Related papers: Solutions to the singular $\sigma_2-$Yamabe proble…

The Yamabe problem in compact closed Riemannian manifolds is concerned with finding a metric with constant scalar curvature in the conformal class of a given metric. This problem was solved by the combined work of Yamabe, Trudinger, Aubin,…

In this paper we produce families of complete non compact Riemannian metrics with positive constant $\sigma_k$-curvature by performing the connected sum of a finite number of given $n$-dimensional Delaunay type solutions, provided $2 \leq…

We continue our previous work studying critical exponent semilinear elliptic (and subelliptic) problems which generalize the classical Yamabe problem. In [3] the focus was on metric-measure spaces with an `almost smooth' structure, with…

In this article, we give nonexistence and nonuniqueness results for the vacuum Einstein conformal constraint equations in the far-from-CMC case and also show that in some cases the equations of the conformal method for positive Yamabe…

A Calabi-Yau orbifold is locally modeled on C^n/G where G is a finite subgroup of SL(n, C). In dimension n=3 a crepant resolution is given by Nakamura's G-Hilbert scheme. This crepant resolution has a description as a GIT/symplectic…

Let (M,g) be a compact Riemannian manifold with boundary. This paper addresses the Yamabe-type problem of finding a conformal scalar-flat metric on M, which has the boundary as a constant mean curvature hypersurface. When the boundary is…

We study existence and uniqueness results for the Yamabe problem on non-compact manifolds of negative curvature type. Our first existence and uniqueness result concerns those such manifolds which are asymptotically locally hyperbolic. In…

Let $(M^n, g)$ and $(X^m, h)$ be closed manifolds $m, n>2$, such that $(X, h)$ has constant positive scalar curvature. We consider the one parameter family of products $(M\times X, g+\epsilon^2 h)$, $\epsilon>0$. We prove that if either the…

In this paper, we investigate the prescribed scalar curvature problem on a non-compact Riemannian manifold $(M, \langle \, , \, \rangle)$, namely the existence of a conformal deformation of the metric $\langle \, , \, \rangle$ realizing a…

In this article, we investigate deformation problems of $Q$-curvature on closed Riemannian manifolds. One of the most crucial notions we use is the $Q$-singular space, which was introduced by Chang-Gursky-Yang during 1990's. Inspired by the…

In this article the degenerate warped products of singular semi-Riemannian manifolds are studied. They were used recently by the author to handle singularities occurring in General Relativity, in black holes and at the big-bang. One main…

Conformally St{\"a}ckel manifolds can be characterized as the class of n-dimensional pseudo-Riemannian manifolds (M, G) on which the Hamilton-Jacobi equation G($\nabla$u, $\nabla$u) = 0 for null geodesics and the Laplace equation --$\Delta$…

An orbit-like foliation is a singular foliation on a complete Riemannian manifold $M$ whose leaves are locally equidistant (i.e., a singular Riemannian foliation) and (transversely) infinitesimally homogenous. This class of singular…

In this paper, we apply various methods to establish the uniqueness of solutions to some classes of anisotropic and isotropic curvature problems. Firstly, by employing integral formulas derived by S. S. Chern \cite{Ch59}, we obtain the…

We concern $C^2$-compactness of the solution set of the boundary Yamabe problem on smooth compact Riemannian manifolds with boundary provided that their dimensions are $4$, $5$ or $6$. By conducting a quantitative analysis of a linear…

We study both analytically and numerically a coupled system of spherically symmetric SU(2) Yang-Mills-dilaton equation in 3+1 Minkowski space-time. It has been found that the system admits a hidden scale invariance which becomes transparent…

We consider a compact pseudo-hermitian manifold (M,\theta, J), that is a manifold equipped with a contact form \theta and CR structure J. We consider a conformal deformation of the contact form to obtain a complete, singular contact form…

We prove that in conformal classes of metrics near the class of an Einstein metric (other than the standard round metric on a sphere) the Yamabe problem has a unique solution up to scaling. This is a local extension, in the space of…

It is shown that on every closed oriented Riemannian 4-manifold $(M,g)$ with positive scalar curvature, $$\int_M|W^+_g|^2d\mu_{g}\geq 2\pi^2(2\chi(M)+3\tau(M))-\frac{8\pi^2}{|\pi_1(M)|},$$ where $W^+_g$, $\chi(M)$ and $\tau(M)$ respectively…

We show that sets of conformal data on closed manifolds with the metric in the positive or zero Yamabe class, and with the gradient of the mean curvature function sufficiently small, are mapped to solutions of the Einstein constraint…