Related papers: The axisymmetric $\sigma_k$-Nirenberg problem

We study the existence of conformal metrics on non-compact Riemannian manifolds with non-compact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature…

We investigate the existence of non-topological solutions $(u_1,u_2)$ satisfying $$u_{i}(x)=-2\beta_i\ln|x|+O(1),\quad\text{as }|x|\rightarrow +\infty,$$ such that $\beta_i>1$ and $$(\beta_1-1)(\beta_2-1)>(N_1+1)(N_2+1),$$ for a…

We prove the existence of a smooth complete conformal metric with prescribed kth elementary symmetric function of negative Ricci curvature under certain condition on general domain in Euclidean space. We then formulate this problem for more…

Exact solutions of Einstein equations with null Riemman-Christoffel curvature tensor everywhere, except on a hypersurface, are studied using quantum particles obeying the Klein-Gordon equation. We consider the particular cases when the…

We study the problem of conformal deformation of Riemannian structure to constant scalar curvature with zero mean curvature on the boundary. We prove compactness for the full set of solutions when the boundary is umbilic and the dimension…

In this note, we prove that for every $0<\sigma<1$, there exists a smooth complete hypersurface $\Sigma$ in $\mathbb{H}^{n+1}$ with prescribed asymptotic boundary $\partial \Sigma=\Gamma$ at infinity, whose principal curvatures…

We study conformally flat surfaces with prescribed Gaussian curvature, described by solutions $u$ of the PDE: $\Delta u(x)+K(x)\exp(2u(x))=0$, with $K(x)$ the Gauss curvature function at $x\in\RR^2$. We assume that the integral curvature is…

This talk is based on a recent paper$^{1}$ of ours. In an attempt to understand three-dimensional conformal field theories, we study in detail one such example --the large $N$ limit of the $O(N)$ non-linear sigma model at its non-trivial…

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…

For each odd sphere $S^n$ with $n=2m+1\ge 5$, we consider the Sasaki volume functional $\mathrm{Vol}^S(V)=\int_{S^n}\sqrt{\det(I+(\nabla V)^\top(\nabla V))}\,d\mathrm{vol}$ on smooth unit tangent vector fields $V$. Using the Gluck--Ziller…

In this paper we study the problem of prescribing a fourth order conformal invariant (the Paneitz curvature) on the $n$-sphere, with $n\geq 5$. Using tools from the theory of critical points at infinity, we provide some topological…

In this paper, we investigate the existence and uniqueness of convex, entire, spacelike hypersurfaces of constant $\sigma_k$ curvature with prescribed set of lightlike directions $\mathcal{F}\subset\mathbb{S}^{n-1}$ and perturbation $q$ on…

Motivated by the prescribing scalar curvature problem, we study the equation $\Delta_g u +Ku^p=0 (1+\zeta \leq p \leq \frac{n+2}{n-2})$ on locally conformally flat manifolds $(M,g)$ with $R(g)=0$. We prove that when $K$ satisfies certain…

This paper is concerned with the existence of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures. Being more specific, given nonnegative smooth functions $K: \overline{\mathbb{D}} \to \mathbb{R}$ and $h: \partial…

In this paper, we study the non-existence of positive solutions for the following conformal $Q$-curvature equation \begin{equation*} (-\Delta)^\sigma u = K(x) u^{\frac{n+2\sigma}{n-2\sigma}} \quad \text{in } \mathbb{R}^n, \end{equation*}…

In hep-th/9910245, Witten and Yau consider the AdS/CFT correspondence in the context of a Riemannian Einstein manifold $M^{n+1}$ of negative Ricci curvature which admits a conformal compactification with conformal boundary $N^n$. They prove…

Given a complete Riemannian metric of nonnegative scalar curvature on $\Sigma \times (-\infty, 0 ] $, where $\Sigma$ denotes a $2$-sphere, we exhibit conditions that imply the existence of a closed minimal surface homologous to the…

Let $(M, g)$ be a compact Riemannian manifold of dimension $N \geq 5$ and $Q_g$ be its $Q$ curvature. The prescribed $Q$ curvature problem is concerned with finding metric of constant $Q$ curvature in the conformal class of $g$. This…

We prove the existence of a smooth complete $3$-convex hypersurface which satisfies prescribed curvature equation $\prod\limits_{i = 1}^n (H - \kappa_i) = \big( (n - 1) \sigma \big)^n$ for $n = 4$ and has prescribed asymptotic boundary…

Let $X$ be a compact connected Riemann surface of genus at least two. The main theorem of arxiv:1010.1488 says that for any positive integer $n \leq 2({\rm genus}(X)-1)$, the symmetric product $S^n(X)$ does not admit any K\"ahler metric…