Compact Manifolds with Positive Einstein Curvature

In this paper we study positive Einstein curvature which is a condition on the Riemann curvature tensor intermediate between positive scalar curvature and positive sectional curvature. We prove some constructions and obstructions for positive Einstein curvature on compact manifolds generalizing similar well known results for the scalar curvature. Finally, because our problem is relatively new, many open questions are included.     

Ricci flow on a 3-manifold with positive scalar curvature

In this paper we consider Hamilton's Ricci flow on a 3-manifold with a metric of positive scalar curvature. We establish several a priori estimates for the Ricci flow which we believe are important in understanding possible singularities of the Ricci flow. For Ricci flow with initial metric of positive scalar curvature, we obtain a sharp estimate on the norm of the Ricci curvature in terms of the scalar curvature (which is not trivial even if the initial metric has non-negative Ricci curvature, a fact which is essential in Hamilton's estimates [R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255-306]), some L²-estimates for the gradients of the Ricci curvature, and finally the Harnack type estimates for the Ricci curvature. These results are established through careful (and rather complicated and lengthy) computations, integration by parts and the maximum principles for parabolic equations.     

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.     

The Ricci flow on the two ball with a rotationally symmetric metric

In this paper we study a boundary-value problem for the Ricci flow in the two-dimensional ball endowed with a rotationally symmetric metric of positive Gaussian curvature and prove short and long time existence results. We construct families of metrics for which the flow uniformizes the curvature along a sequence of times. Finally, we show that if the initial metric has positive Gaussian curvature and the boundary has positive geodesic curvature then the flow uniformizes the curvature along a sequence of times. The text was submitted by the author in English.     

Four-manifolds with positive isotropic curvature

We give a survey on 4-dimensional manifolds with positive isotropic curvature. We will introduce the work of B. L. Chen, S. H. Tang and X. P. Zhu on a complete classification theorem on compact four-manifolds with positive isotropic curvature (PIC). Then we review an application of the classification theorem, which is from Chen and Zhu’s work. Finally, we discuss our recent result on the path-connectedness of the moduli spaces of Riemannian metrics with positive isotropic curvature.     

Strongly positive curvature

We begin a systematic study of a curvature condition (strongly positive curvature) which lies strictly between positive-definiteness of the curvature operator and positivity of sectional curvature, and stems from the work of Thorpe (J Differ Geom 5:113–125, 1971; Erratum. J Differ Geom 11:315, 1976). We prove that this condition is preserved under Riemannian submersions and Cheeger deformations and that most compact homogeneous spaces with positive sectional curvature satisfy it.     

An optimal parabolic estimate and its applications in prescribing scalar curvature on some open manifolds with Ricci \ge 0

By establishing an optimal comparison result on the heat kernel of the conformal Laplacian on open manifolds with nonnegative Ricci curvature, (a) we show that many manifolds with positive scalar curvature do not possess conformal metrics with scalar curvature bounded below by a positive constant; (b) we identify a class of functions with the following property: If the manifold has a scalar curvature in this class, then there exists a complete conformal metric whose scalar curvature is any given function in this class. This class is optimal in some sense; (c) we have identified all manifolds with nonnegative Ricci curvature, which are “uniformly” conformal to manifolds with zero scalar curvature. Even in the Euclidean case, we obtain a necessary and sufficient condition under which the main existence results in [Ni1] and [KN] on prescribing nonnegative scalar curvature will hold. This condition had been sought in several papers in the last two decades. Received: 11 November 1998 / Revised: 7 April 1999     

Minimal surfaces in a unit sphere pinched by intrinsic curvature and normal curvature

We establish a nice orthonormal frame field on a closed surface minimally immersed in a unit sphere Sn, under which the shape operators take very simple forms. Using this frame field, we obtain an interesting property K + K~N= 1 for the Gauss curvature K and the normal curvature K~N if the Gauss curvature is positive. Moreover, using this property we obtain the pinching on the intrinsic curvature and normal curvature, the pinching on the normal curvature, respectively.     

Metrics of positive Ricci curvature on vector bundles over nilmanifolds

We construct metrics of positive Ricci curvature on some vector bundles over tori (or more generally, over nilmanifolds). This gives rise to the first examples of manifolds with positive Ricci curvature which are homotopy equivalent but not homeomorphic to manifolds of non-negative sectional curvature. Submitted: January 2001, Revised: June 2001.     

A priori estimates for the scalar curvature equation on <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

We obtain a priori estimates for solutions to the prescribed scalar curvature equation on S ³. The usual non-degeneracy assumption on the curvature function is replaced by a new condition, which is necessary and sufficient for the existence of a priori estimates, when the curvature function is a positive Morse function.     

