An integer degree for asymptotically conical self-expanders

TL;DR

This paper introduces an integer degree for a projection map related to asymptotically conical self-expanders, providing new insights into their stability and existence in geometric analysis.

Contribution

It establishes the existence of an integer degree for the projection map of asymptotically conical self-expanders, linking geometric properties to topological invariants.

Findings

Existence of an integer degree for the projection map.

Identification of an open set of cones with unstable self-expanding annuli.

Application to stability analysis of asymptotically conical self-expanders.

Abstract

We establish the existence of an integer degree for the natural projection map from the space of parameterizations of asymptotically conical self-expanders to the space of parameterizations of the asymptotic cones when this map is proper. As an application we show that there is an open set in the space of cones in the three-dimensional Euclidean space for which each cone in the set has a strictly unstable self-expanding annuli asymptotic to it.