Lower bound of Ricci flow's existence time

Let $(M^n, g)$ be a compact $n$-dim ($n\geq 2$) manifold with nonnegative Ricci curvature, and if $n\geq 3$ we assume that $(M^n, g)\times \mathbb{R}$ has nonnegative isotropic curvature. The lower bound of the Ricci flow's existence time on $(M^n, g)$ is proved. This provides an alternative proof for the uniform lower bound of a family of closed Ricci flows' maximal existence times, which was firstly proved by E. Cabezas-Rivas and B. Wilking. We also get an interior curvature estimates for $n= 3$ under $Rc\geq 0$ assumption among others. Combining these results, we proved the short time existence of the Ricci flow on a large class of $3$-dim open manifolds, which admit some suitable exhaustion covering and have nonnegative Ricci curvature.Comment: 13 pages, to appear on Bulletin of the London Mathematical Societ

The growth rate of harmonic functions

We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete Riemannian manifolds with non-negative Ricci curvature and maximal volume growth, further assume the dimension of the manifold is not less than three, we prove that quantitative strong unique continuation yields the existence of nonconstant polynomial growth harmonic functions. Also the uniform bound of frequency for linear growth harmonic functions on such manifolds is obtained, and this confirms a special case of Colding-Minicozzi conjecture on frequency.Comment: to appear in J. Lond. Math. Soc, reference adde

Integral of scalar curvature on non-parabolic manifolds

Using the monotonicity formulas of Colding and Minicozzi, we prove that on any complete, non-parabolic Riemannian manifold $(M^3, g)$ with non-negative Ricci curvature, the asymptotic weighted scaling invariant integral of scalar curvature has an explicit bound in form of asymptotic volume ratio.Comment: to appear in J. Geom. Ana