In this note, we prove the sharp Davies-Gaffney-Grigor’yan lemma for minimal heat kernels on graphs.

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We consider an ancient solution g(·, t) of the Ricci flow on a compact surface that exists for t 2 (−1, T ) and becomes spherical at time t = T . We prove that the metric g(·, t) is either a family of contracting spheres, which is a type I ancient solution, or a King–Rosenau solution, which is a type II ancient solution.

Let M be a complete Riemannian manifold with Riemannian volume volg and f be a smooth function on M. A sharp upper bound estimate on the first eigenvalue of symmetric diffusion operator 􏰂 f = 􏰂 − ∇ f · ∇ was given by Wu (J Math Anal Appl 361:10–18, 2010) and Wang (Ann Glob Anal Geom 37:393–402, 2010) under a condition that finite dimensional Bakry–Émery Ricci curvature is bounded below, independently. They propounded an open problem is whether there is some rigidity on the estimate. In this note, we will solve this problem to obtain a splitting type theorem, which generalizes Li–Wang’s result in (J Differ Geom 58:501–534, 2001, J Differ Geom 62:143–162, 2002). For the case that infinite dimensional Bakry–Emery Ricci curvature of M is bounded below, we donotexpectanyupperboundestimateonthefirsteigenvalueof􏰂f withoutanyadditional assumption (see the example in Sect. 2). In this case, we will give a sharp upper bound estimate on the first eigenvalue of􏰂f under the additional assuption that |∇f| is bounded.We also obtain the rigidity result on this estimate, as another Li–Wang type splitting theorem.

The LaplaceBeltrami operator of a smooth Riemannian manifold is determined by the Riemannian metric. Conversely, the heat kernel constructed from the eigenvalues and eigenfunctions of the LaplaceBeltrami operator determines the Riemannian metric. This work proves the analogy on Euclidean polyhedral surfaces (triangle meshes), that the discrete heat kernel and the discrete Riemannian metric (unique up to a scaling) are mutually determined by each other. Given a Euclidean polyhedral surface, its Riemannian metric is represented as edge lengths, satisfying triangle inequalities on all faces. The LaplaceBeltrami operator is formulated using the cotangent formula, where the edge weight is defined as the sum of the cotangent of angles against the edge. We prove that the edge lengths can be determined by the edge weights unique up to a scaling using the variational approach.