Related papers: The Nirenberg problem and its generalizations: A u…

This paper is devoted to establishing some results on the density and multiplicity of solutions to the fractional Nirenberg problem which is equivalent to studying the conformally invariant equation $P_\sigma(v)=K…

We study the problem of prescribing $\sigma_k$-curvature for a conformal metric on the standard sphere $\mathbb{S}^n$ with $2 \leq k < n/2$ and $n \geq 5$ in axisymmetry. Compactness, non-compactness, existence and non-existence results are…

In this paper we study a Nirenberg type problem on standard half spheres $(\mathbb{S}^n_+,g_0)$ consisting of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary $\partial…

We consider the problem of prescribing the $\sigma_k$-curvature on the standard sphere $\mathbb{S}^n$ with $n \geq 3$. We prove existence and compactness theorems when $k \geq n/2$. This extends an earlier result of Chang, Han and Yang for…

In this paper we study the Nirenberg problem on standard half spheres $(\mathbb{S}^n_+,g), \, n \geq 5$, which consists of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary. This…

We propose a new approach to the question of prescribing Gaussian curvature on the 2-sphere with at least three conical singularities and angles less than $2\pi$, the main result being sufficient conditions for a positive function of class…

This paper is devoted to establishing the compactness and existence results of the solutions to the prescribing fractional $Q$-curvatures problem of order $2\sigma$ on $n$-dimensional standard sphere when $ n-2\sigma=2$, $\sigma=1+m/2,$…

In this paper we study the problem of prescribing fractional $Q$-curvature of order $2\sigma$ for a conformal metric on the standard sphere $\Sn$ with $\sigma\in (0,n/2)$ and $n\geq2$. Compactness and existence results are obtained in terms…

Given a smooth positive function $K$ on the standard sphere $(\mathbb{S}^n,g_0)$, we use Morse theoretical methods and counting index formulae to prove that, under generic conditions on the function $K$, there are arbitrarily many metrics…

In this paper we study the problem, posed by Troyanov, of prescribing the Gaussian curvature under a conformal change of the metric on surfaces with conical singularities. Such geometrical problem can be reduced to the solvability of a…

In this paper we perform a refined blow up analysis of finite energy approximated solutions to a Nirenberg type problem on half spheres. The later consists of prescribing, under minimal boundary conditions, the scalar curvature to be a…

We prove some results on the density and multiplicity of positive solutions to the prescribed Webster scalar curvature problem on the $(2n+1)$-dimensional standard unit CR sphere $(\mathbb{S} ^{2n+1},\theta_0)$. Specifically, we construct…

Prescribing $\sigma_k$ curvature equations are fully nonlinear generalizations of the prescribing Gaussian or scalar curvature equations. Given a positive function $K$ to be prescribed on the 4-dimensional round sphere. We obtain asymptotic…

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincar\'e metrics (i.e., complete metrics of constant negative curvature)…

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 establish theorems on the existence and compactness of solutions to the $\sigma_2$-Nirenberg problem on the standard sphere $\mathbb S^2$. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear…

We study a fractional conformal curvature flow on the standard unit sphere and prove a perturbation result of the fractional Nirenberg problem with fractional exponent $\sigma \in (1/2,1)$. This extends the result of Chen-Xu (Invent. Math.…

We first establish local $C^2$ estimates of solutions to the $\sigma_2$-curvature equation with nonlinear Neumann boundary condition. Then, under assumption that the mean curvature of a background metric is nonnegative on totally…

In this paper, we are devoted to establishing the compactness and existence results of the solutions to the fractional Nirenberg problem for $n=3,$ $\sigma=1/2,$ when the prescribing $\sigma$-curvature function satisfies the…

In this paper, the fractional order curvature equation $(-\Delta)^\gamma u = (1 + \varepsilon K(x))u^{\frac{N + 2\gamma}{N - 2\gamma}}$ in $\mathbb{R}^N$ is considered. Assuming $K(x)$ has two critical points satisfying certain local…