Computationally-assisted proof of a novel O(3)× O(10)-invariant Einstein metric on S¹²

2024/12/02 by Timothy Buttsworth, Buttsworth, Timothy, Liam Hodgkinson +1
Mathematics · Physics and Astronomy · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Black Holes and Theoretical Physics

paper · pdf · doi:10.48550/arxiv.2412.01184

Abstract

We prove existence of a non-round Einstein metric g on S¹² that is invariant under the usual cohomogeneity one action of O(3)\timesO(10) on S¹²⊂ ℝ¹³= ℝ³⊕ ℝ¹⁰. The proof involves using several rigorous numerical analysis techniques to produce a Riemannian metric g which approximately satisfies the Einstein condition to known high precision, and then demonstrating that g can be perturbed into a true Einstein metric g.

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