Pucci eigenvalues on geodesic balls

TL;DR

This paper investigates bounds for Pucci eigenvalues on geodesic balls in Riemannian manifolds, extending classical eigenvalue comparison theorems and analyzing geometric influences on these eigenvalues.

Contribution

It extends Cheng's eigenvalue comparison theorem to Pucci eigenvalues, providing bounds based on curvature and geometric conditions for geodesic balls.

Findings

Cheng's bounds hold for Pucci eigenvalues under curvature constraints.

Pucci eigenvalues are smaller for certain hypersurfaces compared to Euclidean balls.

Bounds depend on sectional and Ricci curvature conditions.

Abstract

We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng's eigenvalue comparison theorem for the Laplace-Beltrami operator. For manifolds with bounded sectional curvature, we prove Cheng's bounds hold for Pucci eigenvalues on geodesic balls of radius less than the injectivity radius. For manifolds with Ricci curvature bounded below, we prove Cheng's upper bound holds for Pucci eigenvalues on certain small geodesic balls. We also prove that the principal Pucci eigenvalues of an $O(n)$-invariant hypersurface immersed in $\mathbb{R}^{n+1}$ with one smooth boundary component are smaller than the eigenvalues of an $n$-dimensional Euclidean ball with the same boundary.