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

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} -Δ_{M}u-s\langle \nabla_M f, \nabla_M u\rangle_g +c u=λ(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 $\underset{s\to +\infty }{\lim }\lambda (s)$ 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.