Related papers: Four-dimensional Einstein manifolds with Heisenber…

Motivated by the work of Li and Mantoulidis, we study singular metrics which are uniformly Euclidean $(L^\infty)$ on a compact manifold $M^n$ ($n\ge 3$) with negative Yamabe invariant $\sigma(M)$. It is well-known that if $g$ is a smooth…

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…

This work concerns the non-flat metrics on the Heisenberg Lie group of dimension three $\Heis_3(\RR)$ and the bi-invariant metrics on the solvable Lie groups of dimension four. On $\Heis_3(\RR)$ we prove that the property of the metric…

We study existence of invariant Einstein metrics on complex Stiefel manifolds $G/K = \SU(\ell+m+n)/\SU(n) $ and the special unitary groups $G = \SU(\ell+m+n)$. We decompose the Lie algebra $\frak g$ of $G$ and the tangent space $\frak p$ of…

We classify complete curvature homogeneous metrics on simply connected four dimensional manifolds which are invariant under a cohomogeneity one action. We show that they are either isometric to a symmetric space with one of its…

Almost hypercomplex manifolds with Hermitian and Norden metrics and more specially the corresponding quaternionic Kaehler manifolds are considered. Some necessary and sufficient conditions the investigated manifolds be isotropic…

An almost Einstein manifold satisfies equations which are a slight weakening of the Einstein equations; Einstein metrics, Poincare-Einstein metrics, and compactifications of certain Ricci-flat asymptotically locally Euclidean structures are…

With a f-left-invariant Riemannian metric on a Lie group $G$, we mean a Riemannian metric which is conformally equivalent to a left-invariant Riemannian metric, with the conformal factor $f$. In this article, we study the geometry of such…

We make classifications of gradient Ricci solitons $(M, g, f)$ with harmonic Weyl curvature. As a local classification, we prove that the soliton metric $g$ is locally isometric to one of the following four types: an Einstein manifold, the…

Thanks to the ambitious project initiated by Catino, Mastrolia, Monticelli and Rigoli, which aims to provide a unified viewpoint for various geometric solitons, many classes, including Ricci solitons, Yamabe solitons, $k$-Yamabe solitons,…

For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group G with compact, smooth orbit space, we show the following rigidity result: The nilradical N of G acts polarly, and the N-orbits can be…

The four-dimensional gauge group of general relativity corresponds to arbitrary coordinate transformations on a four-manifold. Theories of gravity with a dynamical structure remarkably like Einstein's theory can be obtained on the basis of…

We classify all left-invariant pseudo-Riemannian Einstein metrics on $\mathrm{SL}(2,\mathbb{R})\times \mathrm{SL}(2,\mathbb{R})$ that are bi-invariant under a one-parameter subgroup. We find that there are precisely two such metrics up to…

The purpose of the present expository paper is to give an account of the recent progress and present status of the classification of solvable Lie groups admitting an Einstein left invariant Riemannian metric, the only known examples so far…

Given any compact homogeneous space $H/K$ with $H$ simple, we consider the new space $M=H\times H/\Delta K$, where $\Delta K$ denotes diagonal embedding, and study the existence, classification and stability of $H\times H$-invariant…

Any oriented $4$-dimensional Einstein metric with semi-definite sectional curvature satisfies the pointwise inequality \[ \frac{|s|}{\sqrt{6}}\geq|W^+|+|W^-|, \] where $s$, $W^+$ and $W^-$ are respectively the scalar curvature, the…

We prove that a complete K\"ahler manifold with holomorphic curvature bounded between two negative constants admits a unique complete K\"ahler-Einstein metric. We also show this metric and the Kobayashi-Royden metric are both uniformly…

Seiberg-Witten theory leads to a delicate interplay between Riemannian geometry and smooth topology in dimension four. In particular, the scalar curvature of any metric must satisfy certain non-trivial estimates if the manifold in question…

The twistor space of self-dual positive Einstein manifolds naturally admits two 1-parameter families of Riemannian metrics, one is the family of canonical deformation metrics and the other is the family introduced by B. Chow and D. Yang in…

In this paper, we investigate nilpotent and unimodular solvable Lie groups that admit quasi-Einstein metrics $(M,g,X)$ with $X$ a left-invariant vector field, which we call totally left-invariant quasi-Einstein metrics. We give a complete…