Non-existence of Yamabe minimizers on singular spheres

TL;DR

This paper demonstrates the non-existence of Yamabe minimizers on certain singular spheres with large cone angles, and characterizes Yamabe metrics for smaller cone angles, revealing the impact of singularities on geometric optimization.

Contribution

It proves the non-existence of Yamabe minimizers on singular spheres with large cone angles and characterizes Yamabe metrics for smaller angles, advancing understanding of singular geometric structures.

Findings

No Yamabe minimizers exist for cone angles ≥ 4π.

Yamabe metrics for smaller cone angles are obtained by scaling and conformal diffeomorphisms.

Characterization of Yamabe metrics in the presence of edge-cone singularities.

Abstract

We prove that a minimizer of the Yamabe functional does not exist for a sphere $\mathbb{S}^n$ of dimension $n \geq 3$, endowed with a standard edge-cone spherical metric of cone angle greater than or equal to $4\pi$, along a great circle of codimension two. When the cone angle along the singularity is smaller than $2\pi$, the corresponding metric is known to be a Yamabe metric, and we show that all Yamabe metrics in its conformal class are obtained from it by constant multiples and conformal diffeomorphisms preserving the singular set.