https://neckpinch.com/authors/jiawei-liu/New arXiv papers by Jiawei Liu, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2410.08667arxiv:2410.08667Fri, 11 Oct 2024 00:00:00 GMTJiawei Liu, Miles Simon — In this paper we study n-dimensional Ricci flows (Mⁿ,g(t))_(t∈ [0,T)), where T< ∞ is a potentially singular time, and for which the spatial L^p norm, p>n/2, of the scalar curvature is uniformly bounded on [0,T). In the case that M is closed and four dimensional, we explain why non-collapsing estimates hold and how they can be combined with integral bounds on the Ricci and full curvature tensor of the prequel paper of the authors, as well as non-inflating estimates (already known due to works of Bamler), to obtain an improved space time integral bound of the Ricci curvature. As an application of these estimates, we show that if we further restrict to n=4, then the solution convergences to an orbifold as t → T and that the flow can be extended using the Orbifold Ricci flow to the time interval [0,T+σ) for some σ>0. We also prove local versions of many of the results mentioned above.Ricci flow(Non)collapsing4-manifoldsScalar curvature https://arxiv.org/abs/2406.08778arxiv:2406.08778Thu, 13 Jun 2024 00:00:00 GMTJiawei Liu, Shiyu Zhang, Xi Zhang — In this paper, we show the regularity and uniqueness of the twisted conical Kähler-Ricci flow running from a positive closed current with zero Lelong number, which extends the regularizing property of the smooth twisted Kähler-Ricci flow, known as Guedj-Zeriahi's existence theorem and Di Nezza-Lu's uniqueness theorem, to the conical singularity case.Ricci flowKähler–Ricci flowSingularitiesKähler geometryStability & uniqueness https://arxiv.org/abs/2406.04591arxiv:2406.04591Fri, 07 Jun 2024 00:00:00 GMTXishen Jin, Jiawei Liu — In this paper, we consider the stability of the generalized Lagrangian mean curvature flow of graph case in the cotangent bundle, which is first defined by Smoczyk-Tsui-Wang. By new estimates of derivatives along the flow, we weaken the initial condition and remove the positive curvature condition in Smoczyk-Tsui-Wang's work. More precisely, we prove that if the graph induced by a closed 1-form is a special Lagrangian submanifold in the cotangent bundle of a Riemannian manifold, then the generalized Lagrangian mean curvature flow is stable near it.Mean curvature flowLagrangian mean curvature flowStability & uniqueness https://arxiv.org/abs/2406.04590arxiv:2406.04590Fri, 07 Jun 2024 00:00:00 GMTJiawei Liu, Xi Zhang — In this paper, we study the limit behavior of the conical Kähler-Ricci flow as its cone angle tends to zero. More precisely, we prove that as the cone angle tends to zero, the conical Kähler-Ricci flow converges to a unique Kähler-Ricci flow, which is smooth outside the divisor and admits cusp singularity along the divisor.Ricci flowKähler–Ricci flowSingularitiesKähler geometry https://arxiv.org/abs/2406.02351arxiv:2406.02351Tue, 04 Jun 2024 00:00:00 GMTJiawei Liu, Miles Simon — In this paper we prove localised weighted curvature integral estimates for solutions to the Ricci flow in the setting of a smooth four dimensional Ricci flow or a closed n-dimensional Kähler Ricci flow. These integral estimates improve and extend the integral curvature estimates shown by the second author in an earlier paper. If the scalar curvature is uniformly bounded in the spatial L^p sense for some p>2, then the estimates imply a uniform bound on the spatial L² norm of the Riemannian curvature tensor. Stronger integral estimates are shown to hold if one further assumes a weak non-inflating condition, or we restrict to closed manifolds.Ricci flowKähler–Ricci flow4-manifoldsKähler geometryScalar curvature