https://neckpinch.com/authors/xiang-dong-li/New arXiv papers by Xiang-Dong Li, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2511.23330arxiv:2511.23330Fri, 28 Nov 2025 00:00:00 GMTXiang-Dong Li, Qi Yan — In this paper, we prove a Li-Yau-Hamilton type Harnack estimate for the f-mean curvature flow in Euclidean space, which can be viewed as a gradient flow of the weighed area functional with the measure density function e^(−f).Mean curvature flowHarnack inequalities https://arxiv.org/abs/2511.23074arxiv:2511.23074Fri, 28 Nov 2025 00:00:00 GMTXiang-Dong Li, Qi Yan — In this paper, we study Perelman' s 𝓦 entropy for mean curvature flow in ℝⁿ⁺¹. Analogously to Perelman's 𝓦-entropy defined for Ricci flow, K. Ecker in defined a functional 𝓦 for the mean curvature flow in ℝⁿ⁺¹ and the region it encloses, and made the conjecture that this functional is monotonically increasing in time. We modify K. Ecker's definition and, using Hamilton's Harnack inequality for mean curvature flow, prove that our redefined 𝓦-entropy is monotonically decreasing in time. Additionally, we provide a rigidity theorem for this 𝓦-entropy.Ricci flowMean curvature flowHarnack inequalitiesStability & uniqueness https://arxiv.org/abs/2510.09383arxiv:2510.09383Fri, 10 Oct 2025 00:00:00 GMTQi Yan, Xiang-Dong Li — We are concerned with a stochastic mean curvature flow of graphs with extra force over a periodic domain of any dimension. Based on compact embedding method of variational SPDE, we prove the existence of martingale solution. Moreover, we derive the small perturbation limit of the stochastic weighted mean curvature flow.Mean curvature flow https://arxiv.org/abs/2505.03202arxiv:2505.03202Tue, 06 May 2025 00:00:00 GMTXiang-Dong Li — In this paper, we extend Perelman's W-entropy formula and the concavity of the Shannon entropy power from smooth Ricci flow to super Ricci flows on metric measure spaces. Moreover, we prove the Li-Yau-Hamilton-Perelman Harnack inequality on super Ricci flows. As a significant application, we prove the equivalence between the volume non-local collapsing property and the lower boundedness of the W-entropy on RCD(0, N) spaces. Finally, we use the W-entropy to study the logarithmic Sobolev inequality with optimal constant on super Ricci flows on metric measure spaces.Ricci flow(Non)collapsingHarnack inequalitiesWeak & singular flowsLimit spaces & RCD