Symmetric Convex Sets with Minimal Gaussian Surface Area

Abstract

Let $\Omega \subset {\mathbb{R}}^{n+1}$ have minimal Gaussian surface area among all sets satisfying $\Omega =-\Omega$ with fixed Gaussian volume. Let $A=A_x$ be the second fundamental form of $\partial \Omega$ at $x$, i.e. $A$ is the matrix of first order partial derivatives of the unit normal vector at $x\in \partial \Omega$. For any $x=(x_1,\dots ,x_{n+1})\in {\mathbb{R}}^{n+1}$, let ${\gamma }_n(x)=(2\pi {)}^{-n/2}{e}^{-(x_1^2+\cdots +x_{n+1}^2)/2}$. Let $\Vert A{\Vert }^2$ be the sum of the squares of the entries of $A$, and let $\Vert A{\Vert }_{2\to 2}$ denote the ${\ell }_2$ operator norm of $A$. It is shown that if $\Omega$ or ${\Omega }^c$ is convex, and if either $$\int_{\partialΩ}(\|A_{x}\|^{2}-1)γ_{n}(x)dx>0\qquad\mbox{or}\qquad \int_{\partialΩ}\Big(\|A_{x}\|^{2}-1+2\sup_{y\in\partialΩ}\|A_{y}\|_{2\to 2}^{2}\Big)γ_{n}(x)dx<0,$$ then $\partial \Omega$ must be a round cylinder. That is, except for the case that the average value of $\Vert A{\Vert }^2$ is slightly less than $1$, we resolve the convex case of a question of Barthe from 2001. The main tool is the Colding-Minicozzi theory for Gaussian minimal surfaces, which studies eigenfunctions of the Ornstein-Uhlenbeck type operator $L=\Delta -⟨x,\nabla ⟩+\Vert A{\Vert }^2+1$ associated to the surface $\partial \Omega$. A key new ingredient is the use of a randomly chosen degree 2 polynomial in the second variation formula for the Gaussian surface area. Our actual results are a bit more general than the above statement. Also, some of our results hold without the assumption of convexity.

52 pages, typos corrected, new section added