We study the Weyl formula for the asymptotic number of eigenvalues of the Laplace-Beltrami operator with Dirichlet boundary condition on a Riemannian manifold in the context of geometric flows. Assuming the eigenvalues to be the energies of some associated statistical system, we show that geometric flows are directly related with the direction of increasing entropy chosen. For a closed Riemannian manifold we obtain a volume preserving flow of geometry being equivalent to the increment of Gibbs entropy function derived from the spectrum of Laplace-Beltrami operator. Resemblance with Arnowitt, Deser, and Misner (ADM) formalism of gravity is also noted by considering open Riemannian manifolds, directly equating the geometric flow parameter and the direction of increasing entropy as time direction.

翻译：我们研究在几何流背景下，具有Dirichlet边界条件的Laplace-Beltrami算子在黎曼流形上特征值渐近数量的Weyl公式。假设特征值为相关统计系统的能量，我们证明几何流与所选取的熵增方向存在直接关联。对于闭黎曼流形，我们得到了体积保持的几何流，该流等价于由Laplace-Beltrami算子谱导出的Gibbs熵函数的增量。通过考虑开黎曼流形，我们注意到与Arnowitt-Deser-Misner（ADM）引力形式体系的相似性，直接建立了几何流参数与作为时间方向的熵增方向之间的等价关系。