闭定向黎曼流形上椭圆算子主特征值的渐近性：小扩散
Asymptotics of the principal eigenvalue of an elliptic operator on closed and orientable Riemannian manifolds: small diffusion
AI总结：
本文研究闭定向黎曼流形上椭圆算子主特征值在小扩散极限下的渐近性，在Morse函数假设下，确定其极限值由函数临界点及对应黎曼Hessian的相关参数完全刻画。
AI中文摘要：
本文研究闭定向黎曼流形$(M,g)$上椭圆特征值问题$-D\nabla_M u - a\nabla_M f,\nabla_M u\rangle_g + c u = \nabla(D)u$的主特征值$\nabla(D)$在小扩散极限$D\to0^+$下的渐近行为。在$f$是$M$上的Morse函数的假设下，我们证明极限值$\nabla(D)$完全由$f$的临界点及相关黎曼Hessian刻画，具体通过临界点处的$c$值和黎曼Hessian特征值实现。
英文摘要：
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.
