https://neckpinch.com/authors/tianbo-liu/New arXiv papers by Tianbo Liu, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2203.13178/arxiv:2203.13178Thu, 24 Mar 2022 00:00:00 GMTHuai-Dong Cao, Tianbo Liu, Junming Xie — Let (Mⁿ, g, f), n≥ 5, be a complete gradient expanding Ricci soliton with nonnegative Ricci curvature Rc≥ 0. In this paper, we show that if the asymptotic scalar curvature ratio of (Mⁿ, g, f) is finite (i.e., limsup_(r→ ∞) R r²< ∞), then the Riemann curvature tensor must have at least sub-quadratic decay, namely, limsup_(r→ ∞) |Rm| r^α< ∞ for any 0<α<2.Ricci solitonsScalar curvature https://neckpinch.com/papers/2111.09848/arxiv:2111.09848Thu, 18 Nov 2021 00:00:00 GMTHuai-Dong Cao, Tianbo Liu — 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 ∇Rm can be bounded by the scalar curvature R by |Rm|≤ Cₐ R^a and |∇Rm| ≤ Cₐ R^a (on Mbackslash K), for any 0≤ a <1 and some constant Cₐ >0. Moreover, if the scalar curvature has at most polynomial decay at infinity, then |Rm| ≤ C R (on Mbackslash K). As an application, it follows that that if a 4-dimensional complete gradient expanding Ricci soliton (M⁴, 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].Ricci solitons4-manifoldsScalar curvature