A Quantitative Isoperimetric Inequality in Higher Codimension

In this talk we establish a quantitative isoperimetric inequality in higher co-dimension, which can be interpreted as a stability result (second order Taylor approximation) of Almgren's Optimal Isoperimetric Inequality in higher codimension. In particular, we establish for any closed $(n{-}1)$-dimensional manifold $\Gamma$ in $\mathbb R^{n+k}$ the following inequality: $$ \mathbf D(\Gamma)\ge C \mathbf d^2(\Gamma) $$ Here, $\mathbf D(\Gamma)$ denotes the isoperimetric gap of $\Gamma$, i.e.~the deviation in Hausdorff-measure of $\Gamma$ from being a round sphere, while $\mathbf d(\Gamma )$ is a natural generalization of the Fraenkel asymmetry index to higher codimension surfaces $\Gamma$. It measures In a generalized sense the distance of $\Gamma$ to spheres with the same volume. As a by-product we obtain the following estimate from below for the Isoperimetric Quotient: $$ \boldsymbol \gamma (\Gamma) := \frac{\mathcal H^{n-1}(\Gamma)^\frac{n}{n-1}}{\mathcal H^{n}(Q(\Gamma))} \ge \boldsymbol\gamma (S^{n-1})+ C\, \mathbf d^2(\Gamma), $$ where $Q(\Gamma)$ denotes an $n$-dimensional area minimizing surface with boundary $\Gamma$.