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Chuanhuan Li

Publications and source records attributed to Chuanhuan Li.

9 recordsLinked to original sources
arXiv2026

Total scalar curvature under a curvature operator lower bound

Let $(M^{n}, g)$ be a complete, simply connected Riemannian manifold without boundary, of dimension $n\ge3$, with curvature operator at least that of the unit sphere. We prove that $$\int_M {\rm scal}(x)\ d {\rm Vol}_x\le n(n-1)ω_n,$$ where $ω_n$ 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.

math.DG↗
arXiv2026

A Gap Theorem for the Möbius Cross Energy Near the Hopf Link

We prove that the critical value $2π^2$ is isolated for the Möbius cross energy of two-component links in the round three dimensional sphere $\mathbb{S}^{3}$. Specifically, there exists $\varepsilon_0>0$ such that any non-split, regular $H^2$ pair of curves with disjoint images, having vanishing first variation and energy at most $2π^2+\varepsilon_0$, must lie in the Möbius-reparametrization orbit of the standard Hopf link.

math.DG↗
arXiv2026

Long-time existence of some geometric flows with bounded scalar curvature

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.

math.DG↗
arXiv2026

Curvature pinching estimate under the Laplacian G_{2} flow

In this paper, we derive a pinching estimate on the traceless Ricci curvature in term of scalar curvature and the C^{1} norm of the Weyl tensor under the Laplacian G_{2} flow for closed G_{2} structures. Then we apply this estimate to study the long time existence of the Laplacian G_{2} flow and prove that the C^{1} norm of the Weyl tensor has to blow up at least at a certain rate under bounded scalar curvature.

math.DG↗
arXiv2026

Real analyticity of the modified Laplacian coflow

Let (M,ψ(t))_{t\in[0, T]} be a solution of the modified Laplacian coflow (1.3) with coclosed G_{2}-structures on a compact 7-dimensional M. We improve Chen's Shi-type estimate [5] for this flow, and then show that (M,ψ(t),g_ψ(t)) is real analytic, where g_ψ(t) is the associate Riemannian metric to ψ(t), which answers a question proposed by Grigorian in [13]. Consequently, we obtain the unique-continuation results for this flow.

math.DG↗
arXiv2025

Parabolic frequency monotonicity for two nonlinear equations under Ricci flow

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.

math.DG↗
arXiv2025

Gradient estimates and parabolic frequency under the Laplacian G_2 flow

In this paper, we consider the Laplacian G_2 flow on a closed seven-dimensional manifold M with a closed G_2-structure. We first obtain the gradient estimates of positive solutions of the heat equation under the Laplacian G_2 flow and then we get the Harnack inequality on spacetime. As an application, we prove the monotonicity for positive solutions of the heat equation with bounded Ricci curvature, and get the integral-type Harnack inequality. Besides, we prove the monotonicity of parabolic frequency for positive solutions of the linear heat equation with bounded Bakry-Emery Ricci curvature, and then obtain the backward uniqueness.

math.DG↗
arXiv2022

Parabolic frequency monotonicity on Ricci flow and Ricci-harmonic flow with bounded curvatures

In this paper, we study the monotonicity of parabolic frequency motivated by \cite{frequency on RF} 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.

math.DG↗