https://neckpinch.com/authors/luen-fai-tam/New arXiv papers by Luen-Fai Tam, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2406.04667arxiv:2406.04667Fri, 07 Jun 2024 00:00:00 GMTLuen-Fai Tam — Motivated by previous study on mean curvature flow and prescribed mean curvature flow on spatially compact space or asymptotically flat spacetime, in this work we will find sufficient conditions for the short time existence of prescribed mean curvature flow on a Lorentz manifold with a smooth time function starting from a complete noncompact spacelike hypersurface. Long time existence and convergence will also be discussed. Results will be applied to study some prescribed mean curvature flows inside the future of the origin in the Minkowski spacetime. Examples of spacetime related to the existence and convergence results near the future null infinity of the Schwarzschild spacetime are also discussed.Mean curvature flowBoundaries & non-compactLong-time existence & convergenceHarmonic maps & DeTurck trickPhysics https://arxiv.org/abs/2312.10940arxiv:2312.10940Mon, 18 Dec 2023 00:00:00 GMTMan-Chun Lee, Luen-Fai Tam, Jingbo Wan — In this work, we consider the area non-increasing map between manifolds with positive curvature. By exploring the strong maximum principle along the graphical mean curvature flow, we show that an area non-increasing map between certain positively curved manifolds is either homotopy trivial, Riemannian submersion, local isometry or isometric immersion. This implies that an area non-increasing self map of ℂP}ⁿ, n≥ 2 is either homotopically trivial or is an isometry. This confirms a speculation of Tsai-Tsui-Wang. We also use Brendle's sphere Theorem and mean curvature flow coupled with Ricci flow to establish related results on manifolds with positive 1-isotropic curvature.Ricci flowMean curvature flowSphere theoremsStability & uniqueness https://arxiv.org/abs/2207.11017arxiv:2207.11017Fri, 22 Jul 2022 00:00:00 GMTMan-Chun Lee, Luen-Fai Tam — Motivated by the Lipschitz rigidity problem in scalar curvature geometry, we prove that if a closed smooth spin manifold admits a distance decreasing continuous map of non-zero degree to a sphere, then either the scalar curvature is strictly less than the sphere somewhere or the map is a distance isometry. Moreover, the property also holds for continuous metrics with scalar curvature lower bound in some weak sense. This extends a result in the recent work of Cecchini-Hanke-Schick and answers a question of Gromov. The method is based on studying the harmonic map heat flow coupled with the Ricci flow from rough initial data to reduce the case to smooth metrics and smooth maps so that results by Llarull can be applied.Ricci flowScalar curvatureWeak & singular flowsStability & uniquenessHarmonic maps & DeTurck trick https://arxiv.org/abs/2110.07142arxiv:2110.07142Thu, 14 Oct 2021 00:00:00 GMTShaochuang Huang, Luen-Fai Tam — In this work, we obtain a short time existence result for harmonic map heat flow coupled with a smooth family of complete metrics in the domain manifold. Our results generalize short time existence results for harmonic map heat flow by Li-Tam [The heat equation and harmonic maps of complete manifolds, Invent. Math., 1991] and Chen-Zhu [Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geometry, 2006]. In particular, we prove the short time existence of harmonic map heat flow along a complete Ricci flow g(t) on M into a complete manifold with curvature bounded from above with a smooth initial map of uniformly bounded energy density, under the assumptions that |Rm(g(t))|≤ a/t and g(t) is uniformly equivalent to g(0).Ricci flowStability & uniquenessBoundaries & non-compactHarmonic maps & DeTurck trick