Sign-changing solutions to the Yamabe problem on manifolds with boundary

TL;DR

This paper proves the existence of least-energy sign-changing solutions to the Yamabe problem on certain manifolds with boundary, using variational methods and conformal invariants.

Contribution

It establishes the existence of nodal solutions for the Yamabe problem on manifolds with boundary under specific geometric conditions, which was previously largely open.

Findings

Existence of least-energy sign-changing solutions when n ≥ 7 and boundary has a nonumbilic point.

Solutions are found for manifolds with positive scalar curvature and non-negative boundary mean curvature.

The approach uses variational methods and analysis of conformal invariants.

Abstract

Let $(M, g)$ be a compact Riemannian manifold with boundary. The Yamabe problem concerning the existence of a metric conformally equivalent to $g$ having constant scalar curvature on $M$ and constant mean curvature on its boundary is equivalent, in analytic terms, to finding a positive solution to a nonlinear boundary-value problem with critical growth. While the existence of positive solutions to this problem is by now well understood, the existence of sign-changing (nodal) solutions remains largely open. In this work we establish the existence of least-energy sign-changing solutions when the manifold is positive and the mean curvature of the boundary is a non-negative constant. More precisely, we prove that if $n\ge7$ and $M$ has a nonumbilic boundary point, then the problem admits least-energy nodal solutions. Our approach is variational and relies on the analysis of suitable conformal invariants and sharp energy estimates.