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     

The Webster scalar curvature problem on the three dimensional CR manifolds

In this paper, we prove some existence results for the Webster scalar curvature problem on the three dimensional CR compact manifolds locally conformally CR equivalent to the unit sphere S³ of C². Our methods are based on the techniques related to the theory of critical points at infinity.     

Scalar curvature type problem on the three dimensional bounded domain

In this paper we prove an existence result for the nonlinear elliptic problem:-△u = Ku~5,u 0 in Ω,u = 0 on?Ω,where Ω is a smooth bounded domain of R~3 and K is a positive function in Ω.Our method relies on studying its corresponding subcritical approximation problem and then using a topological argument.     

Complete hypersurfaces with constant scalar curvature in spheres

To a given immersion i:Mⁿ? \mathbb Sⁿ⁺¹{i:M^n\to \mathbb S^{n+1}} with constant scalar curvature R, we associate the supremum of the squared norm of the second fundamental form sup |A|². We prove the existence of a constant C _n (R) depending on R and n so that R ≥ 1 and sup |A|² = C _n (R) imply that the hypersurface is a H(r)-torus \mathbb S¹(?{1-r²})×\mathbb S^n-1 (r){\mathbb S^1(\sqrt{1-r^2})\times\mathbb S^{n-1} (r)}. For R > (n − 2)/n we use rotation hypersurfaces to show that for each value C > C _n (R) there is a complete hypersurface in \mathbb Sⁿ⁺¹{\mathbb S^{n+1}} with constant scalar curvature R and sup |A|² = C, answering questions raised by Q. M. Cheng.     

Complete hypersurfaces with constant scalar curvature in spheres

To a given immersion ${i:M^n\to \mathbb S^{n+1}}$ with constant scalar curvature R, we associate the supremum of the squared norm of the second fundamental form sup |A|². We prove the existence of a constant C _n (R) depending on R and n so that R ≥ 1 and sup |A|² = C _n (R) imply that the hypersurface is a H(r)-torus ${\mathbb S^1(\sqrt{1-r^2})\times\mathbb S^{n-1} (r)}$ . For R > (n ? 2)/n we use rotation hypersurfaces to show that for each value C > C _n (R) there is a complete hypersurface in ${\mathbb S^{n+1}}$ with constant scalar curvature R and sup |A|² = C, answering questions raised by Q. M. Cheng.     

Prescribed scalar curvature problem on complete manifolds

Conditions on the geometric structure of a complete Riemannian manifold are given to solve the prescribed scalar curvature problem. In some cases, the conformal metric obtained is complete. A minimizing sequence is constructed which converges strongly to a solution.     

Topological tools for the prescribed scalar curvature problem on S n

In this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere S ⁿ , n ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16].     

Harnack type inequalities for conformal scalar curvature equation

We derive a Harnack type inequality for the conformal scalar curvature equation on B _3R . If the positive scalar curvature function K(x) is sub-harmonic in a neighborhood of each critical point and the maximum of u over B _R is comparable to its maximum over B _3R , then the Harnack type inequality can be obtained. Zhang is supported by NSF-DMS-0600275.     

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.     

A perturbation result for the Webster scalar curvature problem on the CR sphere

We consider the problem of prescribing the Webster scalar curvature on the unit sphere of $\text{C}{}^{\mathrm{n+1}}$. Using a perturbation method, we obtain existence results for curvatures close to a positive constant and satisfying an assumption of Bahri–Coron type.     

Sequences of three dimensional manifolds with positive scalar curvature

We develop two new methods of constructing sequences of manifolds with positive scalar curvature that converge in the Gromov-Hausdorff and Intrinsic Flat sense to limit spaces with “pulled regions”. The examples created rigorously using these methods were announced a few years ago and have influenced the statements of some of Gromov's conjectures concerning sequences of manifolds with positive scalar curvature. Both methods extend the notion of “sewing along a curve” developed in prior work of the authors with Dodziuk to create limits that are pulled string spaces. The first method allows us to sew any compact set in a fixed initial manifold to create a limit space in which that compact set has been scrunched to a single point. The second method allows us to edit a sequence of regions or curves in a sequence of distinct manifolds.     

A gap theorem for hypersurfaces of the sphere with constant scalar curvature one

We consider closed hypersurfaces of the sphere with scalar curvature one, prove a gap theorem for a modified second fundamental form and determine the hypersurfaces that are at the end points of the gap. As an application we characterize the closed, two-sided index one hypersurfaces with scalar curvature one in the real projective space. Received: October 12, 2001     

On the Gauss map of hypersurfaces with constant scalar curvature in spheres

In this work we consider connected, complete and orientable hypersurfaces of the sphere with constant nonnegative -mean curvature. We prove that under subsidiary conditions, if the Gauss image of is contained in a closed hemisphere, then is totally umbilic.

     

A gap theorem on submanifolds with finite total curvature in spheres

We study a complete noncompact submanifold Mⁿ$M^n$ in a sphere S^n+p${\mathbb{S}}^{n+p}$. We prove that there admit no nontrivial L²$L^2$-harmonic 1-forms on M if the total curvature is bounded from above by a constant depending only on n. The gap theorem is a generalized version of Carron?s, Yun?s, Cavalcante?s and the first author?s results on submanifolds in Euclidean spaces and Seo?s result on submanifolds in hyperbolic space without the condition of minimality.     

Rigidity theorem for complete spacelike hypersurfaces with constant scalar curvature in Lorentz spaces

In this paper, a complete space-like hypersurface with constant normal scalar curvature in a locally symmetric Lorentz space is discussed. The rigidity theorem is proved by using the operator □ introduced by S. Y. Cheng and S. T. Yau, and the result is a partially affirmative answer to the question posed by Haizhong Li in 1997.     

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.