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arXiv · 2408.14375

Rigidity theorems for the area widths of Riemannian manifolds

Lucas Ambrozio·Fernando C. Marques·André Neves
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Abstract

The volume spectrum of a compact Riemannian manifold is a sequence of critical values for the area functional, defined in analogy with the Laplace spectrum by Gromov. In this paper we prove that the canonical metric on the two-dimensional projective plane is determined modulo isometries by its volume spectrum. We also prove that the surface Zoll metrics on the three-dimensional sphere are characterized by the equality of the spherical area widths. These widths generalize to the surface case the Lusternik-Schnirelmann lengths of closed geodesics. We prove a new sharp area systolic inequality for metrics on the three-dimensional projective space.

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Lucas Ambrozio, Fernando C. Marques, André Neves. 2024-08-26. Rigidity theorems for the area widths of Riemannian manifolds. https://arxiv.org/abs/2408.14375

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