Classification of 3-dimensional complete rectifiable steady and expanding gradient Ricci solitons

2019/11/10 by Maeta, Shun
#53C20 #53C21 #53C25 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1911.03902

Abstract

Let (M,g,f) be a 3-dimensional complete steady gradient Ricci soliton. Assume that M is rectifiable, that is, the potential function can be written as f=f(r), where r is a distance function. Then, we prove that M is isometric to (1) a quotient of ℝ³, or (2) the Bryant soliton. In particular, we show that any 3-dimensional complete rectifiable steady gradient Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. Furthermore, we show that any 3-dimensional complete rectifiable expanding gradient Ricci soliton with positive Ricci curvature is rotationally symmetric.

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