个人简介

      1990年在复旦大学获博士学位,随后经谷超豪院士、胡和生院士推荐成为浙江省第一位理科博士后。1990年任浙江大学讲师;1993年任浙江大学副教授;1994年任日本九州大学数学系客座教授;1996年任浙江大学教授;2000年被评为博士生导师。1993年起,先后担任几何与代数教研室主任、数学研究所党支部书记、数学系副主任、数学系常务副主任等职,现任浙江大学数学中心副主任。2016年8月应邀在第7届世界华人数学家大会上作一小时大会报告,是浙江省第一位获此荣誉的全职数学家。担任美国著名SCI数学期刊《Pure and Applied Math. Quarterly》执行主编、丘成桐大学生数学竞赛组委会副主席。主持承担国家自然科学基金重点项目《子流形与曲率流》,先后八次获得国家自然科学基金科研项目资助。入选国家教育部跨世纪优秀人才(2000)、浙江省151人才工程第一层次(1998)。

        长期从事微分几何研究,在《J. Differential Geom.》, 《Geom. Funct. Anal.》, 《Math. Ann.》, 《J. Math. Pures Appl.》, 《J. Funct. Anal.》, 《Compositio Math.》, 《Trans. Amer. Math. Soc.》, 《Comm. Anal. Geom.》, 《J. Geom. Anal.》, 《Math. Res. Lett.》等国内外重要刊物上发表论文70余篇, 有关工作被S. Brendle, J. Cheeger, M. do Carmo, K. Ecker, P. Gilkey, A. Naber, H. Rosenberg, N. Sesum, K. Shiohama, K. Smoczyk, S.-T. Yau, F. Zheng等多位国际著名微分几何学家引用。与季理真、励建书、丘成桐合作编著一本《Lie Groups and Automorphic Forms》(AMS/IP, Vol.37, 2006)。先后主持承担国家自然科学基金项目“黎曼流形上的几何与分析研究” 、“整体黎曼几何中的若干问题研究”、“黎曼流形的几何与拓扑研究”、“流形上的几何与拓扑的若干问题研究”等10余项国家和省部级科研基金项目。应邀在纪念陈省身先生诞辰一百周年北京微分几何国际会议、纪念苏步青先生诞辰一百一十周年上海国际数学会议、美国哈佛大学几何热流国际研讨会、中日友好微分几何国际会议等一系列国际学术会议上作大会报告和特邀报告。先后访问了哈佛大学等10余所国外著名大学。

        在承担大量行政工作的同时,坚持每年主讲本科生、研究生课程4门以上。指导培养微分几何方向硕士、博士、博士后40多名。指导的多位在读研究生在国际第一流学术期刊上发表了高水平学术论文,其中一位研究生的博士论文发表在《J. Differential Geom.》, 《Geom. Funct. Anal.》, 《Math. Ann.》等多个国际顶尖数学期刊上。有多位近年毕业的研究生获得了国家自然科学基金和省部级科研基金项目的资助,受邀担任美国《数学评论》评论员,应邀在重要国际学术会议上作特邀报告。指导丘成桐数学英才班、竺可桢学院、数学系优秀本科生毕业论文多篇,其中两篇本科生毕业论文获得新世界数学奖学士学位论文银奖,两篇本科生毕业论文入选浙江大学百篇特优本科生毕业论文。所指导的两名优秀本科生被谷超豪院士和胡和生院士遴选为免试直博生,作为关门弟子培养,其中一位获得了以数学大师谷超豪先生命名的首届“谷超豪奖”。

       在主持数学中心和数学系日常工作期间,组织实施了2007数学国家重点学科考核评估工作,并亲自执笔撰写了基础数学国家重点学科考核评估报告、教育部第二轮一级学科排名评估报告、数学国家重点学科建设与发展规划、数学211三期项目申报书与总结评估报告、数学985二期科技创新平台申报书与总结评估报告、数学系人才队伍建设规划等数十份重要报告,为浙大数学学科入选一级学科国家重点学科、教育部第二轮学科排名评估列全国第4位作出了至关重要的贡献。积极推进了数学学科千人计划特聘教授、教育部长江讲座教授与长江特聘教授、浙大光彪特聘教授与求是特聘教授、省特级专家、国家杰出青年基金的申报工作,并取得了成效。牵头引进了一大批优秀的全职教师。参与组织了杭州弦理论国际会议、纪念波莱尔教授核心数学国际会议、中国数学科学与教育发展论坛、第四届世界华人数学家大会等一系列高端学术会议。为浙大数学中心和数学系的建设与发展作出了重要贡献。

