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.     

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.     

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 conformal metrics with prescribed scalar curvature on the four dimensional half sphere

In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the four dimensional half sphere. Using dynamical and topological methods involving the study of the critical points at infinity of the associated variational structure, we prove some existence results like Bahri-Coron theorem. Furthermore, we consider the approximate subcritical problem and we construct some solutions which blow up at two different points, one of them lay on the boundary and the other one is an interior point.     

Riemannian metrics of positive scalar curvature on compact manifolds with boundary

We prove that any metric of positive scalar curvature on a manifold X extends to the trace of any surgery in codim > 2 on X to a metric of positive scalar curvature which is product near the boundary. This provides a direct way to construct metrics of positive scalar curvature on compact manifolds with boundary. We also show that the set of concordance classes of all metrics with positive scalar curvature on S ⁿ is a group.     

Conformal metrics with prescribed nonpositive Gaussian curvature on {\mathbb R}^2

In this paper, we consider the equation where is a nonpositive function in . A solution u is said to be complete if the conformal metric is complete in . Let Assuming only that , we prove that equation (0.1) possesses infinitely many complete solutions. If in addition, K is assumed to satisfy for some positive constant m, then is also necessary for equation (0.1) to have a complete solution with finite total curvature. We are also able to classify the solution set of equation (0.1) for a wider class of the curvature function K than those considered in [5, 6]. Received October 1, 1997 / Revised version August 10, 1999 / Published online April 6, 2000     

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].     

Conformal metrics with prescribed Q-curvature on S n

In this paper we prescribe a fourth order conformal invariant on the standard n-sphere, with n????5, and study the related fourth order elliptic equation. We prove new existence results based on a new type of Euler?CHopf type formula. Our argument gives an upper bound on the Morse index of the obtained solution. We also give a lower bound on the number of conformal metrics having the same Q-curvature.     

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.     

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.     

H-bubbles with prescribed large mean curvature

Given a smooth function , we call H-bubble a conformally immersed surface in ³ parametrized on the sphere ² with mean curvature H at every point. We prove that if is a nondegenerate stationary point for H with , then there exists a curve of embedded H-bubbles, defined for large, which become round and concentrate at as . Also the case of topologically stable extremal points for H is considered.Work supported by M.U.R.S.T. progetto di ricerca Metodi variazionali ed equazioni differenziali nonlineari (cofin. 2002) Mathematics Subject Classification (2000):53A10 (49J10)     

Hypersurfaces with constant scalar curvature and constant mean curvature

Non-spherical hypersurfaces inE ⁴ with non-zero constant mean curvature and constant scalar curvature are the only hypersurfaces possessing the following property: Its position vector can be written as a sum of two non-constant maps, which are eigenmaps of the Laplacian operator with corresponding eigenvalues the zero and a non-zero constant.