Related papers: Prescribing integral curvature equation

This paper deals with the question of prescribing the Gaussian curvature on a disk and the geodesic curvature of its boundary by means of a conformal deformation of the metric. We restrict ourselves to a symmetric setting in which the…

In this paper, we give a sufficient condition for a positive constant scalar curvature metric on a manifold with boundary to be a relative Yamabe metric, which is a natural relative version of the classical Yamabe metric. We also give…

Given a prescription of unparametrised paths on a manifold $M$, one path for each tangent direction, we may ask whether these paths agree with the geodesics of a Riemannian metric on $M$. Generically, this is not the case. Motivated by this…

In this article, we first show that given a smooth function $ S $ either on closed manifolds $ (M, g) $ or compact manifolds $ (\bar{M}, g) $ with non-empty boundary, both for dimensions at least $ 3 $, the condition $ S \equiv 0 $, or $ S…

We study a prescribing functions problem of a conformally invariant integral equation involving Poisson kernel on the unit ball. This integral equation is not the dual of any standard type of PDE. As in Nirenberg problem, there exists a…

Prescribed mean curvature problems on the torus has been considered in one dimension. In this paper, we prove the existence of a graph on the $n$-dimensional torus $\mathbb {T}^n$, the mean curvature vector of which equals the normal…

In a conformal class of metrics with positive Yamabe invariant, we derive a necessary and sufficient condition for the existence of metrics with positive Q curvature. The condition is conformally invariant. We also prove some inequalities…

Let \((M^n,g)\) be a smooth closed Riemannian manifold of dimension \(n \ge 5\) with positive Yamabe invariant and semi-positive \(Q\)-curvature. We establish a precompactness result in the \(C^{\alpha}\)-H\"older topologie on the space of…

This paper develops a technique for applying one-parameter prescribed mean curvature min-max theory in certain non-compact manifolds. We give two main applications. First, fix a dimension $3\le n+1 \le 7$ and consider a smooth function…

In this paper, we consider the problem of existence and multiplicity of conformal metrics on a riemannian compact $4-$dimensional manifold $(M^4,g_0)$ with positive scalar curvature. We prove new exitence criterium which provides existence…

In this paper, we introduce a definition of discrete conformality for triangulated surfaces with flat cone metrics and describe an algorithm for solving the problem of prescribing curvature, that is to deform the metric discrete conformally…

Given a smooth function $f(x)$ on $\mathbb{R}^n$ which is positive somewhere and satisfies $f(x)=O(|x|^{-l})$ for any $l>\frac{n}{2}$, we show that there exists a complete and conformal metric $g=e^{2u}|dx|^2$ with finite total Q-curvature…

We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove…

In this essay we give an introduction to conformal symmetry, based on the example of the Yamabe operator and its use in conformal differential geometry, and in representation theory.

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…

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…

The paper studies the problem of prescribing positive cross curvature on the three-dimensional sphere. We produce several existence results and an example of non-uniqueness, disproving a conjecture of Hamilton's.

In this paper, studying the inverse problem, we establish a curvature compatibility condition on a spherically symmetric Finsler metric. As an application, we characterize the spherically symmetric metrics of scalar curvature. We construct…

Motivated by the theory of isoparametric hypersurfaces, we study submanifolds whose tubular hypersurfaces have some constant "higher order mean curvatures". Here a $k$-th order mean curvature $Q_k$ ($k\geq1$) of a hypersurface $M^n$ is…

In this paper, we obtain the isoperimetric inequality on conformally flat manifold with finite total $Q$-curvature. This is a higher dimensional analogue of Li and Tam's result \cite{L-T} on surfaces with finite total Gaussian curvature.…