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John Man Shun Ma

Publications and source records attributed to John Man Shun Ma.

11 recordsLinked to original sources
arXiv2026

Comparison and Rigidity Theorems for geodesic curvatures in two dimensional Alexandrov spaces

In this work, we study geodesic curvature of the boundary of a two dimensional Alexandrov space of curvature bounded below (CBB). We prove several comparison and globalization theorems for the geodesic curvature, generalizing the known results for curves in space of curvature bounded above (CBA) by Alexander and Bishop (Differ. Geom. Appl. 6, No. 1, 67-86 (1996)). We also prove a rigidity theorem for boundary with corners and geodesic curvature lower bound. This generalizes the known rigidity result by Grove and Peterson (Geom. Topol. 26 (4) 1635 - 1668, (2022)) in 2d.

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arXiv2024

Backward Uniqueness of Extrinsic Geometric Flow in general ambient manifolds

In this paper we prove two backward uniqueness theorems for extrinsic geometric flow of possibly non-compact hypersurfaces in general ambient complete Riemannian manifolds. These are applicable to a wide range of extrinsic geometric flow, including the mean curvature flow, inverse mean curvature flow, Gauss curvature flow and so on.

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arXiv2023

Entropy Bounds for Self-shrinkers with Symmetries

In this work we derive explicit entropy bounds for two classes of closed self-shrinkers: the class of embedded closed self-shrinkers recently constructed in arXiv:2207.04851 using isoparametric foliations of spheres, and the class of compact non-spherical immersed rotationally symmetric self-shrinkers. These bounds generalize the entropy bounds found in arXiv:2202.08641 on the space of complete embedded rotationally symmetric self-shrinkers.

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arXiv2022

Entropy Bounds, Compactness and Finiteness Theorems for Embedded Self-shrinkers with Rotational Symmetry

In this work, we study the space of complete embedded rotationally symmetric self-shrinking hypersurfaces in $\mathbb{R}^{n+1}$. First, using comparison geometry in the context of metric geometry, we derive explicit upper bounds for the entropy of all such self-shrinkers. Second, as an application we prove a smooth compactness theorem on the space of all such shrinkers. We also prove that there are only finitely many such self-shrinkers with an extra reflection symmetry.

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arXiv2022

Rigidity results for mean curvature flow graphical translators moving in non-graphical direction

In this paper, we study the rigidity results of complete graphical translating hypersurfaces when the translating direction is not in the graphical direction. We proved that any entire graphical translating surface in the translating direction not parallel to the graphical one is flat if either the translating surface is mean convex or the entropy of the translating surface is smaller than $2$. For higher dimensional case, we show that the same conclusion holds if the graphical translating hypersurface satisfies certain growth condition.

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arXiv2017

Parabolic Omori-Yau maximum principle for mean curvature flow and some applications

We derive a parabolic version of Omori-Yau maximum principle for a proper mean curvature flow when the ambient space has lower bound on $\ell$-sectional curvature. We apply this to show that the image of Gauss map is preserved under a proper mean curvature flow in euclidean spaces with uniform bounded second fundamental forms. This generalizes the result of Wang \cite{Wang} for compact immersions. We also prove a Omori-Yau maximum principle for properly immersed self-shrinkers, which improves a result in \cite{CJQ}.

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arXiv2016

Geometry of Lagrangian self-shrinking tori and applications to the Piecewise Lagrangian Mean Curvature Flow

We study geometric properties of the Lagrangian self-shrinking tori in $\mathbb R^4$. When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Łojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori and then combining with the compactness theorem in \cite{CMa}. When the area bound is small, we show that any Lagrangian self-shrinking torus in $\mathbb R^4$ with small area is embedded with uniform curvature estimates, and the space of such tori is compact. Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori in $\mathbb R^4$, along which the Lagrangian condition is preserved, area is decreasing, and the type I singularities that are compact with a fixed area upper bound can be perturbed away in finite steps. This is a Lagrangian version of the construction for embedded surfaces in $\mathbb R^3$ in \cite{CM}.

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arXiv2014

The space of compact self-shrinking solutions to the Lagrangian Mean Curvature Flow in $\mathbb C^2$

Let $F_n :(Σ, h_n) \to \mathbb C^2$ be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound to the mean curvature flow, and suppose that the sequence of metrics $\{h_n\}$ converges smoothly to a Riemannian metric $h$. We show that a subsequence of $\{F_n\}$ converges smoothly to a branched conformally immersed Lagrangian self-shrinker $F_\infty : (Σ, h)\to \mathbb C^2$. When the area bound is less than $16π$, the limit $F_\infty$ is an embedded torus. When the genus of $Σ$ is one, we can drop the assumption on convergence $h_n\to h$. When the genus of $Σ$ is zero, we show that there is no branched immersion of $Σ$ as a Lagrangian shrinker, generalizing the rigidity result of Smoczyk in dimension two by allowing branch points.

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