Manifolds with Density, Poincaré, and Isoperimetric Problems

Williams College

Giving a Riemannian manifold a density (as in freshman physics), such as the Gaussian density used by probabilists, has attracted more attention since its appearance in Perelman's original paper proving the Poincaré Conjecture. Of particular importance is the isoperimetric problem, seeking to enclose prescribed weighted volume with minimum weighted perimeter. We'll discuss some general theory and interesting examples, including recent work by undergraduates. No specific prerequisites, undergraduates welcome (although they'll have to take parts on faith).

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