The Hyperplane is the Only Stable, Smooth Solution to the Isoperimetric Problem in Gaussian Space

TL;DR

This paper proves that hyperplanes are the only stable, smooth solutions to the Gaussian isoperimetric problem in Euclidean space, establishing curvature conditions, stability criteria, and decay estimates for such hypersurfaces.

Contribution

It characterizes stable solutions to the Gaussian isoperimetric problem, proving hyperplanes are unique among smooth, stable solutions and analyzing their stability and decay properties.

Findings

Hyperplanes are the only stable smooth solutions.

No hypersurfaces of index one exist for this problem.

Stable solutions with good area growth approach hyperplanes as the domain expands.

Abstract

We study stable smooth solutions to the isoperimetric type problem for a Gaussian weight on Euclidean Space. That is, we study hypersurfaces $\Sigma^n \subset \mathbb R^{n+1}$ that are second order stable critical points of compact variations that minimize Gaussian weighted area and preserve Gaussian weighted volume. We show that such $\Sigma$ satisfy a curvature condition, and derive the Jacobi operator $L$ for the second variation of such $\Sigma$. Our first main result is that for non-planar $\Sigma$, bounds on the index of $L$, acting on volume preserving variations, gives us that $\Sigma$ splits off a linear space. A corollary of this result is that hyperplanes are the only stable smooth complete solutions to this Gaussian isoperimetric type problem, and that there are no hypersurfaces of index one. Finally, we show that for the case of $\Sigma^2 \subset \mathbb R^3$, there is a gradient decay estimate depending on bounds for the curvature condition and an appropriate area growth bound. This shows that, in the limit as $R \to \infty$, stable $(\Sigma, \partial\Sigma) \subset (B_{2R}(0), \partial B_{2R}(0))$ with good area growth bounds approach hyperplanes.