Glimpses of Geometry

While the notion of mass is well understood in mathematical general relativity (positive mass theorem, Penrose inequality, etc.), that of angular momentum seems to be considerably less so. In this talk I will review the definition of the total angular momentum J of an initial data set, and outline proofs of inequalities involving J that have been proved recently: Dain's angular momentum bound for axisymmetric space-times (arXiv:0707.3118 [gr-qc], arXiv:0712.4064 [gr-qc]), Maerten's angular momentum bound for space-times with a cosmological constant (see, e.g., gr-qc/0606064), and the surprising role of J for non-existence of constant mean curvature hypersurfaces with large mean curvature (PTC, Tod: arXiv:0706.4057 [gr-qc]).

The effect of curvature on the behaviour of solutions of the heat equation on a Riemannian manifold is a classical problem. A quantitative measurement of this behaviour is encoded most directly in terms of gradient estimates and Harnack inequalities involving constants depending only on lower Ricci curvature bounds and the dimension of the manifold. In this talk we focus on localized versions of such estimates (linear and nonlinear) and show that Brownian motion on a Riemannian manifold may serve as a unifying tool. The talk is based on joined work with Marc Arnaudon, Bruce Driver and Feng-Yu Wang.

We introduce a notion of Ricci curvature "at a given scale" valid on any metric space, inspired by ideas of Dobrushin about Markov chains. The notion states that balls are closer, in transportation distance, than their centers are. It is simple to test on concrete examples such as graphs or Riemannian manifolds, and consistent with Bakry-Émery theory. Positive Ricci curvature in this sense allows to prove analogues of the Lichnerowicz spectral gap theorem, of the Lévy-Gromov concentration of measure theorem, and of the Bakry-Émery log-Sobolev inequality.

This talk is concerned with the relationship between isoperimetric inequalities and notions of non-positive and negative curvature for (singular) metric spaces. We first give an optimal characterization of Gromov hyperbolicity via isoperimetric and filling radius inequalites for curves, generalizing and strengthening results of Gromov. We then focus on higher dimensional isoperimetric inequalities in the setting of metric spaces of non-positive curvature in a weak sense (including non-positively curved spaces in the sense of Busemann) and show that they detect the asymptotic rank of such spaces. Among other things, this allows to prove higher rank analogs of well-known results from hyperbolic geometry such as for example the stability of quasi-geodesics. Such analogs were exhibited by B. Kleiner and U. Lang.

The main theme of the lecture should be J. Taylor's regularity theorem for Almgren almost-minimal sets of dimension 2 in 3-space, which says that they are locally $C^1$-equivalent to one of the three possible minimal cones (the ones that are easily seen in soap films). I will try to discuss some motivations and simple topologial arguments in the proof.

A compact Riemannian manifold with boundary is said to be boundary rigid if its metric is uniquely determined (up to a isometry) by distances between boundary points. To visualize that, imagine that one wants to find out what the Earth is made of. More generally, one wants to find out what is inside a solid body made of different materials (in other words, properties of the medium change from point to point). The speed of sound depends on the material. One can "tap" at some points of the surface of the body and "listen when the sound gets to other points". The question is if this information is enough to determine what is inside. This problem has been studied a lot, mainly from the PDE viewpoint. We suggest a completely different approach based on "minimality". As a matter of fact, we embed our manifolds into a certain normed space and show that they happen to be minimal surfaces, and then prove certain uniqueness results for such surfaces. We say that a metric $d$ on a manifold with boundary is a "minimal filling" if every manifold with the same boundary and such that all distances between boundary points are greater than or equal to those of $d$ has volume no less than that of $d$. We will discuss the following result: Euclidean regions with Riemannian metrics sufficiently close to a Euclidean one are minimal fillings and boundary rigid. This is the first result in dim>2 showing the boundary rigidity of metrics other than extremely special ones (products and symmetric spaces). The talk is based on a joint work with S. Ivanov.

Symmetrization inequalities and optimal transport methods are used in the proof of many geometric-functional inequalities in sharp form. Sharp quantitative refinements of several isoperimetric principles, including the Euclidean and Gaussian isoperimetric inequalities, as well as the Wulff theorem and the Brunn-Minkowski inequality on convex sets, can be attacked on starting from these methods. The content of this talk originates from a series of joint works with A. Cianchi (U. Firenze), A. Figalli (U. Nice), N. Fusco (U. Napoli) and A. Pratelli (U. Pavia).

We will discuss an alternative approach to Perelman's collapsing theorem which is the last step of his proof of the geometrization conjecture for aspherical 3-manifolds.

Analyzing the singularities can be useful in applying Ricci flow to study the geometry of manifolds. We shall report on several results in this direction.

We give a brief survey on the geometry of metric measure spaces (M,d,m) satisfying a generalized lower Ricci bound. This notion of curvature bound, independently developed by Lott/Villani and Sturm, is based on convexity properties of the relative entropy regarded as a function on the L2-Wasserstein space of probability measures. One of the main results is that this lower curvature bound is stable under convergence. Furthermore, we introduce a (more restrictive) curvature-dimension condition CD(K,N) which implies sharp versions of the Brunn-Minkowski inequality, of the Bishop-Gromov volume comparison theorem and of the Bonnet-Myers theorem. Moreover, we present some recent developments in the analysis on singular spaces with lower Ricci bounds. For Finsler spaces we analyze the heat flow, defined as the gradient flow on L2(M,m) for the energy -- or equivalently as the gradient flow on the Wasserstein space for the relative entropy. In all non-riemannian cases, the heat flow is non-linear and non-smooth. Nevertheless, under appropriate N-Ricci bounds we deduce Bochner inequalities, Bakry-Emery gradient estimates and Li-Yau Harnack inequalities.

We show that for $H(q,p)$ a Hamiltonien on the torus cotangent bundle $T^n\times {\mathbb R}^n$ the sequence $H(k\cdot q,p)$ converges to the "effective Hamiltonian". The convergence is in fact a convergence of the corresponding flows for a metric defined on the group of Hamiltonian maps. The proof uses some new estimate of the "symplectic size" of a Lagrangian contained in a tube. We shall present some of the many applications, that range from extending classical results in the convex case (i.e.for $H$ convex in $p$) about Hamilton-Jacobi equations, Mather theory, etc. to new unexpected results, like the symplectic invariance of the effective Hamiltonian.