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

Abstract

This paper is concerned with the asymptotic behavior of the principal eigenvalue $λ(D)$ of the elliptic eigenvalue problem \[ -DΔ_{M}u - a\langle \nabla_M f, \nabla_M u\rangle_g + c u = λ(D)u, \] posed on a closed orientable Riemannian manifold $(M,g)$, in the small-diffusion limit $D \to 0^+$. Under the assumption that $f$ is a Morse function on $M$, we establish that the limiting value $\lim_{D\to 0}λ(D)$ is completely characterized by the critical points of $f$ and the associated Riemannian Hessian, specifically through the values of $c$ and the Riemannian Hessian eigenvalues at those points.