Related papers: Poincare-Einstein metrics and the Schouten tensor
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…
On a closed manifold, consider the space of all Riemannian metrics for which -Delta + kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature…
In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact, aspherical Riemannian manifold (M,g) is compact. Established in the locally conformally flat case by Schoen [43,44] and for n\leq…
We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds $(M,g)$ when the conformal boundary $\pl M$ has dimension $n$ even. Its definition depends on the choice of metric $h_0$ on $\partial M$ in the…
We develop methods for constructing and computing conformal invariants of submanifolds, with a particular emphasis on conformal submanifold scalars and conformally invariant integrals of natural submanifold scalars. These methods include a…
We prove that a certain class of ALE spaces always has a Kahler conformal compactification, and moreover provide explicit formulas for the conformal factor and the Kahler potential of said compactification. We then apply this to give a new…
In the first part of this thesis, we study the Yamabe problem with singularities, that we can announce as follow: Given a compact Riemannian manifold $(M,g)$, find a constant scalar curvature metric, conformal to $g$, when $g$ has not…
For a conformally compact Poincar\'{e}-Einstein manifold $(X,g_+)$, we consider two types of compactifications for it. One is $\bar{g}=\rho^2g_+$, where $\rho$ is a fixed smooth defining function; the other is the adapted (including…
In this dissertation, we prove a number of results regarding the conformal method of finding solutions to the Einstein constraint equations. These results include necessary and sufficient conditions for the Lichnerowicz equation to have…
In this paper we show that for a Berger metric $\hat{g}$ on $S^3$, the non-positively curved conformally compact Einstein metric on the $4$-ball $B_1(0)$ with $(S^3, [\hat{g}])$ as its conformal infinity is unique up to isometries and it is…
We obtain some rigidity results for metrics whose Schouten tensor is bounded from below after conformal transformations. Liang Cheng recently proved that a complete, nonflat, locally conformally flat manifold with Ricci pinching condition…
Let \((M^n,g)\) be a smooth closed Riemannian manifold of dimension \(n \ge 5\) with positive Yamabe invariant and semi-positive \(Q\)-curvature. We establish a precompactness result in the \(C^{\alpha}\)-H\"older topologie on the space of…
Let $(M^{n+1},g_+)$ be an asymptotically hyperbolic manifold. We compute the Cheeger constant of conformally compact asymptotically constant mean curvature submanifolds $ \iota : Y^{k+1} \to (M^{n+1},g_+)$ with arbitrary codimension. As an…
In this paper, we study constant scalar curvature K\"ahler (cscK) metrics on complete non-compact K\"ahler--Einstein manifolds. We give sufficient conditions under which a cscK perturbation of a K\"ahler--Einstein metric must remain…
This paper describes the connection between scattering matrices on conformally compact asymptotically Einstein manifolds and conformally invariant objects on their boundaries at infinity. The conformally invariant powers of the Laplacian…
In this expository article we review the problem of finding Einstein metrics on compact K\"ahler manifolds and Sasaki manifolds. In the former half of this article we see that, in the K\"ahler case, the problem fits better with the notion…
Our aim in this paper is to study local rigidity for metrics defined on a compact manifold $M$ with boundary satisfying constant scalar curvature on $M$ and constant mean curvature on $\partial M$. We present some geometrical hypotheses…
On a given compact complex manifold or orbifold $(M,J)$, we study the existence of Hermitian metrics $\tilde g$ in the conformal classes of K\"ahler metrics on $(M,J)$, such that the Ricci tensor of $\tilde g$ is of type $(1,1)$ with…
We describe and partially solve a natural Yamabe-type problem on smooth metric measure spaces which interpolates between the Yamabe problem and the problem of finding minimizers for Perelman's $\nu$-entropy. This problem reduces in all…
In this article, we provide a discussion on a composite class of exact static spherically symmetric vacuum solutions of Einstein's equations. We construct the composite solution of Einstein field equation by match the interior vacuum metric…