Critical points of the total scalar curvature functional on the space of metrics of constant scalar curvature

It is well known that critical points of the total scalar curvature functional ? on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of ? is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is also Einstein or isometric to a standard sphere. In this paper we prove that n-dimensional critical points have vanishing n− 1 homology under a lower Ricci curvature bound for dimension less than 8. Received: 12 July 1999     

Non-negative scalar curvature

We study topological obstructions to the existence of Riemannian metrics of non-negative scalar curvature on almost spin manifolds using the Dirac operator, the Bochner technique, C ^* algebras and von Neumann algebras. We also derive some obstructions in terms of the eta invariants of Atiyah, Patodi and Singer. Next, we prove vanishing theorems for the Atiyah-Milnor genus. Finally, we derive obstructions to the existence of metrics of non-negative scalar curvature along the leaves of a leafwise non-amenable foliation on a spin manifold.     

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.     

On the scalar curvature of complex surfaces

Supported in part by NSF grant DMS-9204093.     

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.     

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.     

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.     

Compact manifolds of constant scalar curvature

We consider a compact non-negatively curved Riemannian manifold M of constant scalar curvature and obtain a sufficient condition for it to be isometric to a sphere.     

Deformation of scalar curvature and volume

We address the vanishing viscosity limit of the regularized problem studied in Smarrazzo and Tesei [Arch Rat Mech Anal 2012 (in press)]. We show that the limiting points in a suitable topology of the family of solutions of the regularized problem can be regarded as suitably defined weak measure-valued solutions of the original problem. In general, these solutions are the sum of a regular term, which is absolutely continuous with respect to the Lebesgue measure, and a singular term, which is a Radon measure singular with respect to the other. By using a family of entropy inequalities, we prove that the singular term is nondecreasing in time. We also characterize the disintegration of the Young measure associated with the regular term, proving that it is a superposition of two Dirac masses with support on the branches of the graph of the nonlinearity ${\varphi}$ .     

The total mixed curvature of open curves in $$E^3$$

The total mixed curvature of a curve in \(E^3\) is defined as the integral of \(\sqrt{\kappa ^2+\tau ^2}\), where \(\kappa \) is the curvature and \(\tau \) is the torsion. The total mixed curvature is the length of the spherical curve defined by the principal normal vector field. We study the infimum of the total mixed curvature in a family of open curves whose endpoints and principal normal vectors at the endpoints are prescribed. In our previous works, we studied similar problems for the total absolute curvature, which is the length of the spherical curve defined by the unit tangent vector, and for the total absolute torsion, which is the length of the spherical curve defined by the binormal vector.     

On the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we study the behavior of the scalar curvature S of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of S. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, Rigoli, Setti (2005) [17].     

Deformations of the hemisphere that increase scalar curvature

Consider a compact Riemannian manifold M of dimension n whose boundary ?M is totally geodesic and is isometric to the standard sphere S ^n?1. A natural conjecture of Min-Oo asserts that if the scalar curvature of M is at least n(n?1), then M is isometric to the hemisphere $S_{+}^{n}$ equipped with its standard metric. This conjecture is inspired by the positive mass theorem in general relativity, and has been verified in many special cases. In this paper, we construct counterexamples to Min-Oo??s Conjecture in dimension n??3.