First Dirichlet eigenvalue and exit time moment spectra comparisons

TL;DR

This paper establishes bounds and comparison theorems for eigenvalues and moment spectra of geodesic balls in Riemannian manifolds, linking geometric curvature controls to spectral properties and characterizing equality cases.

Contribution

It provides explicit bounds for the Poisson hierarchy, moment spectra, and torsional rigidity, and introduces a characterization of eigenvalue equality through moment spectrum comparisons.

Findings

Derived bounds for Poisson hierarchy and moment spectra.

Proved a Dirichlet eigenvalue comparison theorem.

Characterized equality cases via moment spectrum sequences.

Abstract

We prove explicit upper and lower bounds for the Poisson hierarchy, the averaged $L^1$-moment spectra $\{\dfrac{\mathcal{A}_k\left(B_R^M\right)}{\text{vol}\left(S_R^M\right)}\}_{k=1}^\infty$, and the torsional rigidity $\mathcal{A}_1(B^M_R)$ of a geodesic ball $B^M_R$ in a Riemannian manifold $M^n$ which satisfies that the mean curvatures of the geodesic spheres $S^M_r$ included in it, (up to the boundary $S^M_R$), are controlled by the radial mean curvature of the geodesic spheres $S^\omega_r(o_\omega)$ with same radius centered at the center $o_\omega$ of a rotationally symmetric model space $M^n_\omega$. As a consecuence, we prove a first Dirichlet eigenvalue $\lambda_1(B^M_R)$ comparison theorem and show that equality with the bound $\lambda_1(B^\omega_R(o_\omega))$, (where $B^\omega_r(o_\omega)$ is the geodesic $r$-ball in $M^n_\omega$), characterizes the $L^1$-moment spectrum $\{\mathcal{A}_k(B^M_R)\}_{k=1}^\infty$ as the sequence $\{\mathcal{A}_k(B^\omega_R)\}_{k=1}^\infty$ and vice-versa.