 

科研工作:
     长期从事流形上的整体几何、几何分析与几何拓扑研究。代表性研究成果包括:
(1)通过构造反例,解决了关于黎曼流形逐点Pinching问题的丘成桐猜想。运用Ricci流、稳定流以及代数拓扑等工具,证明了空间型中完备子流形的最佳拓扑球面定理和最佳微分球面定理。成功地将著名的Brendle-Schoen微分球面定理拓广到一般黎曼流形中具有任意余维数p(=0,1,2,...)的子流形情形,并将关于超曲面的Huisken微分球面定理推广到高余维子流形的情形。通过引进新的内蕴不变量,将关于正曲率黎曼曲面的阿达玛微分球面定理完整推广到n维黎曼流形的情形。证明了具有正数量曲率的黎曼流形的微分球面定理。获得了具有正数量曲率的爱因斯坦流形的度量刚性定理。
(2)首次在具有正Ricci曲率的最优拼挤条件下,证明了双曲空间中高余维平均曲率流的最佳收敛性定理和子流形的最佳微分球面定理。在优化的曲率拼挤条件下证明了球面和复射影空间中高余维平均曲率流解的收敛性定理和紧致子流形的微分球面定理,改进了Baker、Huisken、Pipoli、Sinestrari等人的收敛性定理和微分球面定理。证明了一般黎曼流形中高余维平均曲率流的收敛性定理。证明了曲率积分拼挤条件下高余维平均曲率流解的收敛性定理和可延拓性定理。系统地改进和发展了Huisken学派的平均曲率流理论。
(3)证明了著名的高斯-博内-陈省身定理、陈省身-莱雪夫定理和威尔默定理的统一定理,发现了几何量、分析量、拓扑量之间新的内在联系,并应用该定理获得了曲率与拓扑方面的新结果,为研究流形的几何、分析与拓扑提供了一种新的有效工具。
(4)证明了球面中平行平均曲率子流形的广义陈省身-do Carmo-Kobayashi定理,并给出了数量曲率达到临界值时流形的几何分类,从而解决了这一公开问题。之后,将此结果推广到一般黎曼流形中平行平均曲率子流形的情形,突破了以往外围流形只能局限于对称空间的框框。实质性地改进了关于极小子流形的丘成桐内蕴刚定理和Ejiri内蕴刚性定理,并将其推广到常曲率空间形式和一般黎曼流形中平行平均曲率子流形的情形。
(5)在彭家贵与滕楚莲工作的基础上,解决了关于球面中6维和7维极小超曲面数量曲率第二空隙的国际公开问题。证明球面中n维小常平均曲率超曲面数量曲率的第二空隙定理。在丁琪与忻元龙工作的基础上,证明了单位球面中n维闭极小超曲面数量曲率第二空隙长度至少为n/22,并将该结果推广到球面中小常平均曲率超曲面的情形。证明了欧氏空间中具有多项式体积增长的n维完备自收缩子第二基本形式模长平方的第二空隙长度至少为1/21。
(6)证明了高维紧致带边极小子流形的高阶特征值估计的广义Polya猜想接近于成立,推进了广义Polya猜想的研究,获得了紧致带边极小子流形上Schrodinger算子的特征值个数的估计,改进和发展了郑绍远、李伟光、丘成桐等人的工作。获得了曲率积分拼挤件下流形的拓扑球面定理、微分球面定理、拓扑有限性定理、几何刚性定理和几何不等式,给出了闭子流形贝蒂数之和上界的几何估计。证明了空间型中具有有限全曲率的完备子流形的端的唯一性定理、有限性定理。 

 

主要论著:  

[1] H. W. Xu and Z. Y. Xu, On Chern’s conjecture for minimal hypersurfaces and rigidity of self-shrinkers, J. Funct. Anal., 2017.

[2] K. F. Liu, H. W. Xu, F. Ye and E. T. Zhao, The extension and convergence of mean curvature flow in higher comdimsion, Trans. Amer. Math. Soc., 2017.

[3] Y. Leng and H. W. Xu, The generalized Lu rigidity theorem for submanifolds with parallel mean curvature, Manus. Math., 2017.

