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New results on the geometry of the moduli space of Riemann surfaces

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Science China Mathematics 2008年第4期51卷 632-651页

作者： yau shing-tung Center of Mathematical Sciences Zhejiang University

We briefly survey our recent results about the Mumford goodness of several canonical metrics on the moduli spaces of Riemann surfaces, including the Weil-Petersson metric, the Ricci metric, the Perturbed Ricci metric and the Kahler-Einstein metric. We prove the dual Nakano negativity of the Weil-Petersson metric. As applications of these results we deduce certain important results about the L 2-cohomology groups of the logarithmic tangent bundle over the compactified moduli spaces.

关键词： moduli spaces canonical metrics good metrics L 2-cohomology 14D20 14H20 32G13 53C55

Canonical metrics on complex manifold

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Science China Mathematics 2008年第4期51卷 503-508页

作者： yau shing-tung(yau S.-T.) Department of Mathematics Harvard University

Complex manifolds are topological spaces that are covered by coordinate charts where the coordinate changes are given by holomorphic *** example,Riemann surfaces are one dimensional complex *** order to understand complex manifolds,it is useful to introduce metrics that are compatible with the complex *** general,we should have a pair(M,ds^2_M)where ds^2_M is the *** metric is said to be canonical if any biholomorphisms of the complex manifolds are automatically *** metrics can naturally be used to describe invariants of the complex structures of the manifold.

关键词：云门

Topology evolutions of silhouettes

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Journal of Zhejiang University-Science A(Applied Physics & Engineering) 2007年第10期8卷 1671-1680页

作者： DAI Jun-fei KIM Junho ZENG Hua-yi GU Xian-feng yau shing-tung Center of Mathematical Sciences Zhejiang University Hangzhou 310027 China Visualization Laboratory State University of New York Stony Brook NY 11794 USA Department of Mathematics Harvard University Boston MA02138 USA

We give the topology changing of the silhouette in 3D space while others study the projections in an image. Silhou- ettes play a crucial role in visualization, graphics and vision. This work focuses on the global behaviors of silhouettes, especially their topological evolutions, such as splitting, merging, appearing and disappearing. The dynamics of silhouettes are governed by the topology, the curvature of the surface, and the view point. In this paper, we work on a more theoretical level to give enu- merative properties of the silhouette including: the integration of signed geodesic curvature along a silhouette is equal to the view cone angle; in elliptic regions, no silhouette can be contained in another one; in hyperbolic regions, if a silhouette is homotopic to a point, then it has at least 4 cusps; finally, critical events can only happen when the view point is on the aspect surfaces (ruled surface of the asymptotic lines of parabolic points with surface itself). We also introduce a method to visualize the evolution of silhouettes, especially all the critical events where the topologies of the silhouettes change. The results have broad applications in computer vision for recognition, graphics for rendering and visualization.

关键词： Topological change Silhouette Geodesic curvature Cusp

从明治维新到二战前后中日数学人才培养之比较

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科技导报 2010年第4期28卷 15-20页

作者：丘成桐哈佛大学数学系香港中文大学数学科学研究所清华大学数学科学中心北京100084

在18世纪到19世纪的200年间,欧洲人才辈出,诞生的大数学家数不胜数。这期间,东方的数学却反常地沉寂。在明治维新以前,日本的数学成就远不如中国;但到了19世纪末,中国的数学反而不如日本。通过比较从明治维新到二战前后中日数学人才的培... 详细信息

在18世纪到19世纪的200年间,欧洲人才辈出,诞生的大数学家数不胜数。这期间,东方的数学却反常地沉寂。在明治维新以前,日本的数学成就远不如中国;但到了19世纪末,中国的数学反而不如日本。通过比较从明治维新到二战前后中日数学人才的培养,分析了中国这期间数学落后于日本和世界的原因。日本政府明治维新要求老百姓全面向西方学习乃原因之一,而中国人由于坚持"中学为体,西学为用"的信念,因而在追求真理的态度上始终不能全面以赴,也不失数学及其他科学落伍的重要原因。此外,中国早期学习西方以应用科技为主,缺乏对数学的热情,一直到20世纪20年代还没有认识到当代最先进的数学,始终未接触到学问的前沿。另外,日本数学的迅速兴起与其学习方法有密切的关系。没有做学问的热情,没有崇高的志愿,不可能产生杰出的数学家。今日中国数学的前途,端赖于年轻一代数学家的培养,研究生的培养则溯源于中学生的教育。期望重新燃烧起中国人对数学的热情。

关键词：数学家明治维新比较研究.数学史

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研学之乐

科技导报 2011年第19期29卷 15-18页

作者：丘成桐哈佛大学数学系美国马萨诸塞州02138 香港中文大学数学科学研究院中国香港特别行政区清华大学数学科学中心北京100084

我年少时，并不喜欢读书，在香港元朗的平原上嬉戏玩耍，也在沙田的山丘和海滨游戏。与同伴在一起，乐也融融．甚至逃学半年之久。真可谓倘佯于山水之间，放浪形骸之外。

关键词：科研快乐培养读书

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庞加莱之梦

科学 2006年第5期58卷 55-55页

作者：丘成桐

Geometric aspects of the moduli space of Riemann surfaces

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Science China Mathematics 2005年第Z1期48卷 97-122页

作者： shing-tung yau Department of Mathematics Harvard University CambridgeMA 02138USA

We describe some recent progress in the study of moduli space of Riemann surfaces in this survey paper. New complete Kahler metrics were introduced on the moduli space and Teichmuller space. Their curvature properties and asymptotic behavior were studied in details. These natural metrics served as bridges to connect all the known canonical metrics, especially the Kahler-Einstein metric. We showed that all the known complete metrics on the moduli space are equivalent and have Poincare type growth. Furthermore,the Kahler-Einstein metric has strongly bounded geometry. This also implied that the logarithm cotangent bundle of the moduli space is stable in the sense of Mumford.

关键词： moduli space Teichmiiller space metric curvature.

Complex geometry: Its brief history and its future

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Science China Mathematics 2005年第Z1期48卷 47-60页

作者： shing-tung yau Department of Mathematics Harvard University CambridgeMA 02138USA

This paper reviews the brief history of complex geometry and looks into itsfuture.

关键词： complex geometry, complex manifold, Kahler manifold, Kahler-Einstein metric.

COMPUTING HARMONIC MAPS AND CONFORMAL MAPS ON POINT CLOUDS

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Journal of Computational Mathematics 2023年第5期41卷 879-908页

作者： Tianqi Wu shing-tung yau Center of Mathematical Sciences and Applications Harvard UniversityCambridgeMA 02138USA Department of Mathematics Harvard UniversityCambridgeMA 02138USA

We use a narrow-band approach to compute harmonic maps and conformal maps for surfaces embedded in the Euclidean 3-space,using point cloud data *** a surface,or a point cloud approximation,we simply use the standard cubic lattice to approximate itsϵ-*** the harmonic map of the surface can be approximated by discrete harmonic maps on *** conformal map,or the surface uniformization,is achieved by minimizing the Dirichlet energy of the harmonic map while deforming the target surface of constant *** propose algorithms and numerical examples for closed surfaces and topological *** the best of the authors’knowledge,our approach provides the first meshless method for computing harmonic maps and uniformizations of higher genus surfaces.

关键词： harmonic maps conformal maps point clouds