共查询到20条相似文献,搜索用时 0 毫秒
1.
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 相似文献
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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. 相似文献
4.
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. 相似文献
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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 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 and where \(\delta _{p}\) designates Dirac mass at the point \(p\).
相似文献
$$\begin{aligned} f \ge 0, \quad f\not \equiv 0, \quad \min _M f = 0. \end{aligned}$$
$$\begin{aligned} u_\lambda (p_j) = -4\log \lambda -2\log \left( \frac{1}{\sqrt{2}} \log \frac{1}{\lambda }\right) +O(1) \end{aligned}$$
$$\begin{aligned} \lambda ^2e^{u_\lambda }\rightharpoonup 8\pi \sum _{i=1}^{n}\delta _{p_i},\quad \text{ as } \lambda \rightarrow 0, \end{aligned}$$
6.
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. 相似文献
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Chang-Shou Lin 《Calculus of Variations and Partial Differential Equations》2000,10(4):291-319
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 相似文献
9.
Shinichiroh Matsuo 《Mathematische Annalen》2014,360(3-4):675-680
We solve the modified Kazdan–Warner problem of finding metrics with prescribed scalar curvature and unit total volume. 相似文献
10.
Matthew Gursky Jeffrey Streets Micah Warren 《Calculus of Variations and Partial Differential Equations》2011,41(1-2):21-43
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. 相似文献
11.
José F. Escobar 《Journal of Functional Analysis》2003,202(2):424-442
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. 相似文献
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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. 相似文献
14.
Wael Abdelhedi Hichem Chtioui Mohameden Ould Ahmedou 《Annals of Global Analysis and Geometry》2009,36(4):327-362
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. 相似文献
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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). 相似文献
16.
Eljazi Salem Najoua Gamara 《Calculus of Variations and Partial Differential Equations》2011,42(1-2):107-136
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. 相似文献
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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.
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19.
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. 相似文献
20.
Gang Tian 《Inventiones Mathematicae》1997,130(1):1-37
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 相似文献