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

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.