Given a closed surface $\Sigma$ in $\mathbb{R}^3$, besides the usual geometric quantities like its area $A$ and the volume $V$ it encloses, one can consider another basic quantity $C$, the {\em capacity} of $\Sigma$. Physically speaking, if we think of $\Sigma$ as an ideal conductor, $C$ represents the amount of the electric charge which, in electrostatic equilibrium on $\Sigma$, raises the constant potential of $\Sigma$ to unity. There have been many interesting inequalities in the literature, giving both upper and lower bounds of $C$ in terms of the various geometric quantities of $\Sigma$. For example, it was proposed by Poincar\'{e} and later proved by Szeg\"{o}, that among all surfaces enclosing the same volume, $C$ is minimized by a sphere. When $\Sigma$ is convex, Szeg\"{o} also showed that $C \leq \frac{M}{4 \pi}$, where $M$ is the integral of the mean curvature $H$ over $\Sigma$. In this talk, we discuss a new upper bound on $C$ in terms of the area $A$ and the integral of $H^2$, without assuming $\Sigma$ is convex. Furthermore, the same inequality holds in the context of asymptotically flat $3$-manifolds with non-negative scalar curvature, hence giving applications in mathematical relativity. This is joint work with Hugh Bray.

Colloquia are designed to be of interest to a general mathematical audience and to be accessible without specialist knowledge. There will be wine and cheese afterwards.
 

Convenor: Ian Wanless - firstname.lastname@sci.monash.edu.au