Curvature estimates for four-dimensional complete gradient expanding Ricci solitons

Abstract

In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci solitons with nonnegative Ricci curvature (outside a compact set $K$). More precisely, we prove that the norm of the curvature tensor $Rm$ and its covariant derivative $\nabla Rm$ can be bounded by the scalar curvature $R$ by $|Rm|\le C_a{R}^a$ and $|\nabla Rm|\le C_a{R}^a$ (on $M\backslash K$), for any $0\le a<1$ and some constant $C_a>0$. Moreover, if the scalar curvature has at most polynomial decay at infinity, then $|Rm|\le CR$ (on $M\backslash K$). As an application, it follows that that if a 4-dimensional complete gradient expanding Ricci soliton $(M^4,g,f)$ has nonnegative Ricci curvature and finite asymptotic scalar curvature ratio then it has finite asymptotic curvature ratio, and $C^{1,\alpha }$ asymptotic cones at infinity ($0<\alpha <1$) according to Chen-Deruelle [20].

arXiv comment20 pages; to appear in Crelle's Journal

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Cited by in corpus (2)

• Complete gradient expanding Ricci solitons with finite asymptotic scalar curvature ratio
• Four-dimensional gradient Ricci solitons with (half) nonnegative isotropic curvature