On the spectrum and index of expanding and translating solitons of the mean curvature flow in $\mathbb{R}^3$

TL;DR

This paper investigates the spectral properties and topological classification of two-dimensional translating solitons and self-expanders in b3, establishing conditions under which they have finite topology and specific homeomorphisms.

Contribution

It provides new results linking the spectral bounds of the L-stability operator to the topology of translating solitons and self-expanders in b3.

Findings

Finite L-index implies homeomorphism to a plane or cylinder.

Finite L-index and subexponential volume growth imply finite topology.

Spectral bounds ensure finite topology of solitons and expanders.

Abstract

In this paper we prove that two-dimensional translating solitons in $\mathbb{R}^3$ with finite $L$-index are homeomorphic to a plane or a cylinder and that a two-dimensional self-expander with finite $L$-index and sub exponential weighted volume growth has finite topology. We also prove that translating solitons and self-expanders have finite topology, provided the bottom of the spectrum of the $L$-stability operator is bounded from below and their weighted volume have subexponential growth.