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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

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

由精制体积估计以 Ricci 弯曲，二由于 J 结果。Milnor (1968 ) 被概括。

关键词： Ricci曲率基本原理体积生长几何学

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 ×N2 is given.Other applications of the generalized maximum principle are also given.

关键词：子流形产品平均曲率完备流形黎曼流形高原利玛窦广义

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

在这篇论文，在与凸的高斯一起的欧几里德几何学的空间的完全的潜水艇 manifolds 的吝啬的弯曲流动图象；围住的弯曲被学习。在吝啬的弯曲流动下面的高斯图象的可罚款的性质被证明的反对，它接着帮助一个获得弯曲估计。然后，作者证明... 详细信息

在这篇论文，在与凸的高斯一起的欧几里德几何学的空间的完全的潜水艇 manifolds 的吝啬的弯曲流动图象；围住的弯曲被学习。在吝啬的弯曲流动下面的高斯图象的可罚款的性质被证明的反对，它接着帮助一个获得弯曲估计。然后，作者证明很长时间是存在结果。这些答案的 asymptotic 行为 t →∞是什么时候，也学习了。

关键词：平均曲率流体凸高斯图像曲率估计流体力学

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：平均曲率流格拉斯曼流形凸函数曲率估计子流形高斯余维数图像

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：曲率估计余维欧氏空间高斯定理

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

REGULARITY OF HARMONIC MAPS INTO POSITIVELY CURVED MANIFOLDS

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Chinese Annals of Mathematics,Series B 1992年第4期13卷 385-395页

作者：忻元龙 Institute of Mathematics Fudan University Shanghai 200433 China.

Let M be a compact Riemannian manifold of dimension m, N a complete Amply connected δ-pinched Riemannian manifold of dimension n. There exists a constant d(n). It is proved that if m≤d(n), then every minimizing map from M into N is smooth in the interior of M. If m=d(n)+1, such a map has at most diserete singular set and in general the Hausdorff dimension of the singular set is at most m-d(n)-1.

关键词： Riemannian manifold singular minimizing Hausdorff bundle connected interior regularity ordinary

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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