研究員 |   鄭日新
聯絡資訊
  • cheng$\color{red}{@}$math.sinica.edu.tw
  • +886 2 2368-5999 ext. 631
  • +886 2 2368-9771
研究專長
  • 幾何分析
  • 低維拓樸
  • 多複變幾何
學歷
  • Ph.D. in Mathematics Univ. of Notre Dame (1983/5)
  • M.S. 清華大學 (1979/6)
  • B.S. 台灣大學 (1975/6)
相關連結

經歷
  • Research Fellow Institute of Mathematics, Academia Sinica, R.O.C. 1988
  • Associate Research Fellow Institute of Mathematics, Academia Sinica, R.O.C. 1983

獲獎
  • Ministry of Education Academic Award, 2019-12
  • Academic Award of Taiwan Mathematical Society, 2017-12
  • NSC distinguished research awards, 1990, 1992

研究簡介

鄭日新於民國四十二年生(西元一九五三年)。 西元一九七九年經呂輝雄老師推薦,赴美在長野正教授指導下習微分幾何。一九八三年於聖母大學畢業後任職中研院迄今。

在他的博士論文中,他分類了所有具二階性的 李代數,這二階性是所謂柯西黎曼結構的一個特殊屬性。 此後他也證明了柯西黎曼幾何與它不變曲線所構成幾何的 等價性。一九八六年,他獲國科會補助到哈佛大學研究訪 問一年。在那裡透過蕭蔭堂教授介紹認識了 John Lee,開始了三維柯西黎曼結構各種模空間的研究。

一九九二年,第二次獲國科會補助,選擇了史丹佛大學的 Eliashberg 教授,開始認識理解到三維切觸拓樸與三維柯西黎曼幾何的深刻關連及他們在研究三維流形拓樸時所扮演的角色。

往後在一九九七年及二零零一年分別再獲國科會補助訪問哈佛和普林斯敦大學,遂定心於柯西黎曼結構相關幾何分析的深入研究。與同事黃振芳,普林斯敦的楊建平,義大利比薩高師的Andrea Malchiodi共同發展了海(森堡)群中超曲面的幾何,證明了三維海群中的Bernstein型定理,建立奇點及特徵曲線的結構分類定理,奠定了三維Alexandrov型定理證明的基礎。加入清華邱鴻麟後,對高維臍性及Pansu球的刻劃作出了貢獻。

在柯西黎曼正質量定理及Yamabe問題的研究方向上,與楊建平,Andrea Malchiodi及邱鴻麟先後分別獲致各維度的結果。在帶群作用的柯西黎曼或複流形研究上,與同事蕭欽玉及台大蔡宜洵利用熱核方法得到局部指標公式。在三維柯西黎曼幾何中構造了兩個不變面積元,即得兩個Willmore型能量泛函,與楊建平及河肥中國科技大學的張永兵,在這個方向上探討了與奇異Yamabe問題解及其體積重整化的關係。

在其他零星但緊要的課題上,與Penn State的Jeffrey Case等合作,較值一提的是與Taiji Marugame, 德國Jena大學的Vladimir Matveev及加州大學Santa Cruz分校的Richard Montgomery 透過理解柯西黎曼幾何中叫chain的不變曲線為Finsler幾何中Kropina度量的短程線,探討了chain的廣域聯通性及與柯西黎曼結構的投影等價性。