Ricci flow on Kähler-Einstein surfaces

In this paper, we prove that if M is a K?hler-Einstein surface with positive scalar curvature, if the initial metric has nonnegative sectional curvature, and the curvature is positive somewhere, then the K?hler-Ricci flow converges to a K?hler-Einstein metric with constant bisectional curvature. In a subsequent paper [7], we prove the same result for general K?hler-Einstein manifolds in all dimension. This gives an affirmative answer to a long standing problem in K?hler Ricci flow: On a compact K?hler-Einstein manifold, does the K?hler-Ricci flow converge to a K?hler-Einstein metric if the initial metric has a positive bisectional curvature? Our main method is to find a set of new functionals which are essentially decreasing under the K?hler Ricci flow while they have uniform lower bounds. This property gives the crucial estimate we need to tackle this problem. Oblatum 8-IX-2000 & 30-VII-2001?Published online: 19 November 2001     

Curvature and conjugate points in Lorentz symmetric spaces

We give some relations between conjugate points and curvature in a locally symmetric Lorentzian manifold. In the compact case, we show that the sectional curvature of timelike planes is non positive, and the lightlike sectional curvature of null planes is non negative. We also compute the lightlike conjugate loci of Cahen–Wallach manifolds, which are an important family of symmetric Lorentzian spaces.     

An exotic sphere with positive curvature almost everywhere

In this article we show that there is an exotic sphere with positive sectional curvature almost everywhere. In 1974 Gromoll and Meyer found a metric of nonnegative sectional on an exotic 7-sphere. They showed that the metric has positive curvature at a point and asserted, without proof, that the metric has positive sectional curvature almost everywhere [4]. We will show here that this assertion is wrong. In fact, the Gromoll-Meyer sphere has zero curvatures on an open set of points. Never the less, its metric can be perturbed to one that has positive curvature almost everywhere.     

Geometrically Formal Homogeneous Metrics of Positive Curvature

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is either symmetric, or a metric on a rational homology sphere.     

Ricci flow on Kähler manifolds

In this Note, we announce the result that if M is a Kähler–Einstein manifold with positive scalar curvature, if the initial metric has nonnegative bisectional curvature, and the curvature is positive somewhere, then the Kähler–Ricci flow converges to a Kähler–Einstein metric with constant bisectional curvature.     

Rotopulsators of the curved N-body problem

We consider the N-body problem in spaces of constant curvature and study its rotopulsators, i.e. solutions for which the configuration of the bodies rotates and changes size during the motion. Rotopulsators fall naturally into five groups: positive elliptic, positive elliptic–elliptic, negative elliptic, negative hyperbolic, and negative elliptic–hyperbolic, depending on the nature and number of their rotations and on whether they occur in spaces of positive or negative curvature. After obtaining existence criteria for each type of rotopulsator, we derive their conservation laws. We further deal with the existence and uniqueness of some classes of rotopulsators in the 2- and 3-body case and prove two general results about the qualitative behaviour of rotopulsators. More precisely, for positive curvature we show that there is no foliation of the 3-sphere with Clifford tori such that the motion of each body is confined to some Clifford torus. For negative curvature, a similar result is proved relative to foliations of the hyperbolic 3-sphere with hyperbolic cylinders.     

Surfaces of positive curvature inE 3 whose characteristic curves form a Tchebychef net

In this paper, it is proved that the surfaces of positive curvature with no umbilical points in 3-dimensional Euclidean space whose characteristic curves form a Tchebychef net are translation surfaces and that the characteristic curves are represented on the unit sphere by a rhombic net. The determination of these surfaces depends on two elliptic integrals of the first kind. Furthermore, the case where these elliptic integrals reduce to elementary integrals is studied and it is shown that the surfaces corresponding to this case belong to one of the following two classes: (a) Translation surfaces of positive curvature with plane characteristic curves as generators lying in two planes intersecting each other under a constant angle. The special case where these planes are perpendicular gives an analogue of the Scherk's minimal surfaces of translation. (b) Translation surfaces of revolution of positive curvature with characteristic curves as generators which are circular helices.     

On the curvature on G-manifolds with finitely many non-principal orbits

We investigate the curvature of invariant metrics on G-manifolds with finitely many non-principal orbits. We prove existence results for metrics of positive Ricci curvature, and discuss some families of examples to which these existence results apply. In fact, many of our examples also admit invariant metrics of non-negative sectional curvature.     

Flats and submersions in non-negative curvature

We find constraints on the extent to which O??Neill??s horizontal curvature equation can be used to create positive curvature on the base space of a Riemannian submersion. In particular, we study when K. Tapp??s theorem on Riemannian submersions of compact Lie groups with bi-invariant metrics generalizes to arbitrary manifolds of non-negative curvature.