On polar actions invariant solutions of semilinear equations on manifolds

TL;DR

This paper develops a framework combining differential topology and analysis to study invariant solutions of semilinear PDEs on Riemannian manifolds, reducing the problem to equations on submanifolds using group actions.

Contribution

It introduces a method to reduce semilinear PDEs on manifolds to equations on submanifolds via polar actions, expanding analysis tools for existence and properties of solutions.

Findings

Reduction of PDEs to submanifold equations under group actions

Conditions for the existence of such reductions on various manifolds

Application of analysis techniques to one-dimensional submanifold equations

Abstract

In this paper we put together some tools from differential topology and analysis in order to study second order semi-linear partial differential equations on a Riemannian manifold $M$. We look for solutions that are constants along orbits of a given group action. Using some results obtained by Helgason in [J DIFFER GEOM,6(3), 411-419] we are able to write a (reduced) second order semi-linear problem on a submanifold $\Sigma$. This submanifold is, in certain sense, transversal to the orbits of the group actions and its existence is assumed. We describe precise conditions on the Riemannian Manifold $M$ and the submanifold $\Sigma$ in order to be able to write the reduced equation on $\Sigma$. These conditions are satisfied by several particular cases including some examples treated separately in the literature such as the sphere, surfaces of revolution and others. Our framework also includes the setup of polar actions or exponential coordinates. Using this procedure, we are left with a second order semi-linear equation posed on a submanifold. In particular, if the submanifold $\Sigma$ is one-dimensional, we can use suitable tools from analysis to obtain existence and properties of solutions.