Curvature Diffusion of Inverse-weight Lin--Lu--Yau Ricci Flow on Finite Trees

RESUMO

For the inverse-weight Lin--Lu--Yau Ricci flow on a finite tree, we prove that the curvature satisfies an exact heat equation and that every edge curvature converges. The limiting curvature vector is independent of the positive initial metric and is the unique minimum-norm point of the base polyhedron of a submodular function determined by the tree; in particular, it can be recovered by finitely many combinatorial minimizations. We also characterize subsequential limits of the normalized weights and obtain criteria for their convergence. Finally, for any two positive initial metrics, the corresponding unnormalized edgewise ratios converge, and the normalized omega-limit sets are related by a projective homeomorphism.