The prescribed cross curvature problem on the three-sphere

TL;DR

This paper investigates the problem of prescribing positive cross curvature on the three-sphere, providing existence results, an example of non-uniqueness, and disproving a conjecture by Hamilton.

Contribution

It offers new existence results for the prescribed cross curvature problem and presents a counterexample to Hamilton's conjecture on uniqueness.

Findings

Existence results for prescribed positive cross curvature

An example demonstrating non-uniqueness

Disproof of Hamilton's conjecture

Abstract

The paper studies the problem of prescribing positive cross curvature on the three-dimensional sphere. We produce several existence results and an example of non-uniqueness, disproving a conjecture of Hamilton's.