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     

Singular limit laminations, Morse index, and positive scalar curvature

For any 3-manifold M³ and any nonnegative integer g, we give here examples of metrics on M each of which has a sequence of embedded minimal surfaces of genus g and without Morse index bounds. On any spherical space form we construct such a metric with positive scalar curvature. More generally, we construct such a metric with Scal>0 (and such surfaces) on any 3-manifold which carries a metric with Scal>0.     

On the prescribed scalar curvature problem on Sn: The degree zero case

In this Note, 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 the well known results due to Y. Li (1995) [10].     

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.     

The scalar curvature equation on

We give existence results for solutions of the prescribed scalar curvature equation on S³, when the curvature function is a positive Morse function and satisfies an index-count condition.     

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.     

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].     

Manifolds of positive scalar curvature and conformal cobordism theory

For a supergoup , we study closed -manifolds with positive conformal classes. We use the relative Yamabe invariant from [2] to define the conformal cobordism relation on the category of such manifolds. We prove that the corresponding conformal cobordism groups are isomorphic to the cobordism groups defined by Stolz in [19]. As a corollary, we show that the conformal concordance relation on positive conformal classes coincides with the standard concordance relation on positive scalar curvature metrics. Our main technical tools come from analysis and conformal geometry. Received: 22 August 2000 / Published online: 5 September 2002     

The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds

Given (M, g) a smooth compact Riemannian N-manifold, N ≥ 2, we show that positive solutions to the problem

Some results for impulsive problems via Morse theory

We use Morse theory to study impulsive problems. First we consider asymptotically piecewise linear problems with superlinear impulses, and prove a new existence result for this class of problems using the saddle point theorem. Next we compute the critical groups at zero when the impulses are asymptotically linear near zero, in particular, we identify an important resonance set for this problem. As an application, we finally obtain a nontrivial solution for asymptotically piecewise linear problems with impulses that are asymptotically linear at zero and superlinear at infinity. Our results here are based on the simple observation that the underlying Sobolev space naturally splits into a certain finite dimensional subspace where all the impulses take place and its orthogonal complement that is free of impulsive effects.     

Bahri-Coron type theorem for the scalar curvature problem on high dimensional spheres

We consider the existence and multiplicity results for the prescribed scalar curvature problem on the standard spheres of high dimension n ?? 7. Given a C ² positive function K, using the theory of critical points at infinity, we prove an existence result as Bahri-Coron theorem. Our case is a generalization of Li (J Differ Equ 120:319?C410, 1995). Indeed, here the function K is flat near some critical points as in Li (J Differ Equ 120:319?C410, 1995) and it can have some nondegenerate critical points with ?? K ?? 0. Furthermore, using some topological arguments, we prove another kind of result.     

Prescribed scalar curvature on then-sphere

This paper considers the prescribed scalar curvature problem onS ⁿ forn>-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 then show that forn=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 scalar curvature problem onS ³.This article was processed by the author using the style filepljourlm from Springer-Verlag.     

The Dirichlet problem for the equation of prescribed scalar curvature in Minkowski space

We prove a maximum principle for the curvature of spacelike admissible solutions of the equation of prescribed scalar curvature in Minkowski space. This enables us to extend to higher dimensions a recent existence result of Bayard for the Dirichlet problem in three and four dimensions. We also prove an interior curvature bound which permits us to prove the existence of locally smooth solutions in the case of spacelike affine boundary data. Uniform convexity of the boundary data is assumed throughout.Received: 11 November 2002, Accepted: 23 January 2003, Published online: 16 May 2003Mathematics Subject Classification (1991): 35J60, 35J65, 53C50John Urbas: Supported by an Australian Research CouncilSenior Research Fellowship     

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.