William Wylie

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

Noncompact Manifolds with Nonnegative Ricci Curvature

Let (M, d) be a metric space. For 0 < r < R, let G(p, r, R) be the group obtained by considering all loops based at a point p ∈ M whose image is contained in the closed ball of radius r and identifying two loops if there is a homotopy between them that is contained in the open ball of radius R. In this article we study the asymptotic behavior of the G(p, r, R) groups of complete open manifolds of nonnegative Ricci curvature. We also find relationships between the G(p, r, R) groups and tangent cones at infinity of a metric space and show that any tangent cone at infinity of a complete open manifold of nonnegative Ricci curvature and small linear diameter growth is its own universal cover.

Gradient shrinking Ricci solitons of half harmonic Weyl curvature

Gradient Ricci solitons and metrics with half harmonic Weyl curvature are two natural generalizations of Einstein metrics on four-manifolds. In this paper we prove that if a metric has structures of both gradient shrinking Ricci soliton and half harmonic Weyl curvature, then except for three examples, it has to be an Einstein metric with positive scalar curvature. Precisely, we prove that a four-dimensional gradient shrinking Ricci soliton with

New restrictions on the topology of extreme black holes

\delta W^{\pm }=0

Comparison geometry for the Bakry-Emery Ricci tensor

δW±=0 is either Einstein, or a finite quotient of

On gradient Ricci solitons with symmetry

S^3\times \mathbb {R}