17:45 Uhr | Verena Bögelein | Quantitative isoperimetric inequalities
in geometric settings
The aim of this talk is twofold. In the first part we establish a quantitative isoperimetric inequality in
higher codimension. In a certain sense such an inequality can be regarded as a stability result (second order Taylor approximation) of Almgren's optimal isoperimetric.
In particular, we prove that for any closed $(n-1)$-dimensional manifold
$\Gamma$ in $\mathbb R^{n+k}$ the following inequality
$$
\mathbf D(\Gamma)\ge C \mathbf d^2(\Gamma)
$$
holds true. Here, $\mathbf D(\Gamma)$ stands for the isoperimetric gap of $\Gamma$, i.e.
the deviation in measure of $\Gamma$ from being a round $(n-1)$-dimensional sphere and $\mathbf d(\Gamma )$
denotes a natural generalization of the Fraenkel asymmetry index of $\Gamma$ to higher codimensions. It measures in a generalized sense the distance of $\Gamma$ from spheres enclosing the same volume as $\Gamma$.
In the second part of the talk we are interested in the isoperimetric inequality on manifolds. As the simplest manifold we will consider the sphere. It was shown by E.~Schmidt that geodesic balls are the unique isoperimetric sets on the sphere. We will present a strong quantitative version of this result. More precisely, we prove that for any set $E\subset S^n$ of finite perimeter there holds
$$
\mathbf D(E)
:=
\frac{\mathbf P(E)-\mathbf P(B_\vartheta)}{\mathbf P(B_\vartheta)}
\ge C
\boldsymbol\beta^2(E),
$$
where $B_\vartheta\subset S^n$ is a geodesic ball with the same volume as $E$ and $\boldsymbol\beta$ denotes the $L^2$-oscillation index of $\partial E$. In principle the oscillation index is given by
$$
\boldsymbol\beta (E)
:=
\boldsymbol\alpha (E) + \mbox{term counting the $L^2$-oscillation of $\partial E$,}
$$
where $\boldsymbol\alpha$ denotes the Fraenkel asymmetry index.
The advantage of such a quantity stems from the fact that the above quantitative isoperimetric inequality is sharp in the sense that also the reverse inequality holds true. Both results are joint work with Frank Duzaar and Nicola Fusco.
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