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

Weyl formula and thermodynamics of geometric flow