We prove directly without using a density theorem that (i) the ADM mass defined in the usual way on an asymptotically flat manifold is equal to the mass defined intrinsically using Ricci tensor; (ii) the Hamiltonian formulation of center of mass and the center of mass defined intrinsically using Ricci tensor are the same.
Conformal Ricci solitons are self similar solutions of the conformal Ricci flow equation. This paper deals with the study of conformal Ricci solitons within the framework of warped product manifolds which extends the notion of usual Riemannian product manifolds. First, we prove that if a warped product manifold admits conformal Ricci soliton then the base and the fiber also share the same property. In the next section the characterization of conformal Ricci solitons on warped product manifolds in terms of Killing and conformal vector fields has been studied. Next, we prove that a warped produc…
Let $X$ be a compact Kähler manifold. We prove that if $X$ admits a smooth Hermitian metric $ω$ with quasi-positive second Chern-Ricci curvature $\mathrm{Ric}^{(2)}(ω)$, then $X$ is projective and rationally connected. In particular, $X$ is simply connected.
讨论了齐次Ricci孤子的几何性质。在证明了紧致均匀孤子和非紧致稳态均匀孤子的不存在性之后，我们集中研究了左不变Ricci孤子。证明了在幺模情形下，Ricci孤子方程在左不变向量场的集合中不存在解。证明了梯度型左不变孤子必是具有非平凡Euclidean de Rham因子的黎曼积。作为结果的应用，我们证明了任何广义度量Heisenberg李群都是非梯度的膨胀型左不变Ricci孤子.
We prove that for a compact 3-manifold M with boundary admitting an ideal triangulation T with valence at least 10 at all edges, there exists a unique complete hyperbolic metric with totally geodesic boundary, so that T is isotopic to a geometric decomposition of M. Our approach is to use a variant of the combinatorial Ricci flow introduced by Luo [Luo05] for pseudo 3-manifolds. In this case, we prove that the extended Ricci flow converges to the hyperbolic metric exponentially fast.
