$p$-Eigenvalue pinching sphere theorems
Paulo Henryque C. Silva
TL;DR
The paper proves two pinching sphere theorems for the first nonzero eigenvalue of the nonlinear $p$-Laplacian on closed $n$-manifolds with curvature bounds. It combines Schwarz symmetrization, a Polya–Szegö-type energy comparison, and a $p$-Laplacian version of Croke’s isoperimetric framework to relate eigenvalue pinching to diameter, then invokes the Grove–Shiohama diameter sphere theorem and, under Ricci and injectivity-radius hypotheses, a sphere-diffeomorphism result due to Bessa. Concretely, if $K_M \\ge 1$ and $C(n,p)\\lambda_{1,p}( \\mathbb{S}^{n}) \\ge \\lambda_{1,p}(M)$, then $M$ is homeomorphic to $\\mathbb{S}^{n}$; if ${\\rm Ric}_M \\ge (n-1)$ and ${\\rm inj}_M \\ge i_0>0$, the same pinching yields that $M$ is diffeomorphic to $\\mathbb{S}^{n}$. These results extend the classical Laplacian sphere theorems of Croke and Bessa to the nonlinear $p$-Laplacian, highlighting the sharp interplay between nonlinear spectral data and global geometry.
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.