$p$-Eigenvalue pinching sphere theorems

TL;DR

This paper proves new sphere theorems based on the first non-zero p-eigenvalue of the p-Laplacian, showing that manifolds with eigenvalues close to those of a sphere are topologically or smoothly spherical.

Contribution

It extends classical sphere theorems to the p-Laplacian setting, providing conditions under which manifolds are homeomorphic or diffeomorphic to spheres based on eigenvalue pinching.

Findings

Manifolds with eigenvalues close to sphere eigenvalues are homeomorphic to spheres.

Manifolds with Ricci curvature bounds and eigenvalue closeness are diffeomorphic to spheres.

Results generalize classical Laplacian sphere theorems to p-Laplacian case.

Abstract

In this paper, we establish two $p$-eigenvalue pinching sphere theorems, for the \( p \)-Laplacian, $p>1$. The first result states that if the first non-zero $p$-eigenvalue of a closed Riemannian $n$-manifold with sectional curvature $K_{M}\geq 1$ is sufficiently close to the first non-zero $p$-eigenvalue of $\mathbb{S}^{n}$ then $M$ is homeomorphic to $\mathbb{S}^{n}$. The second states that if the first non-zero $p$-eigenvalue of a closed Riemannian $n$-manifold with Ricci curvature ${\rm Ric}_{M}\geq (n-1)$ and injectivity radius ${\rm inj}_{M}\geq i_0>0$ is sufficiently close to the first non-zero $p$-eigenvalue of $\mathbb{S}^{n}$ then $M$ is diffeomorphic to $\mathbb{S}^{n}$. Our results extend sphere theorems originally settled for the Laplacian by S. Croke~\cite{Croke1982} and G.P. Bessa~\cite{bessa} respectively.