# Symmetric Convex Sets with Minimal Gaussian Surface Area

## TL;DR

This paper characterizes symmetric convex sets with minimal Gaussian surface area, showing they are round cylinders under certain conditions, using advanced geometric analysis and eigenfunction techniques related to Gaussian minimal surfaces.

## Contribution

It provides new conditions under which symmetric convex sets with minimal Gaussian surface area are necessarily round cylinders, extending previous results and employing novel second variation methods.

## Key findings

- Convex symmetric sets with minimal Gaussian surface area are round cylinders under specific integral conditions.
- Introduction of a new second variation approach using degree 2 polynomials for Gaussian minimal surface analysis.
- Results extend to some non-convex cases, broadening the scope of Gaussian isoperimetric characterizations.

## 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 $x=(x_{1},\ldots,x_{n+1})\in\mathbb{R}^{n+1}$, let $\gamma_{n}(x)=(2\pi)^{-n/2}e^{-(x_{1}^{2}+\cdots+x_{n+1}^{2})/2}$. Let $\|A\|^{2}$ be the sum of the squares of the entries of $A$, and let $\|A\|_{2\to 2}$ denote the $\ell_{2}$ operator norm of $A$. It is shown that if $\Omega$ or $\Omega^{c}$ is convex, and if either $$\int_{\partial\Omega}(\|A_{x}\|^{2}-1)\gamma_{n}(x)dx>0\qquad\mbox{or}\qquad \int_{\partial\Omega}\Big(\|A_{x}\|^{2}-1+2\sup_{y\in\partial\Omega}\|A_{y}\|_{2\to 2}^{2}\Big)\gamma_{n}(x)dx<0,$$ then $\partial\Omega$ must be a round cylinder. That is, except for the case that the average value of $\|A\|^{2}$ 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 $L= \Delta-\langle x,\nabla \rangle+\|A\|^{2}+1$ 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.