We introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution to this flow, discuss its convergence and possible applications, and its relation to the Navier-Stokes equations on manifolds and Kazdan-Warner-Bourguignon-Ezin identity for conformal Killing vector fields. We also provide two new criterions on the existence of Killing vector fields. A similar flow to finding holomorphic vector fields on Khler manifolds will be studied by Li and Liu (2014).

We prove, in the setting of a measure energy space (M, ,(, )), that if the smallest eigenvalue 1 () of the generator of the Dirichlet form in any precompact open set M admits the estimate 1 () ()- where is a measure absolutely continuous with respect to and > 0 then a similar estimate holds for the kth smallest eigenvalue: k () const (k/ ()) . As an application, we obtain an upper estimate of the stability index of a minimal surface in [inline-graphic xmlns: xlink=" http://www. w3. org/1999/xlink" xlink: href=" 01i"/] via the total curvature.

We derive a gradient estimate for positive functions, in particular for positive solutions to the heat equation, on finite or locally finite graphs. Unlike the well known Li-Yau estimate, which is based on the maximum principle, our estimate follows from the graph structure of the gradient form and the Laplacian operator. Though our assumption on graphs is slightly stronger than that of Bauer et al. (J Differ Geom 99:359–405, 2015), our estimate can be easily applied to nonlinear differential equations, as well as differential inequalities. As applications, we estimate the greatest lower bound of Cheng’s eigenvalue and an upper bound of the minimal heat kernel, which is recently studied by Bauer et al. (Preprint, 2015) by the Li-Yau estimate. Moreover, generalizing an earlier result of Lin and Yau (Math Res Lett 17:343–356, 2010), we derive a lower bound of nonzero eigenvalues by our gradient estimate.

Let M be a compact n-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary ∂M. Assume that the mean curvature H of the boundary ∂M satisfies H≥(n−1)k>0 for some positive constant k. In this paper, we prove that the distance function d to the boundary ∂M is bounded from above by 1/𝑘 and the upper bound is achieved if and only if M is isometric to an n-dimensional Euclidean ball of radius 1/𝑘.

In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a closed submanifold satisfying a pinching condition in a hyperbolic space form to a round point in finite time.