Related papers: The Conformal Laplacian and The Kazdan-Warner Prob…

We give sufficient and "almost" necessary conditions for the prescribed scalar curvature problems within the conformal class of a Riemannian metric $ g $ for both closed manifolds and compact manifolds with boundary, including the…

In this article, we give results of prescribing scalar and mean curvature functions for metrics either pointwise conformal or conformally equivalent to a Riemannian metric that is equipped on a compact manifold with boundary, with…

In this article, we first show that for all compact Riemannian manifolds with non-empty smooth boundary and dimension at least 3, there exists a metric, pointwise conformal to the original metric, with constant scalar curvature in the…

In this article, we show that (i) any smooth function on compact Riemann surface with non-empty smooth boundary $ (M, \partial M, g) $ can be realized as a Gaussian curvature function; (ii) any smooth function on $ \partial M $ can be…

Let $(M^n,g),~n\ge 3$ be a noncompact complete Riemannian manifold with compact boundary and $f$ a smooth function on $\partial M$. In this paper we show that for a large class of such manifolds, there exists a metric within the conformal…

Let $(M, g)$ be a compact Riemannian manifold with boundary $\partial M$. Given a function $f$ on $\partial M$, we consider the problem of finding a conformal metric of $g$ with zero scalar curvature in $M$ and prescribed mean curvature $f$…

Let $M$ be a compact connected surface with boundary. We prove that the signal condition given by the Gauss-Bonnet theorem is necessary and sufficient for a given smooth function $f$ on $\partial M$ (resp. on $M$) to be geodesic curvature…

In this paper we prove that, given a compact four dimensional smooth Riemannian manifold (M,g) with smooth boundary there exists a metric conformal to g with constant T-curvature, zero Q-curvature and zero mean curvature under generic and…

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…

On a closed Riemannian surface $(M,\bar g)$ with negative Euler characteristic, we study the problem of finding conformal metrics with prescribed volume $A>0$ and the property that their Gauss curvatures $f_\lambda= f + \lambda$ are given…

We employ three different methods to prove the following result on prescribed scalar curvature plus mean curvature problem: Let $(M^n,g_0)$ be a $n$-dimensional smooth compact manifold with boundary, where $n \geq 3$, assume the conformal…

Let (M,g) be a compact Riemannian manifold with boundary. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We prove that this…

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…

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension $n\geq 3$. We prove the existence of such…

Let $\Omega$ be a compact surface with smooth boundary and the geodesic curvature $k_g \ge {c > 0}$ along $\partial \Omega$ for some constant $c \in \mathbb{R}$. We prove that, if the Gaussian curvature satisfies $K \ge -\alpha$ for a…

We consider the case with boundary of the classical Kazdan-Warner problem in dimension greater or equal than three, i.e. the prescription of scalar and boundary mean curvatures via conformal deformations of the metric. We deal in particular…

Given an smooth function $K <0$ we prove a result by Berger, Kazhdan and others that in every conformal class there exists a metric which attains this function as its Gaussian curvature for a compact Riemann surface of genus $g>1$. We do so…

A theorem of Escobar asserts that, on a positive three dimensional smooth compact Riemannian manifold with boundary which is not conformally equivalent to the standard three dimensional ball, a necessary and sufficient condition for a $C^2$…

Suppose $(M,g_0)$ is a compact Riemannian manifold without boundary of dimension $n\geq 3$. Using the Yamabe flow, we obtain estimate for the first nonzero eigenvalue of the Laplacian of $g_0$ with negative scalar curvature in terms of the…

This article uses the iterative schemes and perturbation methods to completely solve the Han-Li conjecture, i.e. the general boundary Yamabe problem with prescribed constant scalar curvature and constant mean curvature on compact manifolds…