[4] Y. X. Hu and H. W. Xu, An eigenvalue pinching theorem for compact hypersurfaces in a sphere, J. Geom. Analysis, 2017.

[5] H. J. Wang, H. W. Xu, and E. T. Zhao, Gap theorems for complete $lambda$-hypersurfaces, Pacific J. Math., 2017.

[6] J. R. Gu and H. W. Xu, A sharp differentiable pinching theorem for submanifolds in space forms, Proc. Amer. Math. Soc., 144(2016), 337-346.

[7] H. W. Xu, and D. Y. Yang, Rigidity theorem for Willmore surfaces in a sphere. Proc. Indian Acad. Sci. Math. Sci., 126 (2016), 253–260.

[8] J. R. Gu, Y. Leng, and H. W. Xu, Rigidity of closed submanifolds in a locally symmetric Riemannian manifold. Appl. Math. J. Chinese Univ., 31 (2016), 237–252.

[9] Y. Li, H. W. Xu, and E. T. Zhao, Contracting pinched hypersurfaces in spheres by their mean curvature. Pure Appl. Math. Q. 11, (2015), 329–368.

[10] H. W. Xu, F. Huang, and E. T. Zhao, Differentiable pinching theorem for submanifolds via Ricci flow, Tohoku Math. J., 67(2015), 531-540.

[11] Y. X. Hu, H. W. Xu, and E. T. Zhao, First eigenvalue pinching for Euclidean hypersurfaces via k-th mean curvatures, Ann. Global Anal. Geom., 48(2015), 23-35.

[12] H. W. Xu and Z. Y. Xu, A new characterization of the Clifford torus via scalar curvature pinching, J. Funct. Anal., 267(2014), 3931-3962.

[13] H. W. Xu and J. R. Gu, Rigidity of Einstein manifolds with positive scalar curvature, Math. Ann., 358(2014), 169-193.

[14] C. Y. Xia and  H. W. Xu, Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds, Ann. Global Anal. Geom., 45(2014), 155-166.

[15] H. W. Xu, Y. Leng, and E. T. Zhao, Volume-preserving mean curvature flow of hypersurfaces in space forms,  Intern. J. Math., 25(2014), no. 3, 1450021, 19 pp.

[16] H. W. Xu, Y. Leng, and J. R. Gu, Geometric and Topological rigidity for compact submanifolds of odd dimension, Science China Math., 57(2014), 1525-1538.

[17] H. W. Xu and J. R. Gu, Geometric,topological and differentiable rigidity of
submanifolds in space forms, Geom. Funct. Anal., 23(2013), 1684-1703.

[18] H. W. Xu and Z. Y. Xu, The second pinching theorem for hypersurfaces with constant mean curvature in a sphere, Math. Ann., 356(2013), 869-883.

[19] H. W. Xu, F. Huang, and E. T. Zhao, Geometric and differentiable rigidity of
submanifolds in spheres, J. Math. Pures Appl., 99(2013), 330-342.

[20] K. F. Liu, H. W. Xu, F. Ye, and E. T. Zhao, Mean curvature flow of higher codimension in hyperbolic spaces, Comm. Anal. Geom., 21(2013), 651-669.

[21] J. R. Gu, H. W. Xu, Z. Y. Xu, and E. T. Zhao, A survey on rigidity problems in geometry and topology of submanifolds, Sixth International Congress of Chinese Mathematicians, AMS/IP, 2014.

[22] K. F. Liu, H. W. Xu, and E. T. Zhao, Some recent progress on mean curvature flow of arbitrary codimension, Sixth International Congress of Chinese Mathematicians, AMS/IP, 2014.

[23] J. R. Gu and H. W. Xu, The sphere theorem for manifolds with positive scalar curvature, Journal of Differential Geometry, 92(2012), 507-545.

[24] J. R. Gu and  H. W. Xu, On Yau rigidity theorem for minimal submanifolds in
spheres, Math.Res. Lett., 19(2012), 511-523.

[25] H. W. Xu, F. Huang, J. R. Gu and M. Y. He, L^{n/2} pinching theorem for submanifolds with parallel mean curvature in H^{n+p}(-1), Pure Appl. Math. Q., 8(2012), 1097-1116.

[26] H. W. Xu and F. Ye, Differentiable sphere theorems for submanifolds of positive k-th Ricci curvature, Manuscripta Math., 138(2012), 529-543.

