21 feb 2007

Abstract.

Perelman's proof of the Poincaré Conjecture requires placing a positive, continuous "density" function on the manifold. Manifolds with density appear a number of places in mathematics. The premier example, Gauss space (Euclidean space with Gaussian density), is important to probabilists. The grand goal is to generalize all of Riemannian geometry to manifolds with density.

References:

1. Frank Morgan, Manifolds with density, Notices Amer. Math. Soc. 52 (2005), 853-858.

We discuss the category of Riemannian manifolds with density and present easy generalizations of the volume estimate of Heintze and Karcher and thence of the isoperimetric inequality of Levy and Gromov. Some minor corrections appear in:

2. Frank Morgan, Myers' Theorem with density, Kodai Math. J. 29 (2006), 454-460.

We provide generalizations of theorems of Myers and others to Riemannian manifolds with density.

3. César Rosales, Vincent Bayle, Antonio Cañete, and Frank Morgan, On the isoperimetric problem in Euclidean space with density, ArXiv.org (2006).

In R with unimodal density we characterize isoperimetric regions. In Rⁿ with density we prove existence results and derive stability conditions, leading to the conjecture that for a radial, log-convex density, balls about the origin are isoperimetric. We prove this conjecture for the density e^r2 by symmetrization.