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

This paper is concerned with the asymptotic behavior of the principal eigenvalue $\lambda (D)$ of the elliptic eigenvalue problem $$-D{\mathrm{\Delta }}_{M}u-a⟨{\mathrm{\nabla }}_{M}f,{\mathrm{\nabla }}_{M}u{⟩}_g+cu=\lambda (D)u,$$ 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 ${\lim }_{D\to 0}\lambda (D)$ 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.