部分著作目錄
  1. (with Chang C.-H., Tsai I-H.) "Theta functions and adiabatic curvature on an elliptic curve" , The Journal of Geometric Analysis , to appear. pdf file
  2. (with Hung-Lin Chiu) "Positive mass theorem and the CR Yamabe equation on 5-dimensional contact spin manifolds" , . , preprint. (ArXiv: 2107.04209v1) pdf file
  3. (with Malchiodi A., Yang P.) "On the Sobolev quotient of three-dimensional CR manifolds" , . , preprint. (arXiv: 1904.04665) pdf file
  4. (with Chiu H.-L., Ho P. T.) "Connected sum of CR manifolds with positive CR Yamabe constant" , The Journal of Geometric Analysis , 31 (1), 298-311, 2021. pdf file
  5. "微分幾何學--重力研究的好幫手" , 數學傳播 , 44 (4), 27-33, 2020. (第176期) pdf file
  6. "Submanifolds in Cauchy Riemann Geometry" , Journal of Mathematical Study , 53 (4), 436-470, 2020. pdf file
  7. (with Chiu H.-L.) "Connected sum of spherical CR manifolds with positive CR Yamabe constant" , The Journal of Geometric Analysis , 29 (4), 3113-3123, 2019. pdf file
  8. (with Marugame T., Matveev V.S., Montgomery R.) "Chains in CR geometry as geodesics of a Kropina metric" , Advances in Mathematics , 350, 973-999, 2019. pdf file
  9. (with Case J.S., Yang P.) "An integral formula for the Q-prime curvature in 3-dimensional CR geometry" , Proceedings of AMS , 147, 1577-1586, 2019. pdf file
  10. (with Hsiao C.-Y., Tsai I-H.) "Heat kernel asymptotics, local index theorem and trace integrals for CR manifolds with $S^1$ action" , Memoires de la SMF , 162, 2019. (152 pages) pdf file
  11. (with Yang P., Zhang Y.) "Invariant surface area functionals and singular Yamabe problem in 3-dimensional CR geometry" , Advances in Mathematics , 335, 405-465, 2018. pdf file
  12. (with Chiu H.-L., Hwang J.-F., Yang P.) "Umbilicity and characterization of Pansu spheres in the Heisenberg group" , Journal fur die reine und angewandte Mathematik (738), 203-235, 2018. pdf file
  13. (with Chiu H.-L., Hwang J.-F., Yang P.) "Strong Maximum principle for mean curvature operator on subriemannian manifolds" , Math. Annalen , 372, 1393-1435, 2018. pdf file
  14. "Contact 𝑠𝑝𝑖𝑛^{𝑐} structure, embeddability of CR structure, and positivity of the mass" , Proceedings of the Sixth International Congress of Chinese Mathematicians. Vol. II, Adv. Lect. Math. (ALM), vol. 37, Int. Press, Somerville, MA , 23-35, 2017.
  15. (with Malchiodi A., Yang P.) "A positive mass theorem in three dimensional Cauchy-Riemann geometry" , Advances in Mathematics , 308, 276-347, 2017. pdf file
  16. (with Chiu H.-L., Hwang J.-F., Yang P.) "Umbilic hypersurfaces of constant sigma-k curvature in the Heisenberg group" , Calculus of Variations and Partial Differential Equations , 55 (3), 2016-06. pdf file
  17. (with 夏杼,何南國) "有朋自遠方來--專訪Nigel Hitchin教授" , 數學傳播 , 39 (4), 3-16, 2015-12.
  18. "Flows and a tangency condition for embeddable CR structures in dimension 3" , Methods and Applications of Analysis , 21 (3), 337-356, 2014. pdf file
  19. (with Hwang J.-F.) "Uniqueness of generalized $p$-area minimizers and integrability of a horizontal normal in the Heisenberg group" , Calculus of Variations and PDEns , 50, 579-597, 2014. (published online: 29 June 2013) pdf file
  20. (with Chiu H.-L., Yang P.) "Uniformization of spherical CR manifolds" , Advances in Mathematics , 255, 182-216, 2014. pdf file
  21. (with 劉太平, 蔡忠潤) "有朋自遠方來--專訪Clifford Taubes教授" , 數學傳播 , 37 (1), 3-14, 2013-03.
  22. (with Malchiodi A., Yang P.) "A positive mass theorem in three dimensional Cauchy-Riemann geometry" , Extended abstract Fall 2013, Research Perspectives CRM, Barcelona , 2013. (editors: Maria del Mar Gonzalez, Paul C. Yang, etc.)
  23. (with Jenn-Fang Hwang, Andrea Malchiodi, Paul Yang) "A Codazzi-like equation and the singular set for $C^{1}$ smooth surfaces in the Heisenberg group" , Journal fur die reine und angewandte Mathematik , 671, 131-198, 2012. pdf file
  24. (with Jenn-Fang Hwang) "Variations of generalized area functionals and p-area minimizers of bounded variation in the Heisenberg group" , Bulletin of the Institute of Mathematics, Academia Sinica, New Series , 5 (4), 369-412, 2010. pdf file
  25. (with 劉太平, 江孟蓉) "有朋自遠方來--專訪Kenji Fukaya教授" , 數學傳播 , 33 (3), 3-12, 2009-09.
  26. "The prescribed p-mean curvature equation of low regularity in the Heisenberg group" , Science in China, Series A , 52 (12), 2604-2609, 2009. (2007 International Conference on Several Complex Variables and Complex Geometry, May 28-31, 2007, Xiamen.) pdf file
  27. (with Jenn-Fang Hwang!Paul Yang) "Regularity of $C^{1}$ smooth surfaces with prescribed p-mean curvature in the Heisenberg group" , Math. Annalen , 344 (1), 1-35, 2009. pdf file
