The scalar curvature problem on the four dimensional half sphere

In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We provide an Euler-Hopf type criterion for a given function to be a scalar curvature to a metric conformal to the standard one. Our proof involves the study of critical points at infinity of the associated variational problem.Received: 2 August 2003, Accepted: 10 May 2004, Published online: 16 July 2004Mathematics Subject Classification (2000): 35J60, 35J20, 58J05K. El Mehdi: elmehdik@ictp.trieste.it     

Prescribing scalar and boundary mean curvature on the three dimensional half sphere

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 some existence and compactness results.     

Nonexistence of complete conformal metrics with prescribed scalar curvatures on a manifold of nonpositive curvature

Let (M,g) be a complete simply-connected Riemannian manifold with nonpositive curvature, k its scalar curvature, and K a smooth function on M. We obtain a nonexistence result of complete metrics on M conformal to g and with K as their scalar curvature.     

Large conformal metrics with prescribed sign-changing Gauss curvature

Let \((M,g)\) be a two dimensional compact Riemannian manifold of genus \(g(M)>1\). Let \(f\) be a smooth function on \(M\) such that

$$\begin{aligned} f \ge 0, \quad f\not \equiv 0, \quad \min _M f = 0. \end{aligned}$$

Let \(p_1,\ldots ,p_n\) be any set of points at which \(f(p_i)=0\) and \(D^2f(p_i)\) is non-singular. We prove that for all sufficiently small \(\lambda >0\) there exists a family of “bubbling” conformal metrics \(g_\lambda =e^{u_\lambda }g\) such that their Gauss curvature is given by the sign-changing function \(K_{g_\lambda }=-f+\lambda ^2\). Moreover, the family \(u_\lambda \) satisfies

$$\begin{aligned} u_\lambda (p_j) = -4\log \lambda -2\log \left( \frac{1}{\sqrt{2}} \log \frac{1}{\lambda }\right) +O(1) \end{aligned}$$

and

$$\begin{aligned} \lambda ^2e^{u_\lambda }\rightharpoonup 8\pi \sum _{i=1}^{n}\delta _{p_i},\quad \text{ as } \lambda \rightarrow 0, \end{aligned}$$

where \(\delta _{p}\) designates Dirac mass at the point \(p\).

     

Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary

This paper considers the prescribed zero scalar curvature and mean curvature problem on the n-dimensional Euclidean ball for n?3. We consider the limits of solutions of the regularization obtained by decreasing the critical exponent. We characterize those subcritical solutions which blow-up at the least possible energy level, determining the points at which they can concentrate, and their Morse indices. We show that when n=3 this is the only blow-up which can occur for solutions. We use this in combination with the Morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed zero scalar curvature and mean curvature on the three-dimensional Euclidean ball. In the higher-dimensional case n?4, we give conditions on the function h to guarantee there is only one simple blow-up point.     

Uniqueness of conformal metrics with prescribed total curvature in R^2

In this paper, we investigate the solution structure of solutions of where K(x) is a H?lder function in . For a given positive total curvature, we consider the problem of the uniqueness of solutions with this prescribed total curvature. We apply various methods such as the method of moving spheres and the isoperimetric inequality to show the uniqueness for several classes of K. Received December 15, 1998 / Accepted April 23, 1999     

The prescribed scalar curvature problem for metrics with unit total volume

We solve the modified Kazdan–Warner problem of finding metrics with prescribed scalar curvature and unit total volume.     

Existence of complete conformal metrics of negative Ricci curvature on manifolds with boundary

We show that on a compact Riemannian manifold with boundary there exists ${u \in C^{\infty}(M)}$ such that, u _|?M ?? 0 and u solves the ?? _k -Ricci problem. In the case k = n the metric has negative Ricci curvature. Furthermore, we show the existence of a complete conformally related metric on the interior solving the ?? _k -Ricci problem. By adopting results of (Mazzeo and Pacard, Pacific J. Math. 212(1), 169?C185 (2003)), we show an interesting relationship between the complete metrics we construct and the existence of Poincaré?CEinstein metrics. Finally we give a brief discussion of the corresponding questions in the case of positive curvature.     

Uniqueness and non-uniqueness of metrics with prescribed scalar and mean curvature on compact manifolds with boundary

Let (Mⁿ,g) be a compact manifold with boundary with n?2. In this paper we discuss uniqueness and non-uniqueness of metrics in the conformal class of g having the same scalar curvature and the mean curvature of the boundary of M.     

Conformal metrics on the unit ball with prescribed mean curvature

This paper focuses on the study of the prescribed mean curvature problem on the unit ball. If the difference between the mean curvature candidate f and mean curvature of the standard metric in the supremum norm is sufficiently small, then the existence of positive solutions of conformal mean curvature equation has been known. The purpose of the paper is to investigate quantitatively how large that difference can be by using a flow method.     

Conformal metrics with prescribed boundary mean curvature on balls

We provide a variety of classes of functions that can be realized as the mean curvature on the boundary of the standard n dimensional ball, n ≥ 3, with respect to some scalar flat metric. Because of the presence of some critical nonlinearity, blow up phenomena occur and existence results are highly nontrivial since one has to overcome topological obstructions. Our approach consists of, on one hand, developing a Morse theoretical approach to this problem through a Morse-type reduction of the associated Euler–Lagrange functional in a neighborhood of its critical points at Infinity and, on the other hand, extending to this problem some topological invariants introduced by A. Bahri in his study of Yamabe type problems on closed manifolds.     

Complete intersections with metrics of positive scalar curvature

We give a complete list of complex projective complete intersections admitting Riemannian metrics of positive scalar curvature. To cite this article: F. Fang, P. Shao, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

The Webster scalar curvature revisited: the case of the three dimensional CR sphere

In this paper we consider the problem of prescribing the Webster scalar curvature on the three CR sphere of ${\mathbb{C}^{2}}$ . We use techniques related to the theory of critical points at infinity, and obtain existence results for curvature satisfying an assumption of Bahri?CCoron type.     

Finsler metrics of scalar flag curvature with special non-Riemannian curvature properties

In this paper, we study Finsler metrics of scalar flag curvature. We find that a non-Riemannian quantity is closely related to the flag curvature. We show that the flag curvature is weakly isotropic if and only if this non-Riemannian quantity takes a special form. This will lead to a better understanding on Finsler metrics of scalar flag curvature.      

一些球对称射影平坦的Finsler度量的构造

研究刻画球对称Finsler度量的射影平坦性质的偏微分方程,通过对射影平坦Finsler度量PDE的研究,构造了两类球对称射影平坦Finsler度量,得到了一些球对称的射影平坦Finsler度量,并进一步给出这些Finsler度量的射影因子和旗曲率.     

Kähler-Einstein metrics with positive scalar curvature

In this paper, we prove that the existence of K?hler-Einstein metrics implies the stability of the underlying K?hler manifold in a suitable sense. In particular, this disproves a long-standing conjecture that a compact K?hler manifold admits K?hler-Einstein metrics if it has positive first Chern class and no nontrivial holomorphic vector fields. We will also establish an analytic criterion for the existence of K?hler-Einstein metrics. Our arguments also yield that the analytic criterion is satisfied on stable K?hler manifolds, provided that the partial C ⁰-estimate posed in [T6] is true. Oblatum 12-IV-1996 & 8-XI-1996