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Register a new userPrescribing Ricci curvature on a Product of Spheres
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We prove an existence result for the prescribed Ricci curvature equation for certain doubly warped product metrics on $mathbb{S}^{d_1+1}times mathbb{S}^{d_2}$, where $d_i geq 2$. If $T$ is a metric satisfying certain curvature assumptions, we show that $T$ can be scaled independently on the two factors so as to itself be the Ricci tensor of some metric.
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The notion of the Ricci curvature is defined for sprays on a manifold. With a volume form on a manifold, every spray can be deformed to a projective spray. The Ricci curvature of a projective spray is called the projective Ricci curvature. In this paper, we introduce the notion of projectively Ricci-flat sprays. We establish a global rigidity result for projectively Ricci-flat sprays with nonnegative Ricci curvature. Then we study and characterize projectively Ricci-flat Randers metrics.
Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under this Ricci condition. In particular, two new results, the rigidity result of Bishop-Gromov volume comparison and Lipschitz continuity of heat kernel, are obtained.
Motivated by the recent work of Chu-Lee-Tam on the nefness of canonical line bundle for compact K{a}hler manifolds with nonpositive $k$-Ricci curvature, we consider a natural notion of {em almost nonpositive $k$-Ricci curvature}, which is weaker than the existence of a K{a}hler metric with nonpositive $k$-Ricci curvature. When $k=1$, this is just the {em almost nonpositive holomorphic sectional curvature} introduced by Zhang. We firstly give a lower bound for the existence time of the twisted K{a}hler-Ricci flow when there exists a K{a}hler metric with $k$-Ricci curvature bounded from above by a positive constant. As an application, we prove that a compact K{a}hler manifold of almost nonpositive $k$-Ricci curvature must have nef canonical line bundle.
For $k ge 2,$ let $M^{4k-1}$ be a $(2k{-}2)$-connected closed manifold. If $k equiv 1$ mod $4$ assume further that $M$ is $(2k{-}1)$-parallelisable. Then there is a homotopy sphere $Sigma^{4k-1}$ such that $M sharp Sigma$ admits a Ricci positive metric. This follows from a new description of these manifolds as the boundaries of explicit plumbings.
Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces. We consider Sasakian manifolds with non-negative Tanaka-Webster Ricci curvature equipped with the metric measure space structure defined by the sub-Riemannian metric and the Popp measure. We show that these spaces satisfy the measure contraction property $MCP(0,N)$ for some positive integer $N$. We also show that the same result holds when the Sasakian manifold is equipped with a family of Riemannian metrics extending the sub-Riemannian one.
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