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arXiv · 2311.01342

Gromov-Hausdorff stability of tori under Ricci and integral scalar curvature bounds

Shouhei Honda·Christian Ketterer·Ilaria Mondello·Raquel Perales·Chiara Rigoni
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Abstract

We establish a nonlinear analogue of a splitting map into a Euclidean space, as a harmonic map into a flat torus. We prove that the existence of such a map implies Gromov-Hausdorff closeness to a flat torus in any dimension. Furthermore, Gromov-Hausdorff closeness to a flat torus and an integral bound {on $r_M(x)$, the smallest eigenvalue of the Ricci tensor $\text{ric}_x$ in $x$}, imply the existence of a harmonic splitting map. Combining these results with Stern's inequality, we provide a new Gromov-Hausdorff stability theorem for flat $3$-tori. The main tools we employ include the harmonic map heat flow, Ricci flow, and both Ricci limits and RCD theories.

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Shouhei Honda, Christian Ketterer, Ilaria Mondello, Raquel Perales, Chiara Rigoni. 2023-11-02. Gromov-Hausdorff stability of tori under Ricci and integral scalar curvature bounds. https://arxiv.org/abs/2311.01342

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