A quantitative isoperimetric inequality in higher codimension

Dr. Verena Bögelein / Prof. Frank Duzaar (Universität Erlangen-Nürnberg)

Friday, November 16, 2012, 13.00 - 15.00
Friday, November 23, 2012, 13.00 - 15.00
Tuesday, November 27, 2012,  14.00 - 15.00

HG G 19.1

Abstract

The main goal of the course is to give a proof of the quantitative isoperimetric inequality in higher codimension. For smooth closed (n-1)-dimensional submanifolds $\Gamma\subset R^{n+k}$ this quantitative isoperimetric inequality has the form $\mathbf D(\Gamma)\ge C \mathbf d^2(\Gamma)$.

Here, $\mathbf D(\Gamma)$ stands for the ${\it isoperimetric gap}$ of $\Gamma$, i.e. the deviation in measure of $\Gamma$ from being a round sphere. More precisely, the isoperimetric gap is defined by

$\mathbf D(\Gamma)$:=$\frac{H^{n-1}(\Gamma)- H^{n-1}(\partial D_\rho)}{H^{n-1}(\partial D_\rho)}$,

where $D_\rho$ is an n-dimensional flat disk in $R^{n+k}$ with the same area as an area minimizing n-dimensional surface $Q(\Gamma)$ spanned by the closed surface $\Gamma$, i.e. $H^n(D\rho)= H^n(Q(\Gamma))$. The quantitity $\mathbf d(\Gamma )$ stands for a natural generalization of the Fraenkel asymmetry index of $\Gamma$ to higher codimensions. The precise definition of the ${\it asymmetry index}$ $\mathbf d(\Gamma)$ is more technical and requires the use of a certain seminorm from geometric measure theory. The underlying geometric idea however is quite natural and can be described as follows.

Given any flat disk $D_\rho$ with the same area as an area minimizing n-dimensional surface $Q(\Gamma)$ with boundary $\Gamma$, one considers an area minimizing cylindric type surface $\Sigma (D_\rho)$ spanned by the boundary components $\Gamma$ and $\partial D_\rho$, and afterwards one takes the infimum of the surface area $H^n(\Sigma (D_\rho))$ amongst all such possible disks $D_\rho$; that is one defines

$\mathbf d(\Gamma)$:=$\rho^{-n}\inf\big\{H^n(\Sigma(D_\rho)):H^n(D_\rho)=H^n(Q(\Gamma))\big\}$.

In order to make the above definitions rigorous one has to use the framework of geometric measure theory.

In the lectures all relevant objects from geometric measure theory will be explained. Moreover, the proof of the quantitative isoperimetric inequality will be given in detail. A basic knowledge in measure theory (cf. the book of Evans & Gariepy) is necessary to follow the lectures.