https://neckpinch.com/authors/chuanhuan-li/New arXiv papers by Chuanhuan Li, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2609.19851arxiv:2609.19851Thu, 17 Sep 2026 00:00:00 GMTJian Ge, Chuanhuan Li, Ronggang Li — Let (Mⁿ, g) be a complete, simply connected Riemannian manifold without boundary, of dimension n≥3, with curvature operator at least that of the unit sphere. We prove that ∫_M rm scal(x) d rm Volₓ≤ n(n-1)ωₙ, where ωₙ is the volume of the unit n-sphere. Equality holds if and only if (M,g) is isometric to the unit round sphere. In fact, we obtain a stronger bound containing rm Vol(M, g). In even dimensions, the proof follows from the Chern-Gauss-Bonnet formula. In odd dimensions, we apply the corresponding boundary formula to Deruelle's Ricci expander filling.Ricci solitonsScalar curvature https://arxiv.org/abs/2606.10354arxiv:2606.10354Tue, 09 Jun 2026 00:00:00 GMTChuanhuan Li, Yi Li — The analysis of singularities in scalar curvature is a classical problem. In this survey, we investigate the behavior of scalar curvature under several geometric flows, with a focus on three specific cases: the Ricci flow, the Kähler-Ricci flow coupled with (1,1)-forms, and the Laplacian flow.Ricci flowGeneralized & G₂ flowsSingularitiesKähler geometryScalar curvatureLong-time existence & convergence https://arxiv.org/abs/2310.14912arxiv:2310.14912Mon, 23 Oct 2023 00:00:00 GMTChuanhuan Li, Yi Li, Kairui Xu, Jichun Zhu — In this paper, we consider the parabolic frequency for positive solutions of two nonlinear parabolic equations under the Ricci flow on closed manifolds. We obtain the monotonicity of parabolic frequency for the solution of two nonlinear parabolic equations with bounded Ricci curvature, then we apply the parabolic frequency monotonicity to get some integral type Harnack inequalities and we use -K1 instead of the lower bound 0 of Ricci curvature from Theorem 4.3 in 16, where K1 is any positive constant.Ricci flowHarnack inequalities https://arxiv.org/abs/2205.07702arxiv:2205.07702Mon, 16 May 2022 00:00:00 GMTChuanhuan Li, Yi Li, Kairui Xu — In this paper, we study the monotonicity of parabolic frequency motivated by under the Ricci flow and the Ricci-harmonic flow on manifolds. Here we consider two cases: one is the monotonicity of parabolic frequency for the solution of linear heat equation with bounded Bakry-Émery Ricci curvature, and another case is the monotonicity of parabolic frequency for the solution of heat equation with bounded Ricci curvature.Ricci flowLimit spaces & RCD