[27] H. W. Xu and J. R. Gu, The differentiable sphere theorem for manifolds with positive Ricci curvature, Proc. Amer. Math. Soc., 140 (2012), 1011-1021.

[28] H. W. Xu, Recent developments in differentiable sphere theorem, Fifth International Congress of Chinese Mathematicians, AMS/IP, Studies in Advanced Math., Vol.51, 2012, pp.415-430.

[29] H. W. Xu and L. Tian, A differentiable sphere theorem inspired by rigidity of minimal submanifolds, Pacific J. Math., 254(2011), 499-510.

[30] K. Shiohama and H. W. Xu, An integral formula for Lipschitz-Killing curvature and the critical points of height functions, J. Geom. Analysis, 21(2011), 241–251.

[31] H. W. Xu, F. Ye and E. T. Zhao, Extend mean curvature flow with finite integral curvature, Asian J. Math., 15(2011), 549–556.

[32] H. W. Xu and L. Tian, A new pinching theorem for closed hypersurfaces with constant mean curvature in S^{n+1}, Asian J. Math., 15(2011), 611–630.

[33] H. W. Xu, F. Ye and E. T. Zhao, The extension for mean curvature flow with finite integral curvature in Riemannian manifolds, Science China Math., 54(2011), 2195–2204.

[34] H. W. Xu, F. Huang and F. Xiang, An extrinsic rigidity theorem for submanifolds with parallel mean curvature in a sphere, Kodai Math. J., 34(2011), 85–104.

[35] H. W. Xu and D. Y. Yang, The gap phenomenon for extremal submanifolds in a sphere, Differential Geom. and its Applications, 29(2011), 26–34.

[36] H. W. Xu and D. Y. Yang, A new characterization of Willmore submanifolds, Appl. Math. J. Chinese Univ., 26(2011), 453-463.

[37] H. W. Xu and J. R. Gu, An optimal differentiable sphere theorem for complete manifolds, Math. Res. Lett., 17(2010), 1111-1124.

[38] H. W. Xu and E. T. Zhao, L^p Ricci curvature pinching theorems for conformally flat Riemannian manifolds, Pacific J. Math., 245(2010), 381-396.

[39] H. P. Fu and H. W. Xu, Total curvature and L^2 harmonic 1-forms on complete submanifolds in space forms, Geom. Dedicata., 144(2010), 129-140.

[40] H. W. Xu and E. T. Zhao, Topological and differentiable sphere theorems for complete submanifolds, Comm. Anal. Geom., 17(2009), 565-585.

[41] H. W. Xu and E. T. Zhao, Complete hypersurfaces in a 4-dimensional hyperbolic space, Appl. Math. J. Chinese Univ., 24(2009), 370-378.

[42] H. P. Fu and H. W. Xu, Weakly stable constant mean curvature hypersurfaces, Appl. Math. J. Chinese Univ., 24(2009), 119-126.

[43] H. P. Fu and H. W. Xu, Vanishing and topological sphere theorems for submanifolds in a hyperbolic space, Intern. J. Math., 19(2008), 811-822.

[44] H. W. Xu and X. Ren, Closed hypersurfurfaces with constant mean curvature in a symmetric manifold, Osaka J. Math., 45(2008), 747-756.

[45] H. W. Xu and J. F. Zhu, An optimal rigidity theorem for complete submanifolds in a sphere, Appl. Math. J. Chinese Univ., 23(2008), 219-226.

[46] H. W. Xu and J. R. Gu, A general gap theorem for submanifolds with parallel mean curvature in R^{n+p}, Comm. Anal. Geom., 15(2007),175-193.

[47] S. M. Wei and H. W. Xu, Scalar curvature of minimal hypersurfaces in a sphere, Math. Res. Lett., 14(2007),423-432.

[48] H. W. Xu and J. R. Gu, L^2-isolation phenomenon for complete surfaces arising from Yang-Mills theory, Lett. Math. Phys, 80(2007),115-126.

[49] N. Q. Xie and H. W. Xu, Geometric inequalities for certain submanifolds in a pinched Riemannian manifold, Acta Math. Scientia, 27(2007),611-618.

[50] Y. Leng and H. W. Xu, On complete submanifolds with parallel mean curvature in negative pinched manifolds, Appl. Math. J. Chinese Univ., 22(2007),153-162.

