Asymptotics of the principal eigenvalue of an elliptic operator on closed and orientable Riemannian manifolds

TL;DR

This paper studies the asymptotic behavior of the principal eigenvalue of a specific elliptic operator on closed Riemannian manifolds as a parameter tends to infinity, revealing that the limit depends on the minimum of a potential function over the maxima of a Morse function.

Contribution

It establishes that the limit of the principal eigenvalue as the parameter grows large is determined solely by the minimum of the potential over the maximum points of a Morse function, independent of manifold curvature.

Findings

Limit of eigenvalue depends on minimum of c over maxima of f

Result is independent of the manifold's curvature

Provides asymptotic characterization of eigenvalues for large s

Abstract

This paper investigates the asymptotic behavior of the principal eigenvalue $\lambda(s)$, as $s\to+\infty$, for the following elliptic eigenvalue problem \begin{equation*}\label{E} -\Delta_{M}u-s\langle \nabla_M f, \nabla_M u\rangle_g +c u=\lambda(s)u, \end{equation*} defined on an orientable and closed Riemannian manifold $(M,g)$. Assuming $f$ is a Morse function defined on $M$, we find that the limit $\lim\limits_{s\to+\infty} \lambda(s)$ is determined by the minimum value of the function $c$ over the set of the maximum points of $f$, a result that is independent of the curvature of manifold.