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 $|Rm| \le C R$ (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 $C^{1,α}$ asymptotic cones at infinity ($0 < α< 1$) according to Chen-Deruelle [20].

Theorems & Definitions (40)

• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Theorem 1.2
• Remark 1.3
• Lemma 2.1
• Lemma 2.2
• Remark 2.1
• Lemma 2.3
• Remark 2.2