We establish an analytic interpretation for the index of certain transversally elliptic symbols on non-compact manifolds. By using this interpretation, we establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a non-compact symplectic manifold with proper moment map. In particular, we present a solution to a conjecture of Michèle Vergne in her ICM 2006 plenary lecture.

The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regularity theorem of Ecker and Huisken for Lipschitz hypersurfaces. In particular, any submanifold of the Euclidean space with a continuous induced metric can be smoothed out by the mean curvature flow. The smallness assumption is necessary in the higher codimension case in view of an example of Lawson and Osserman. The stationary phase of the mean curvature flow corresponds to minimal submanifolds. Our result thus generalizes Morrey’s classical theorem on the smoothness of $C_1$ minimal submanifolds.

This is the second in a series of papers on a new equivariant cohomology that takes values in a vertex algebra. In an earlier paper, the first two authors gave a construction of the cohomology functor on the category of O(sg) algebras. The new cohomology theory can be viewed as a kind of "chiralization'' of the classical equivariant cohomology, the latter being defined on the category of $G^*$ algebras a la H. Cartan. In this paper, we further develop the chiral theory by first extending it to allow a much larger class of algebras which we call sg[t] algebras. In the geometrical setting, our principal example of an O(sg) algebra is the chiral de Rham complex Q(M) of a G manifold M. There is an interesting subalgebra of Q(M) which does not admit a full O(sg) algebra structure but retains the structure of an sg[t] algebra, enough for us to define its chiral equivariant cohomology. The latter then turns out to have many surprising features that allows us to delineate a number of interesting geometric aspects of the G manifold M, sometimes in ways that are quite different from the classical theory.

An almost K\"ahler structure on a symplectic manifold (N,ω) consists of a Riemannian metric g and an almost complex structure J such that the symplectic form ω satisfies ω(⋅,⋅)=g(J(⋅),⋅) . Any symplectic manifold admits an almost K\"ahler structure and we refer to (N,ω,g,J) as an almost K\"ahler manifold. In this article, we propose a natural evolution equation to investigate the deformation of Lagrangian submanifolds in almost K\"ahler manifolds. A metric and complex connection $\hn$ on TN defines a generalized mean curvature vector field along any Lagrangian submanifold M of N . We study the evolution of M along this vector field, which turns out to be a Lagrangian deformation, as long as the connection $\hn$ satisfies an Einstein condition. This can be viewed as a generalization of the classical Lagrangian mean curvature flow in K\"ahler-Einstein manifolds where the connection $\hn$ is the Levi-Civita connection of g . Our result applies to the important case of Lagrangian submanifolds in a cotangent bundle equipped with the canonical almost K\"ahler structure and to other generalization of Lagrangian mean curvature flows, such as the flow considered by Behrndt \cite{b} in K\"ahler manifolds that are almost Einstein.

Sharp Lp affine isoperimetric inequalities are established for the entire class of Lp projection bodies and the entire class of Lp centroid bodies. These new inequalities strengthen the Lp Petty projection and the Lp Busemann–Petty centroid inequality.