Min-max theory and Yamabe metrics on conical four-manifolds

TL;DR

This paper establishes the existence of Yamabe metrics on four-manifolds with conical singularities using a novel min-max variational approach tailored for singular spaces.

Contribution

It introduces the first min-max scheme for Yamabe metrics on conical four-manifolds, handling singularities by deforming regular bubbles into singular ones while controlling energy.

Findings

Proves existence of Yamabe metrics on conical four-manifolds.

Develops a new min-max method in the singular setting.

Analyzes mass divergence near singular points.

Abstract

We prove existence of Yamabe metrics on four-manifolds possessing finitely-many conical points with $\mathbb{Z}_2$-group, using for the first time a min-max scheme in the singular setting. In our variational argument we need to deform continuously regular bubbles into singular ones, while keeping the Yamabe energy sufficiently low. For doing this, we exploit recent positive mass theorems in the conical setting and study how the mass of the conformal blow-up diverges as the blow-up point approaches the singular set.