Peter Petersen

On the Classification of Gradient Ricci Solitons

We show that the only shrinking gradient solitons with vanishing Weyl tensor and Ricci tensor satisfying a weak integral condition are quotients of the standard ones S n , S n 1 R and R n . This gives a new proof of the Hamilton‐Ivey‐Perelman classification of 3‐dimensional shrinking gradient solitons. We also show that gradient solitons with constant scalar curvature and suitably decaying Weyl tensor when noncompact are quotients of H n , H n 1 R, R n , S n 1 R or S n . 53C25

A radius sphere theorem

The purpose of this paper is to present an optimal sphere theorem for metric spaces analogous to the celebrated Rauch-Berger-Klingenberg Sphere Theorem and the Diameter Sphere Theorem in riemannian geometry. There has lately been considerable interest in studying spaces which are more singular than riemannian manifolds. A natural reason for doing this is because Gromov-Hausdorff limits of riemannian manifolds are almost never riemannian manifolds, but usually only inner metric spaces with various nice properties. The kind of spaces we wish to study here are the so-called Alexandrov spaces. Alexandrov spaces are finite dimensional inner metric spaces with a lower curvature bound in the distance comparison sense. This definition might seem a little ambiguous since there are many ways in which one can define finite dimensionality and lower curvature bounds. The foundational work by Plaut in [PI], however, shows that these different possibilities for definitions are equivalent. The structure of Alexandrov spaces was studied in [BGP], [PI] and [P]. In particular if X is an Alexandrov space and p e X then the space of directions Ep at p is an Alexandrov space of one less dimension and with curvature > 1. Furthermore a neighborhood of p in X is homeomorphic to the linear cone over £„ . One of the important implications of this is that the local structure of n-dimensional Alexandrov spaces is determined by the structure of (n-l)-dimensional Alexandrov spaces with curvature > 1. Sphere theorems in this context seem to be particularly interesting. For if one can give geometric characterizations of

Examples of Riemannian manifolds with positive curvature almost everywhere

We show that the unit tangent bundle of S 4 and a real cohomology CP 3 admit Riemannian metrics with positive sectional curvature almost everywhere. These are the only examples so far with positive curvature almost everywhere that are not also known to admit positive curvature.

Analysis and geometry on manifolds with integral Ricci curvature bounds. II

We extend several geometrical results for manifolds with lower Ricci curvature bounds to situations where one has integral lower bounds. In particular we generalize Colding’s volume convergence results and extend the Cheeger-Colding splitting theorem.

Classification of almost quarter-pinched manifolds

We show that if a simply connected manifold is almost quarter-pinched, then it is diffeomorphic to a CROSS or a sphere.

Aspects of global Riemannian geometry

In this article we survey some of the developments in Riemannian geometry. We place special emphasis on explaining the relationship between curvature and topology for Riemannian manifolds with lower curvature bounds.

The Bochner Technique

Aside from the variational techniques we’ve used in prior sections one of the oldest and most important techniques in modern Riemannian geometry is that of the Bochner technique. In this chapter we prove the classical theorem of Bochner about obstructions to the existence of harmonic 1-forms. We also explain in detail how the Bochner technique extends to forms and other tensors by using Lichnerowicz Laplacians. This leads to a classification of compact manifolds with nonnegative curvature operator in chapter 10 To establish the relevant Bochner formula for forms, we have used a somewhat forgotten approach by Poor. It appears to be quite simple and intuitive. It can, as we shall see, also be generalized to work on other tensors including the curvature tensor.

Spaces on and beyond the boundary of existence

In this note we discuss various questions on whether or not quotients of Riemannian manifolds by Lie groups can be the Gromov-Hausdorff limits of manifolds with certain curvature bounds. In particular we show that any quotient of a manifold by a Lie group is a limit of manifolds with a lower curvature bound; this answers a question posed by Burago, Gromov, and Perelman. On the other hand, we prove that not all such spaces are limits of manifolds with absolute curvature bounds. We also give examples of spaces with curvature ≥1 that are not limits of manifolds with curvature ≥δ > 1/4.

On Frankel's Theorem

In this paper we show that two minimal hypersurfaces in a manifold with positive Ricci curvature must intersect. This is then generalized to show that in manifolds with positive Ricci cur- vature in the integral sense two minimal hypersurfaces must be close to each other. We also show what happens if a manifold with nonnegative Ricci curvature admits two nonintersecting minimal hypersurfaces.

Small Excess and Ricci Curvature

It is proved that a Riemanniann-manifold with Ricci curvature ≥ (n − 1) and a lower injectivity radius bound is a sphere provided the diameter is sufficiently close to π.