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Mean Curvature Flow with Convex Gauss Image

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Chinese Annals of Mathematics,Series B 2008年第2期29卷 121-134页

作者： yuanlong xin Key Laboratory of Mathematics for Nonlinear Sciences School of Mathematical Sciences Fudan University Shanghai 200433 China

In this paper,the mean curvature flow of complete submanifolds in Euclidean space with convex Gauss image and bounded curvature is *** confinable property of the Gauss image under the mean curvature flow is proved,which in turn helps one to obtain the curvature *** the author proves a long time existence *** asymptotic behavior of these solutions when t→∞is also studied.

关键词： Mean curvature flow Convex Gauss image Curvature estimates Long time existence

Ricci Curvature and Fundamental Group

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Chinese Annals of Mathematics,Series B 2006年第2期27卷 113-120页

作者： yuanlong xin Dedicated to the memory of Shiing-Shen Chern

Abstract By refined volume estimates in terms of Ricci curvature, the two results due to J. Milnor （1968） are generalized.

关键词： Ricci curvature Fundamental group Volume growth

Boundary Regularity for Minimal Graphs of Higher Codimensions

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Chinese Annals of Mathematics,Series B 2022年第5期43卷 869-876页

作者： Qi DING yuanlong xin Shanghai Center for Mathematical Sciences Fudan UniversityShanghai 200438China Institute of Mathematics Fudan UniversityShanghai 200433China

In this paper,the authors derive H¨older gradient estimates for graphic functions of minimal graphs of arbitrary codimensions over bounded open sets of Euclidean space under some suitable conditions.

关键词： Boundary regularity Minimal graphs Higher condimension Bernstein type theorem

Proper Submanifolds in Product Manifolds

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Chinese Annals of Mathematics,Series B 2012年第1期33卷 1-16页

作者： Hongbing QIU yuanlong xin School of Mathematical Sciences Fudan UniversityShanghai 200433China School of Mathematical Sciences Fudan UniversityShanghai 200433China

The authors obtain various versions of the Omori-Yau＇s maximum principle on complete properly immersed-submanifolds with controlled mean curvature in certain product manifolds, in complete Riemannian manifolds whose k-Ricci curvature has strong quadratic decay, and also obtain a maximum principle for mean curvature flow of complete manifolds with bounded mean curvature. Using the generalized maximum principle, an estimate on the mean curvature of properly immersed submanifolds with bounded projection in N1 in the product manifold N1 x N2 is given. Other applications of the generalized maximum principle are also given.

关键词： Calabi-Chern problem Omori-Yau maximum principle Properlyimmersed submanifold Mean curvature flow Stochastic completeness

Mean Curvature Flow via Convex Functions on Grassmannian Manifolds

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Chinese Annals of Mathematics,Series B 2010年第3期31卷 315-328页

作者： yuanlong xin Ling YANG Institute of Mathematics Fudan University Shanghai 200433 China. Max Planck Institute for the Mathematics in Sciences Inselstr. 22-24 Leibzig 04103 Germany.

Using the convex functions on Grassmannian manifolds, the authors obtain the interior estimates for the mean curvature flow of higher codimension. Confinable properties of Gauss images under the mean curvature flow have been obtained, which reveal that if the Gauss image of the initial submanifold is contained in a certain sublevel set of the v-function, then all the Gauss images of the submanifolds under the mean curvature flow are also contained in the same sublevel set of the v-function. Under such restrictions, curvature estimates in terms of v-function composed with the Gauss map can be carried out.

关键词： Mean curvature flow Convex function Gauss map

Curvature Estimates for Minimal Submanifolds of Higher Codimension

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Chinese Annals of Mathematics,Series B 2009年第4期30卷 379-396页

作者： yuanlong xin Ling YANG Key Laboratory of Mathematics for Nonlinear Sciences School of Mathematical Sciences Fudan University Shanghai 200433 China School of Mathematical Sciences Fudan University Shanghai 200433 China

The authors derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau’s results and Ecker-Huisken’s results are generalized ... 详细信息

关键词： Curvature estimates Minimal submanifold Bernstein type theorem

Boundary regularity for mean curvature flows of higher codimension

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Science China Mathematics 2024年

作者： Qi Ding Jürgen Jost yuanlong xin Shanghai Center for Mathematical Sciences Fudan University Max Planck Institute for Mathematics in the Sciences Institute of Mathematics Fudan University

In this paper, we derive global bounds for the H?lder norms of the gradient of solutions of graphic mean curvature flows with boundaries of arbitrary codimension.

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Harmonic Maps from Kahler Manifolds

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Acta Mathematica Sinica,English Series 1999年第2期15卷 277-292页

作者： yuanlong xin, Institute of Mathematics and LMNS, Fudan University, Shanghai 200433, P. R. ChinaE-mail: ylxin@fudan, edu. cn Institute of Mathematics and LMNS Fudan University Shanghai P. R. China

Some Liouville type theorems for harmonic maps from Kahler manifolds are obtained. The main result is to prove that a harmonic map from a bounded symmetric domain (except IV（2）) to any Riemannian manifold with finit... 详细信息

关键词： Harmonic map Kahler manifold Bounded symmetric domain

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极小曲面Gauss像的值分布

中国科学：数学 2018年第6期48卷 843-848页

作者：忻元龙复旦大学数学科学学院上海200433

本文概述了欧氏空间中极小子流形Gauss像值分布问题的最新研究进展,包括高维极小超曲面Gauss像分布的丘成桐问题,以及高余维极小子流形的Lawson-Osserman问题;介绍了作者及其合作者近年来在这两方面的研究结果和研究方法,并提出了进一... 详细信息

本文概述了欧氏空间中极小子流形Gauss像值分布问题的最新研究进展,包括高维极小超曲面Gauss像分布的丘成桐问题,以及高余维极小子流形的Lawson-Osserman问题;介绍了作者及其合作者近年来在这两方面的研究结果和研究方法,并提出了进一步的相关研究问题.

关键词：极小子流形 Guass映照 Bernstein定理