Gaussian Harmonic Forms and Two-Dimensional Self-Shrinkers

TL;DR

This paper introduces Gaussian Harmonic Forms (GHFs) on 2D self-shrinkers in mean curvature flow, using them to derive bounds on geometric quantities like curvature, eigenvalues, and index related to the surface's genus.

Contribution

It defines GHFs as minimizers of a certain energy in cohomology and applies them to establish new bounds on curvature, eigenvalues, and index for self-shrinkers.

Findings

Lower bound on the supremum norm of $A^2$ for genus $ extgreater= 1$ self-shrinkers.

Upper bound for the lowest eigenvalue of the operator $L$ using GHFs.

Lower bounds on the index and $ ext{inf}|x|^2$ in the codimension one case.

Abstract

We consider 2-dimensional orientable self-shrinkers $\Sigma$ for the Mean Curvature Flow of polynomial volume growth immersed in $\mathbb R^n$. We look at closed one forms minimizing the norm $\int_\Sigma \eterm |\omega|^2$ in their cohomology class. Any closed form satisfying the Euler-Lagrange equation for this minimization will be called a Gaussian Harmonic one Form (GHF). We then use these forms to show that if such a $\Sigma$ has genus $\geq 1,$ then we have a lower bound on the supremum norm of $A^2$. GHF's may also be applied to create an upperbound for the lowest eigenvalue of the operator $L$. In the codimension one case $\Sigma \to \mathbb R^3$, for certain conditions on the principal curvatures, we use GHF's to get a lower bound on the index of $L$ depending on the genus $g$. Likewise, in the compact codimension one case we obtain an estimate of the lowest eigenvalue of $L$ and also on $\inf |x|^2$.