Existence and uniqueness of radial solutions of semi-linear equations on manifolds

TL;DR

This paper establishes the existence and uniqueness of solutions for semi-linear PDEs on Riemannian manifolds by reducing the problem to a one-dimensional ODE setting using geometric and analytical techniques.

Contribution

It introduces a novel reduction method transforming PDE problems on manifolds into equivalent ODE problems on submanifolds, broadening the scope of solvable semi-linear equations.

Findings

Existence and uniqueness of solutions on various manifolds.

Reduction of PDE problems to ODE problems via group actions.

Applicable to spheres, surfaces of revolution, and similar geometries.

Abstract

We investigate the existence and uniqueness of solutions for second-order semi-linear partial differential equations defined on a Riemannian manifold $M$. By combining differential geometry and analysis techniques, we establish the existence and uniqueness of constant solutions through the orbits of a group action. Our approach transforms such problems into equivalent ones over a submanifold $\Sigma$ of dimension one, which is transversal to the group action. This reduction leads us to a one-dimensional setting, where we can apply different results from the theory of ordinary differential equations. Our framework is versatile and includes the setups of polar actions or exponential coordinates, with particular examples such as the sphere, surfaces of revolution, and others.