[51] H. W. Xu and W. Zhang, Geometric properties for Gaussian image of submanifolds in $S^{n+p}$, Appl. Math. J. Chinese Univ., 22(2007),371-377.

[52] H. W. Xu, Mean value theorem for critical points and sphere theorems, Proceedings of the Fourth ICCM, Vol.2, Higher Education Press & International Press, 2007, pp.203-217.

[53] H. W. Xu, W. Fang and F. Xiang, A generalization of Gauchman's rigidity theorem, Pacific J. Math., 228(2006),185-199.

[54] L. Ji, J. S. Li, H. W. Xu and S. T. Yau, Lie Groups and Automorphic Forms, AMS/IP, Studies in Anvanced Math., Vol.37, 2006.

[55] H. W. Xu and W. Han, Geometric rigidity theorem for submanifolds with positive curvature, Appl. Math. J. Chinese Univ., 20(2005), 475-482.

[56] X. Ren and H. W. Xu, A lower bound for the first eigenvalue with mixed boundary condition, Appl. Math. J. Chinese Univ., 19(2004), 223-228.

[57] K. Shiohama and H. W. Xu, Rigidity and sphere theorems for submanifolds II, Kyushu J. Math., 54 (2000), 103-109.

[58] K. Shiohama and H. W. Xu, A general rigidity theorem for complete submanifolds, Nagoya Math. J., 150 (1998), 105-134.

[59] K. Shiohama and H. W. Xu, Lower bound for L^{n/2} curvature norm and its application, J. Geom. Analysis, 7(1997), 377-386.

[60] K. Shiohama and H. W. Xu, The topological sphere theorem for complete submanifolds, Compositio Math., 107 (1997), 221-232.

[61] H. W. Xu, On closed minimal submanifolds in pinched Riemannian manifolds, Trans. Amer.  Math. Soc., 347 (1995), 1743-1751.

[62] H. W. Xu, L_{n/2} pinching theorems for submanifolds with parallel mean curvature in a sphere, J. Math. Soc. Japan, 46 (1994), 503-515.

[63] K. Shiohama and H. W. Xu, Rigidity and sphere theorems for submanifolds, Kyushu J. Math., 48 (1994), 291-306.

[64] H. W. Xu, Quantization phenomena for Riemannian submanifolds in Euclidean space, Interface between Phys. and Math., World Scientific, Singapore, 1994, pp.392-397.

[65] H. W. Xu, A rigidity theorem for submanifolds with parallel mean curvature in a sphere, Arch. der Math., 61 (1993), 489-496.

[66] H. W. Xu, Estimates of higher eigenvalues for minimal submanifolds, Differential Geometry, World Scientific, Singapore, 1993, pp.288-300.

[67] H. W. Xu, A pinching constant of Simons' type and the problem of isometric  immersion. Chinese Ann. Math. Ser. A, 12(1991), 261–269.

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预印本:

1. Li Lei, Hongwei Xu: Mean curvature flow of arbitrary codimension in complex projective spaces, arXiv:1605.07963 [pdf, ps, other]

2.  Li Lei, Hongwei Xu: Mean curvature flow of arbitrary codimension in spheres and sharp differentiable sphere theorem, arXiv:1506.06371 [pdf, ps, other]

3.  Li Lei, Hongwei Xu: A new version of Huisken's convergence theorem for mean curvature flow in spheres, arXiv:1505.07217 [pdf, ps, other]

4.  Li Lei, Hongwei Xu: An optimal convergence theorem for mean curvature flow of arbitrary codimension in hyperbolic spaces, arXiv:1503.06747 [pdf, ps, other]

5.  Kefeng Liu, Hongwei Xu, Entao Zhao: Mean curvature flow of higher codimension in Riemannian manifolds, arXiv:1204.0107 [pdf, ps, other]

6.  Kefeng Liu, Hongwei Xu, Entao Zhao: Deforming submanifolds of arbitrary codimension in a sphere, arXiv:1204.0106 [pdf, ps, other]

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主讲科目:
       
本科生课程:微分几何、微分流形、解析几何、高等代数、前沿数学专题讨论(I, II)等。
研究生课程:整体微分几何(博)、几何分析(博)、几何曲率流(博)、整体子流形几何(博)、
微分流形与流形几何(硕)、整体黎曼几何(硕)等。

 

工作研究领域

整体微分几何/几何分析/流形拓扑学

联系方式

电话:87953121
电子信箱:xuhw@cms.zju.edu.cn