The paper considers a manifold $M$ evolving under the Ricci flow and
establishes a series of gradient estimates for positive solutions of the heat
equation on $M$. Among other results, we prove Li-Yau-type inequalities in this
context. We consider both the case where $M$ is a complete manifold without
boundary and the case where $M$ is a compact manifold with boundary.
Applications of our results include Harnack inequalities for the heat equation
on $M$.Comment: 21 pages, 2 figure
Let G be a compact connected Lie group and H a closed subgroup of G. Suppose the homogeneous space G/H is effective and has dimension 3 or higher. Consider a G-invariant, symmetric, positivesemidefinite, nonzero (0,2)-tensor field T on G/H. Assume that H is a maximal connected Lie subgroup of G. We prove the existence of a G-invariant Riemannian metric g and a positive number c such that the Ricci curvature of g coincides with cT on G/H. Afterwards, we examine what happens when the maximality hypothesis fails to hold.
Consider a compact Lie group G and a closed subgroup H < G. Suppose M is the set of Ginvariant Riemannian metrics on the homogeneous space M = G/H. We obtain a sufficient condition for the existence of g ∈ M and c > 0 such that the Ricci curvature of g equals cT for a given T ∈ M. This condition is also necessary if the isotropy representation of M splits into two inequivalent irreducible summands. Immediate and potential applications include new existence results for Ricci iterations.
Let M be a domain enclosed between two principal orbits on a cohomogeneity one manifold M 1 . Suppose that T and R are symmetric invariant (0, 2)-tensor fields on M and ∂ M, respectively. The paper studies the prescribed Ricci curvature equation Ric(G) = T for a Riemannian metric G on M subject to the boundary condition G ∂ M = R (the notation G ∂ M here stands for the metric induced by G on ∂ M). Imposing a standard assumption on M 1 , we describe a set of requirements on T and R that guarantee global and local solvability.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.