Non-uniqueness and symmetries for the Nirenberg problem using computer assistance

TL;DR

This paper uses verified numerical methods to prove the existence and symmetry properties of solutions to the Nirenberg problem, confirming predictions and identifying solution symmetries exactly.

Contribution

It introduces a computer-assisted proof approach to establish solution existence and symmetry classification for the Nirenberg problem.

Findings

Proved the existence of a solution near two approximate solutions.

Determined the symmetry groups of the solutions exactly.

Confirmed a predicted solution existence that was previously unproven.

Abstract

We apply verified numerics to the Nirenberg problem, proving that a genuine solution exists near two given computer-generated approximate solutions. This proves existence of a solution for a particular prescribed curvature that was previously predicted, but not proved, to exist. We are also able to determine the symmetry groups of the genuine solutions exactly, which in one case is different from the symmetry of the prescribed curvature. We expect the computer code for this proof can be reused to study other aspects of the Nirenberg problem.