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mean curvature flow of Arbitrary Codimension in Complex Projective Spaces

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Chinese Annals of Mathematics,Series B 2023年第6期44卷 857-892页

作者： Li LEI Hongwei XU School of Mathematical Sciences Chongqing Normal UniversityChongqing 401331China. Center of Mathematical Sciences Zhejiang UniversityHangzhou 310027China

Recently,Pipoli and Sinestrari[Pipoli,G.and Sinestrari,C.,mean curvature flow of pinched submanifolds of CPn,Comm.Anal.Geom.,25,2017,799-846]initiated the study of convergence problem for the mean curvature flow of small codimension in the complex projective space CPm.The purpose of this paper is to develop the work due to Pipoli and Sinestrari,and verify a new convergence theorem for the mean curvature flow of arbitrary codimension in the complex projective space.Namely,the authors prove that if the initial submanifold in CPm satisfies a suitable pinching condition,then the mean curvature flow converges to a round point in finite time,or converges to a totally geodesic submanifold as t→∞.Consequently,they obtain a differentiable sphere theorem for submanifolds in the complex projective space.

关键词： mean curvature flow Submanifolds of arbitrary codimension Complex projective space Convergence theorem Differentiable sphere theorem

A STABILITY RESULT FOR TRANSLATINGSPACELIKE GRAPHS IN LORENTZ MANIFOLDS

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Acta Mathematica Scientia 2024年第2期44卷 474-483页

作者：高雅毛井吴传喜 Faculty of Mathematics and Statistics Key Laboratory of Applied Mathematics of Hubei ProvinceHubei UniversityWuhan430062China

In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.

关键词： mean curvature flow spacelike graphs translating spacelike graphs maximal spacelike graphs constant mean curvature Lorentz manifolds

Translating Surfaces of the Non-parametric mean curvature flow in Lorentz Manifold M^(2)×R^(*)

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Chinese Annals of Mathematics,Series B 2021年第2期42卷 297-310页

作者： Li CHEN Dan-Dan HU Jing MAO Ni XIANG Faculty of Mathematics and Statistics Key Laboratory of Applied Mathematics of Hubei ProvinceHubei UniversityWuhan 430062China

In this paper, for the Lorentz manifold M^(2)× R with M^(2) a 2-dimensional complete surface with nonnegative Gaussian curvature, the authors investigate its spacelike graphs over compact, strictly convex domains in M^(2), which are evolving by the nonparametric mean curvature flow with prescribed contact angle boundary condition, and show that solutions converge to ones moving only by translation.

关键词： Translating surfaces mean curvature flow Lorentz manifolds

Two-Dimensional Graphs Moving by mean curvature flow

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Acta Mathematica Sinica,English Series 2002年第2期18卷 209-224页

作者： CHEN Jing Yi Department of Mathematics.The University of British Columbia.Vancouver.B.C..Canada V6T 1Z2 E-mail:jychen@math.ubc.caLI Jia Yu Institute of Mathematics.Academy of Mathematics and System Sciences.Chinese Academy of Sciences.Beijing 100080.P.R.China Department of Mathematics.Fudan University.Shanghai 200433.P.R.China E-mail:lijia@math03.math.ac.cnTIAN Gang Department of Mathematics,MIT.Cambridge.MA 02139.U.S.A.E-mail:tian@math,mit.edu Department of Mathematics The University of British Columbia Vancouver B. C. Canada V6T 1Z2 Institute of Mathematics Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing P. R. China Department of Mathematics Fudan University Shanghai P. R. China Department of Mathematics MIT Cambridge U.S.A.

A surface E is a graph in R^4 if there is a unit constant 2-form ω on R^4 such that ≥v_0>0 where{e_1.e_2}is an orthonormal frame on Σ.We prove that.if v_0≥on the initial snrface,then the mean curvature flow has a global solution and the scaled surfaces converge to a self-similar solution.A surface Σ is a graph in M_1×M_2 where M_1 and M_2 are Riemann surfaces. if≥v_0>0 where w_1 is a Khler form on M_1.We prove that.if M is a Khler-Einstein surface with scalar curvature R.v_0≥ on the initial surface,then the mean curvature flow has a global solution and it sub-converges to a minimal surface,if.in addition.R≥0 it converges to a totally geodesic surface which is holomorphic.

