Related papers: Escobar-Yamabe compactifications for Poincare-Eins…

In this paper we continue an earlier study of ends non-compact manifolds. The over-arching goal is to investigate and obtain generalizations of Siebenmann's famous collaring theorem that may be applied to manifolds having non-stable…

In this paper we show that for an $\text{Sp}(k+1)$ invariant metric $\hat{g}$ on $\mathbb{S}^{4k+3}$ $(k\geq 1)$ close to the round metric, the conformally compact Einstein (CCE) manifold $(M, g)$ with $(\mathbb{S}^{4k+3}, [\hat{g}])$ as…

Let $k\ge1$ be a positive integer and let $P_g$ be the GJMS operator $P_{g}$ of order $2k$ on a closed Riemannian manifold $(M,g)$ of dimension $n>2k$. We investigate the compactness of the set of conformal metrics to $g$ with prescribed…

We provide a general B\"ochner type formula which enables us to prove some rigidity results for $V$-static spaces. In particular, we show that an $n$-dimensional positive static triple with connected boundary and positive scalar curvature…

For a smooth compact Riemannian manifold with positive Yamabe invariant, positive Q curvature and dimension at least 5, we prove the existence of a conformal metric with constant Q curvature. Our approach is based on the study of extremal…

Let $\Omega$ be a domain on the unit $n$-sphere $ \mathbb S^n$ and $\mathring{g}$ the standard metric of $\mathbb S^n$, $n\ge 3$. We show that there exists a conformal metric $g$ with vanishing scalar curvature $R(g)=0$ such that $(\Omega,…

Let $(M,g)$ be an asymptotically hyperbolic manifold with a smooth conformal compactification. We establish a general correspondence between semilinear elliptic equations of scalar curvature type on $\del M$ and Weingarten foliations in…

We introduce new invariants of a Riemannian singular space, the local Yamabe and Sobolev constants, and then go on to prove a general version of the Yamabe theorem under that the global Yamabe invariant of the space is strictly less than…

Let $(M,g)$ be a compact connected spin manifold of dimension $n\geq 3$ whose Yamabe invariant is positive. We assume that $(M,g)$ is locally conformally flat or that $n \in \{3,4,5\}$. According to a positive mass theorem of Witten, the…

Let $ (M, g) $ be a compact manifold or a complete non-compact manifold without boundary, $ \dim M \geqslant 4 $, and not locally conformally flat. In this article, we introduce a new local method to resolve the Yamabe problem on compact…

In principle, global properties of solution of Einstein equations need to be addressed using the conformal Einstein equations, because this conformal compactification allows a clean definition of the `infinities' (spacelike, timelike and…

In this paper we prove classification results to elliptic fully nonlinear conformal equations on certain subdomains of the sphere with prescribed constant mean curvature on its boundary. Such subdomains are the hemisphere (or a geodesic…

Let M be a compact Riemannian manifold of dimension n. The k-curvature, for k=1,2,..n, is defined as the k-th elementary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a…

Let $G$ be a discrete group. The topological category of finite dimensional unitary representations of $G$ is symmetric monoidal under direct sum and has an associated $\mathbb{E}_\infty$-space $\mathcal{K}^{\mathrm{def}}(G)$. We show that…

Let $M^n(n\geq3)$ be an $n$-dimensional compact Riemannian manifold with harmonic curvature and positive scalar curvature. Assume that $M^n$ satisfies some integral pinching conditions. We give some rigidity theorems on compact manifolds…

We consider the Yamabe equation on a complete non-compact Riemannian manifold and study the condition of stability of solutions. If $(M^m,g)$ is a closed manifold of constant positive scalar curvature, which we normalize to be $m(m-1)$, we…

Throughout the history of Einstein manifolds, differential geometers have shown great interest in finding the relationships between curvature and the topology of Einstein manifolds. In the paper, first, we prove that a compact Einstein…

In this paper,we prove the following Myers-type theorem: if $(M^n,g)$, $n\geq 3$, is an n-dimensional complete locally conformally flat Riemannian manifold with bounded Ricci curvature satisfying the Ricci pinching condition $Rc\geq…

We produce some explicit examples of conformally compact Einstein manifolds, whose conformal compactifications are foliated by Riemannian products of a closed Einstein manifold with the total space of a principal circle bundle over products…

The goal of the paper is to introduce a version of Schubert calculus for each dihedral reflection group W. That is, to each "sufficiently rich'' spherical building Y of type W we associate a certain cohomology theory and verify that, first,…