The second Yamabe invariant with singularities

TL;DR

This paper introduces a new invariant called the second Yamabe invariant with singularities for certain manifolds, proves its attainment by a generalized metric, and derives nodal solutions to related Yamabe equations with singularities.

Contribution

It defines the second Yamabe invariant with singularities, proves its realization by a generalized metric, and establishes existence of nodal solutions for singular Yamabe equations.

Findings

The second Yamabe invariant with singularities is attained by a generalized metric.

Existence of nodal solutions to Yamabe-type equations with singularities.

Extension of Yamabe problem concepts to metrics with singularities.

Abstract

Let (M,g) be a compact manifold of dimension n greater or equals to 3. We suppose that g is a given metric in a precised Sobolev space and there is a point P in M and d>o such that g is smooth on the ball B(P,d). We define the second Yamabe invariant with singularities a the minimum of the second eigenvalue of the singular Yamabe operator over a generalized class of conformal metrics to g and of volume 1. We show that this operator is attained by a generalized metric, we deduce nodal solutions to a Yamabe type equation with singularities.