Related papers: Unique continuation results for Ricci curvature an…

We investigate the prescribed Ricci curvature problem in the class of left-invariant naturally reductive Riemannian metrics on a non-compact simple Lie group. We obtain a number of conditions for the solvability of the underlying equations…

Suppose $M$ is a manifold with boundary. Choose a point $o\in\partial M$. We investigate the prescribed Ricci curvature equation $\Ric(G)=T$ in a neighborhood of $o$ under natural boundary conditions. The unknown $G$ here is a Riemannian…

It is shown that many results, previously believed to be properties of the Lichnerowicz Ricci curvature, hold for the Ricci curvature of all Gauduchon connections. We prove the existence of $t$--Gauduchon Ricci-flat metrics on the…

Gradient steady Ricci solitons are natural generalizations of Ricci-flat manifolds. In this article, we prove a curvature gap theorem for gradient steady Ricci solitons with nonconstant potential functions; and a curvature gap theorem for…

We prove that metric measure spaces obtained as limits of closed Riemannian manifolds with Ricci curvature satisfying a uniform Kato bound are rectifiable. In the case of a non-collapsing assumption and a strong Kato bound, we additionally…

Typical existence result on Ricci-flat metrics is in manifolds of finite geometry, that is, on $F=\bar F-D$ where $\bar F$ is a compact K\"ahler manifold and $D$ is a smooth divisor. We view this existence problem from a different…

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, positive-semidefinite, nonzero (0,2)-tensor…

Motivated by the long-time behavior of Ricci flows that collapse with bounded curvature, we study expanding Ricci solitons with nilpotent symmetry on vector bundles over a closed manifold. We prove that, under mild assumptions that are…

We present a construction of complete self-dual Einstein metrics of negative scalar curvature on an uncountable family of manifolds of infinite topological type, which are enumerated by continued fraction expansions of irrational numbers.…

We describe the local structure of Riemannian manifolds with harmonic curvature which admit a maximum number, in a well-defined sense, of local warped-product decompositions, and at the same time their Ricci tensor has, at some point, only…

In this paper, we present extensions of the classical Bonnet-Myers theorem for Riemannian manifolds with nonnegative Ricci curvature. Our results provide criteria for compactness and a method for estimating the diameter of such manifolds…

Let $M=G/K$ be a compact homogeneous space and assume that $G$ and $K$ have many simple factors. We show that the topological condition of having maximal third Betti number, in the sense that $b_3(M)=s-1$ if $G$ has $s$ simple factors, so…

We study the prescribed Ricci curvature problem for homogeneous metrics. Given a (0,2)-tensor field $T$, this problem asks for solutions to the equation $\mathrm{Ric}(g)=cT$ for some constant $c$. Our approach is based on examining global…

We classify Einstein metrics on $\mathbb{R}^4$ invariant under a four-dimensional group of isometries including a principal action of the Heisenberg group. The metrics are either Ricci-flat or of negative Ricci curvature. We show that all…

In this paper we prove rigidity results on critical metrics for quadratic curvature functionals, involving the Ricci and the scalar curvature, on the space of Riemannian metrics with unit volume. It is well-known that Einstein metrics are…

We investigate the curvature of invariant metrics on G-manifolds with finitely many non-principal orbits. We prove existence results for metrics of positive Ricci curvature and non-negative sectional curvature, and discuss some families of…

We state that any constant curvature Riemannian metric with conical singularities of constant sign curvature on a compact (orientable) surface $S$ can be realized as a convex polyhedron in a Riemannian or Lorentzian) space-form. Moreover…

We study metric contraction properties for metric spaces associated with left-invariant sub-Riemannian metrics on Carnot groups. We show that ideal sub-Riemannian structures on Carnot groups satisfy such properties and give a lower bound of…

We show that for a complete Ricci shrinker there exists a sequence of points tending to infinity whose norms of the Ricci tensor grow at most linearly.

The linear stability of warped product Einstein metrics as fixed points of the Ricci flow is investigated. We generalise the results of Gibbons, Hartnoll and Pope and show that in sufficiently low dimensions, all warped product Einstein…