  28. "Surfaces of prescribed p-mean curvature in the Heisenberg group" , Proceedings of the 4th ICCM , II, 40-58, 2007. (Dec.17-22, Hangzhou, ed. L. Ji, K. Liu, L. Yang, and S.T. Yau.) pdf file
  29. (with Shu-Cheng Chang ! Hung-Lin Chiu) "A fourth order curvature flow on a CR 3-manifold" , Indiana Univ. Math. J. , 56, 1793-1826, 2007. pdf file
  30. (with Jenn-Fang Hwang ! Paul Yang) "Existence and uniqueness for p-area minimizers in the Heisenberg group" , Math. Annalen , 337, 253-293, 2007. pdf file
  31. (with 葉永南,周文賢) "有朋自遠方來---專訪Gyula O. H. Katona教授" , 數學傳播 , 30 (1), 15-24, 2006.
  32. "The mean curvature equation in pseudohermitian geometry" , Contemporary Mathematics , 400, 89-94, 2006. pdf file
  33. (with Jenn-Fang Hwang!Andrea Malchiodi!Paul Yang) "Minimal surfaces in pseudohermitian geometry" , Annali della Scuola Normale Superiore di Pisa Classe di Scienze V , 4 (1), 129-177, 2005. pdf file
  34. (with Jenn-Fang Hwang) "Properly embedded and immersed minimal surfaces in the Heisenberg group" , Bull. Aus. Math. Soc. , 70, 507-520, 2004. pdf file
  35. (with Hung-Lin Chiu) "Monopoles and contact 3-manifolds. Quademi di Matematica, Selected Topics in Cauchy-Riemann Geometry, ed. Sorin Dragomir, 9(2003), 99-123; math. DG/9905066." , 2003.
  36. "Deformation of fillable CR structures, preprint; math.DG/0202051." , 2002.
  37. (with Shu-Cheng Chang) "The Harnack estimate for the Yamabe flow on CR manifolds of dimension 3, Annals of Global Analysis and Geometry, 21(2002), 111-121; math. DG/0003197." , 2002.
  38. "淺談場量子化與低維拓樸." , 數學傳播 , 25 (4), 4-7, 2001-12.
  39. "Rick Schoen, Yamabe 問題與正質量定理." , 數學傳播 , 24 (4), 63-67, 2000-12.
  40. (with 李瑩英, 蔡宜洵) "有朋自遠方來 ---Rick Schoen 訪談紀要." , 數學傳播 , 24 (4), 58-62, 2000-12.
  41. "Contact topology and CR geometry in three dimensions. Proceedings of the 3rd Asian Mathematical Conference 2000 (Quezon City, Philippines) ed. T. Sunada, P. W. Sy, Yang Lo (2002), 70-80; math.SG/0106071." , 2000/2002.
  42. "Cauchy-Riemann geometry and contact topology in three dimensions. Proc. Natl. Sci. Counc., ROC(A), 24(2000), 87-94; math. DG/0003211." , 24, 87-94, 2000.
  43. "Rigidity of automorphisms and spherical CR structures. Proc. Amer. Math. Soc., 128(2000), 1825-1832; math. DG/9806108." , Proc. Amer. Math. Soc. , 128, 1825-1832, 2000.
  44. (with I-Hsun Tsai) "Deformation of spherical CR structures and the universal Picard variety. Commun. in Anal. And Geom., 8(2000), 301-346; math. DG/9807035." , Commun. in Anal. And Geom. , 8, 301-346, 2000.
  45. "The geometry of Cauchy-Riemann manifolds in dimension three, Proceedings of the second Asian mathematical conference, Suranaree University of Technology, Nakhon Ratchasima, Thailand, (Oct. 17-20, 1995), ed. S.Tangmanee and E. Schulz (1998), 201-208." , 201-208, 1998.
  46. (with John M. Lee) "A local slice theorem for 3-dimensional CR structures" , Amer. J. Math. , 117, 1249-1298, 1995.
  47. "Some applications of Cartan's theory on three-dimensional Cauchy-Riemann geometry" , Math. Zeit. , 218, 527-548, 1995.
  48. (with Shuenn-Jyi Sheu) "Brownian motion on pseudohermitian manifolds" , incomplete draft , 1994.
  49. "Cauchy-Riemann geometry and contact topology on three-manifolds, Proceedings of conference on the 29th annual meeting of ROC Math. Soc., National Chiao-Tung University, Hsinchu, Taiwan, (Dec. 3-4, 1993), 105-111." , 105-111, 1993.
  50. "On embeddable CR 3-manifolds" , draft , 1992.
  51. "淺談Jones-Witten理論及Floer同調群." , 數學傳播 , 15 (1), 12-22, 1991-03.
  52. "Curvature functions for the sphere in pseudohermitian geometry" , Tokyo J. Math. , 14, 151-163, 1991.
  53. (with John M. Lee) "The Burns-Epstein invariant and deformation of CR structures" , Duke Math. J. , 60, 221-254, 1990.
  54. "Deforming CR structures in the direction of Cartan tensor, Proceedings of the 1989 Conference on nonlinear analysis, Academia Sinica, Taipei, R.O.C. (June 19-24, 1989), 17-22, World Scientific." , 1989.
  55. "Chain-preserving diffeomorphisms and CR equivalence" , Proceedings of AMS , 103, 75-80, 1988.
  56. "On the curvature of CR structures and its covariant derivatives" , Math. Zeit. , 196, 203-209, 1987.
  57. "Graded Lie algebras of the second kind" , Transactions of AMS , 302, 467-488, 1987.
  58. "On the second fundamental form of algebraic hypersurfaces" , Bulletin of IMAS , 10, 25-34, 1982.
  59. "An integral formula on the scalar curvature of algebraic manifolds" , Proceedings of AMS , 81, 451-454, 1981.
  60. "CR umbilicity for certain class of real hypersurfaces in $C^{n+1}$" , draft
  61. (with C.H. Chang, H.P. Lee and A.N. Wang) "Examples of spiralling chains" , draft
  62. "CR structures for a family of homogeneous spaces" , draft