Related papers: Nonuniqueness for a fully nonlinear boundary Yamab…

We present a general existence proof for a wide class of non-linear elliptic equations which can be applied to problems with barrier conditions without specifying any assumptions guaranteeing the uniqueness or local uniqueness of particular…

On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. We prove that this result may be localized to compact…

In this paper we prove that every Riemannian metric on a locally conformally flat manifold with umbilic boundary can be conformally deformed to a scalar flat metric having constant mean curvature. This result can be seen as a generalization…

We consider the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work \cite{DKSaU}, it was shown that a metric in a…

We use the equivariant $\mu$-bubbles technique to prove that for any compact manifold $M^n$ with non-empty boundary, $n\in\{3,5,6\}$, the Yamabe invariant of $M^n$ is positive if and only if the Yamabe invariant of $M^n\times S^1$ is…

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…

Let $(M,g)$ be a closed Riemannian manifold of dimension $n\geq 3$ and $x_0 \in M$ be an isolated local minimum of the scalar curvature $s_g$ of $g$. For any positive integer $k$ we prove that for $\epsilon >0$ small enough the subcritical…

We use PDE methods as developed for the Liouville equation to study the existence of conformal metrics with prescribed singularities on surfaces with boundary, the boundary condition being constant geodesic curvature. Our first result shows…

The problem of prescribing conformally the scalar curvature on a closed Riemannian manifold of negative Yamabe invariant is always solvable, when the function $K$ to be prescribed is strictly negative, while sufficient and necessary…

We present a boundary integral method for solving a certain class of Riemann-Hilbert problems known as the general conjugation problem. The method is based on a uniquely solvable boundary integral equation with the generalized Neumann…

The study of the $k$-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called $\sigma_k$ curvature, has produced many fruitful results in conformal geometry in recent years. In these…

A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study whether these metrics have negative Ricci curvatures. Affirmatively, we prove…

We prove some rigidity results for compact manifolds with boundary. In particular for a compact Riemannian manifold with nonnegative Ricci curvature and simply connected mean convex boundary, it is shown that if the sectional curvature…

In this paper, we study the weak compactness of the set of conformal metrics in any Riemann surface without boundary whose Calabi energy and area are uniformly bounded. We prove that for any sequence of such metrics, there alwasy exists a…

We study the set of curvature functions which a given compact manifold with boundary can possess. First, we prove that the sign demanded by the Gauss-Bonnet Theorem is a necessary and sufficient condition for a given function to be the…

Let $G=(V,E)$ be a locally finite graph, $\Omega\subset V$ be a bounded domain, $\Delta$ be the usual graph Laplacian, and $\lambda_1(\Omega)$ be the first eigenvalue of $-\Delta$ with respect to Dirichlet boundary condition. Using the…

We extend the study of the vacuum Einstein constraint equations on manifolds with ends of cylindrical type initiated by Chru\'sciel and Mazzeo by finding a class of solutions to the fully coupled system on such manifolds. We show that given…

We initiate the study of an analogue of the Yamabe problem for complex manifolds. More precisely, fixed a conformal Hermitian structure on a compact complex manifold, we are concerned in the existence of metrics with constant Chern scalar…

In this article, we consider Einstein-type manifolds with boundary which generalizes important geometric equations, like static vacuum and static perfect fluid. We investigate some geometric inequalities for those manifolds. Then, we…

We study the boundary behaviors of a complete conformal metric which solves the $\sigma_k$-Ricci problem on the interior of a manifold with boundary. We establish asymptotic expansions and also $C^1$ and $C^2$ estimates for this metric…