184 research outputs found
Lower bound of Ricci flow's existence time
Let be a compact -dim () manifold with nonnegative
Ricci curvature, and if we assume that
has nonnegative isotropic curvature. The lower bound of the Ricci flow's
existence time on 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 under assumption among
others. Combining these results, we proved the short time existence of the
Ricci flow on a large class of -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 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
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