Conformally flat manifolds with nonnegative Ricci curvature

Abstract

We show that complete conformally flat manifolds of dimension n 3 with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat, or locally isometric to a product of a sphere and a line; or are globally conformally equivalent to R n or to a spherical spaceform S n /Γ. This extends previous results due to Cheng, Noronha, Chen, Zhu and Zhu.

The classification of locally conformally flat Yamabe solitons

“…Our first result establishes the rotational symmetry of locally conformally flat Yamabe solitons. We will show at the end of section 2 that the result in [5] implies that rotationally symmetric complete Yamabe solitons with nonnegative sectional curvature are globally conformally flat, namely g ij = u 4 n+2 dx 2 , where dx 2 denotes the standard metric on R n and u 4 n+2 is the conformal factor. We have the following result.…”

On conformally flat manifolds with constant positive scalar curvature

“…This result, with a pointwise pinching condition on the Ricci curvature, was generalized by many authors (for instance see [12,24,22,29,7] for results and references). In [5] Carron and Herzlich classify complete conformally flat manifolds of dimension n ≥ 3 with non-negative Ricci curvature: they are either flat, or locally isometric to R×S n−1 with the product metric; or are globally conformally equivalent to R n or to a spherical space form. On the other hand, classification of compact conformally flat manifolds satisfying an integral pinching condition were obtained by Gursky [13] and Hebey and Vaugon [15,16].…”

Extinction profile of complete non-compact solutions to the Yamabe flow

“…This soliton also behaves as a cylinder at infinity. It is singular at the origin and therefore by the classification and rigidity result in [7] it has to have somewhere negative Ricci curvature (since it is not flat, not locally isometric to a cylinder, not globally conformally flat and not conformal to a spherical spaceform; the last is true because our soliton is not compact). It is easy to check the completeness of our solution at the origin where V β,B ∼ |x| −α/β as |x| → 0.…”