’s flow, formulates
a mean-field kernel Stein variational gradient flow; The transport Hessian Hamiltonian
flow of negative Boltzmann-Shannon entropy forms the Shallow water’s equation; The
transport Hessian gradient flow of Fisher information forms the heat equation. Several
examples and closed-form solutions of finite-dimensional transport Hessian metrics and
dynamics are presented.
We define the notion of a Ricci curvature lower bound for parametrized statistical
models. Following the seminal ideas of Lott–Sturm–Villani, we define this notion
based on the geodesic convexity of the Kullback–Leibler divergence in a Wasserstein
statistical manifold, that is, a manifold of probability distributions endowed with a
Wasserstein metric tensor structure. Within these definitions, which are based on
Fisher information matrix and Wasserstein Christoffel symbols, the Ricci curvature
is related to both, information geometry and Wasserstein geometry. These definitions
allow us to formulate bounds on the convergence rate of Wasserstein gradient flows
and information functional inequalities in parameter space. We discuss examples of
Ricci curvature lower bounds and convergence rates in exponential family models.
In this survey, we discuss some recent results on free boundary minimal surfaces in the Euclidean unit-ball. The subject has been a very active field of research in the past few years due to the seminal work of Fraser and Schoen on the extremal Steklov eigenvalue problem. We review several different techniques of constructing examples of embedded free boundary minimal surfaces in the unit ball. Next, we discuss some uniqueness results for free boundary minimal disks and the conjecture about the uniqueness of critical catenoid. We also discuss several Morse index estimates for free boundary minimal surfaces. Moreover, we describe estimates for the first Steklov eigenvalue on such free boundary minimal surfaces and various smooth compactness results. Finally, we mention some sharp area bounds for free boundary minimal submanifolds and related questions.
For any smooth Riemannian metric on an (n+1)-dimensional compact manifold with boundary (M,∂M) where 3≤(n+1)≤7, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min-max theory in the Almgren-Pitts setting. We apply our Morse index estimates to prove that for almost every (in the C-infinity Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in M. If ∂M is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting.
We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more generally in the context of varifolds.