Related papers: A note on the uniqueness of solutions for the Yama…

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…

Let (M,g) be a compact Riemannian manifold of dimension n \geq 3. The Compactness Conjecture asserts that the set of constant scalar curvature metrics in the conformal class of g is compact unless (M,g) is conformally equivalent to the…

In this article, we make a generalization of classical fixed point theorems by using the concept of half-continuity and then apply it to improve the nonuniqueness result for solutions to the vacuum Einstein conformal equations shown by the…

We compute the Yamabe invariants for a new infinite class of closed $4$-dimensional manifolds by using a "twisted" version of the Seiberg-Witten equations, the $\mathrm{Pin}^-(2)$-monopole equations. The same technique also provides a new…

We apply iteration schemes and perturbation methods to provide a complete solution of the boundary Yamabe problem with minimal boundary scenario, or equivalently, the existence of a real, positive, smooth solution of $ -\frac{4(n -1)}{n -…

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. Under some assumptions, we prove that there exists a positive function $\varphi$ solution of the following Yamabe type equation \Delta \varphi+ h\varphi= \tilde h…

A classical theorem in conformal geometry states that on a manifold with non-positive Yamabe invariant, a smooth metric achieving the invariant must be Einstein. In this work, we extend it to the singular case and show that in all…

For a given finite subset $S$ of a compact Riemannian manifold $(M,g)$ whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on $M…

We consider the product of a compact Riemannian manifold without boundary and null scalar curvature with a compact Riemannian manifold with boundary, null scalar curvature and constant mean curvature on the boundary. We use bifurcation…

We provide a full resolution of the Yamabe problem on closed 3-manifolds for Riemannian metrics of Sobolev class $W^{2,q}$ with $q > 3$. This requires developing an elliptic theory for the conformal Laplacian for rough metrics and…

We consider a spinorial Yamabe-type problem on open manifolds of bounded geometry. The aim is to study the existence of solutions to the associated Euler-Lagrange-equation. We show that under suitable assumptions such a solution exists. As…

Given a compact Riemannian manifold, with positive Yamabe quotient, not conformally diffeomorphic to the standard sphere, we prove a priori estimates for solutions to the Yamabe problem. We restrict ourselves to the dimensions less than or…

Given a closed manifold of positive Yamabe invariant and for instance positive Morse functions upon it, the conformally prescribed scalar curvature problem raises the question, whether or not such functions can by conformally changing the…

We consider the Yamabe invariant of a compact orbifold with finitely many singular points. We prove a fundamental inequality for the estimate of the invariant from above, which also includes a criterion for the non-positivity of it.…

In this paper, we consider the scalar curvature of Yamabe solitons. In particular we show that, with natural conditions and non positive Ricci curvature, any complete Yamabe soliton has constant scalar curvature, namely, it is a Yamabe…

We study the set of volumes of constant scalar curvature one metrics on an atoroidal three-manifold.The infinum of this set is believed to be attained at a hyperbolic metric. We prove that the supremum of this set is always infinity. The…

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 provide the classification of locally conformally flat gradient Yamabe solitons with positive sectional curvature. We first show that locally conformally flat gradient Yamabe solitons with positive sectional curvature have to be…

Let M,g a compact Riemannian n-dimensional manifold. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have the boundary of M as a constant mean curvature hypersurface. Also,…

We study two natural problems concerning the scalar and the Ricci curvatures of the Bismut connection. Firstly, we study an analog of the Yamabe problem for Hermitian manifolds related to the Bismut scalar curvature, proving that, fixed a…