关键词： mean curvature flow 2-dimensional graphs in R^4,Self-similar solution

ε_0-Regularity for mean curvature flow from Surface to Flat Riemannian Manifold

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Acta Mathematica Sinica,English Series 2012年第7期28卷 1475-1490页

作者： Xiao Li HAN Jun SUN Department of Mathematical Sciences Tsinghua University Math.Group The Abdus Salam International Centre for Theoretical Physics

In this paper we prove an so-regularity theorem for mean curvature flow from surface to a flat Riemannian manifold. More precisely, we prove that if the initial energy ∫∑0 ｜A｜^2 ≤ ε0 and the initial area u0（∑0） is not large, then along the mean curvature flow, we have ∫∑t ｜A｜^2 ≤ ε0. As an application, we obtain the long time existence and convergence result of the mean curvature flow.

关键词： ε0-Regularity mean curvature flow

Singularities of mean curvature flow

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Science China Mathematics 2021年第7期64卷 1349-1356页

作者： Yuanlong Xin Institute of Mathematics Fudan UniversityShanghai 200433China

mean curvature flow and its singularities have been paid attention extensively in recent years. The present article reviews briefly their certain aspects in the author's interests.

关键词： mean curvature flow self-shrinker translating soliton

SOME EXTENSIONS OF THE mean curvature flow IN RIEMANNIAN MANIFOLDS

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Acta Mathematica Scientia 2013年第1期33卷 171-186页

作者：吴加勇 Department of Mathematics Shanghai Maritime University

Given a family of smooth immersions of closed hypersurfaces in a locally symmetric Riemannian manifold with bounded geometry, moving by mean curvature flow, we show that at the first finite singular time of mean curvature flow, certain subcritical quantities concerning the second fundamental form blow up. This result not only generalizes a result of Le-Sesum and Xu-Ye-Zhao, but also extends the latest work of Le in the Euclidean case

关键词： mean curvature flow Riemannian submanifold integral curvature maximalexistence time

Convex mean curvature flow with a Forcing Term in Direction of the Position Vector

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Acta Mathematica Sinica,English Series 2012年第2期28卷 313-332页

作者： Guang Han LI Jing MAO Chuan Xi WU School of Mathematics and Computer Science Key Laboratory of Applied Mathematics of Hubei ProvinceHubei UniversityWuhan 430062P.R.China Departamento de Matemática Instituto Superior TecnicoTechnical University of LisbonEdifício CiênciaPiso 3Av.Rovisco Pais1049-001 LisboaPortugal Institute of Mathematics Hubei UniversityWuhan 430062P.R.China

A smooth, compact and strictly convex hypersurface evolving in R^n＋1 along its mean curvature vector plus a forcing term in the direction of its position vector is studied in this paper. We show that the convexity is preserving as the case of mean curvature flow, and the evolving convex hypersurfaces may shrink to a point in finite time if the forcing term is small, or exist for all time and expand to infinity if it is large enough. The flow can converge to a round sphere if the forcing term satisfies suitable conditions which will be given in the paper. Long-time existence and convergence of normalization of the flow are also investigated.

关键词： Evolution equation mean curvature flow forcing term normalization

Ginzburg-Landau Vortex and mean curvature flow with External Force Field

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Acta Mathematica Sinica,English Series 2006年第6期22卷 1831-1842页

作者： Huai Yu JIAN Yan Nan LIU Department of Mathematical Sciences Tsinghua University Beijing 100084 P. R. China

This paper is devoted to the study of the vortex dynamics of the Cauchy problem for a parabolic Ginzburg Landau system which simulates inhomogeneous type II superconducting materials and three-dimensional superconducting thin films having variable thickness. We will prove that the vortex of the problem is moved by a codimension k mean curvature flow with external force field. Besides, we will show that the mean curvature flow depends strongly on the external force, having completely different phenomena from the usual mean curvature flow.

关键词： system of parabolic equations Ginzburg-Landau vortex mean curvature flow