Computationally-assisted proof of a novel $\mathsf{O}(3)\times \mathsf{O}(10)$-invariant Einstein metric on $S^{12}$

TL;DR

This paper demonstrates the existence of a new non-round Einstein metric on the 12-sphere, invariant under a specific symmetry group, using computational methods to rigorously approximate and perturb towards an exact solution.

Contribution

It provides the first computational proof of a non-round Einstein metric on $S^{12}$ with $ ext{O}(3) imes ext{O}(10)$ symmetry, combining numerical analysis with geometric perturbation techniques.

Findings

Existence of a non-round Einstein metric on $S^{12}$ proven.

Numerical methods successfully approximated the Einstein condition.

Perturbation techniques confirmed the approximate metric can be made exact.

Abstract

We prove existence of a non-round Einstein metric $g$ on $S^{12}$ that is invariant under the usual cohomogeneity one action of $\mathsf{O}(3)\times\mathsf{O}(10)$ on $S^{12}\subset \mathbb{R}^{13}= \mathbb{R}^3\oplus \mathbb{R}^{10}$. The proof involves using several rigorous numerical analysis techniques to produce a Riemannian metric $\hat{g}$ which approximately satisfies the Einstein condition to known high precision, and then demonstrating that $\hat{g}$ can be perturbed into a true Einstein metric $g$.