## Cheng, Jih-Hsin

Adjunct Research Fellow

Research Fields: Geometric Analysis, low-dimensional topology, Geometry of Several Complex Variables,

Email : cheng@math.sinica.edu.tw

Extension : 748

Dr. Cheng's main research interest is on Cauchy-Riemann(CR for short) geometry and contact topology. In his Ph.D. dissertation, he classified all possible Lie algebras with the 2-gradation property-which is a special feature of CR structure. Later he proved an equivalence between CR geometry and geometry of chains, the distinguished invariant curves in CR geometry.

In 1986 he got a NSC (National Science Council) grant for a one-year visit to the Harvard University. At Harvard, Professor Siu introduced John Lee to him. There John Lee and he started the study of various kinds of moduli spaces of CR structrues in dimension 3. In the fall of 1992, he had a second visit supported by the NSC. This time he visited Professor Eliashberg of the Stanford University and began to realize the deep connection between contact topology and CR geometry and their roles in the study of 3-manifolds. Many interesting problems in this respect are just emerging and are expected to attract his attention for the following years.

Later in 1997 and 2001 he received the support from the NSC again for the visits to Harvard and Princeton respectively. He then decided to work mainly on geometry and analysis related to CR structure. Together with his institute colleague Jenn-Fang Hwang, Paul Yang (Princeton University) and Andrea Malchiodi (SNS, Pisa), he developed the geometry of hypersurfaces in the Heisenberg group, proved a Bernstein-type theorem in 3D Heisenberg group, and built up a structure theorem for the singular set and characteristic curves. Later a proof of the Alexandrov-type theorem in 3D Heisenberg group was given, based on the above mentioned structure theorem. After Hung-Lin Chiu (Tsing-Hua University, Taiwan) joined the team, we could then study the umbilicity in high dimensions and characterize the Pansu spheres.

In the direction of positive mass theorem and the CR Yamabe equation, jointly with Andrea Malchiodi, Paul Yang and Hung-Lin Chiu, he obtains the results in various dimensions. For CR manifolds and complex manifolds with group action, he and his collaborators Chin-Yu Hsiao (institute colleague), I-Hsun Tsai (Taiwan University) obtain a local index formula by using the heat kernel method. For surfaces in a CR manifold of 3D, jointly with Paul Yang and Yongbing Zhang (USTC, Hefei) he expresses two CR invariant area elements in terms of pseudohermitian geometry and studies how the associated Willmore-like energy functionals relate to the singular Yamabe problem and the volume renormalization.

In other sporadic topics, he worked with Jeffrey Case (Penn State University) and Paul Yang on a Q prime integral, a CR invariant, with Taiji Marugame (RIKEN, Japan), Vladimir Matveev (University of Jena, Germany) and Richard Montgomery (UC, Santa Cruz) on the study of CR invariant curves called chains. He and his above mentioned collaborators show that we can view chains as geodesics of a Kropina metric in Finsler geometry and obtain local connectivity by chains in general and global connectivity by chains under some positive curvature condition.

- Ph.D. in Mathematics Univ. of Notre Dame 1983/5
- M.S. National Tsing Hua University 1979/6
- B.S. National Taiwan University 1975/6

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

- 2021, 科技部110年度傑出特約研究員
- 2019-12, Ministry of Education Academic Award
- 2017-12, Academic Award of Taiwan Mathematical Society
- 1990, 1992, NSC distinguished research awards

- (with Claudio Afeltra, Andrea Malchiodi and Paul Yang) "On the variation of the Einstein-Hilbert action in pseudohermitian geometry ",
*Crelle's Journal, to appear, arXiv:2306.07523v1*, 2024 - (with A. Malchiodi and P. Yang) "A characterization of homogeneous CR 3-manifolds",
*.Rev. Mat. Iberoam. to appear, arXiv: 2309.02278v1*, 2024 - (with C.-H. Chang, I-H. Tsai) "Theta functions and adiabatic curvature on an Abelian variety",
*.The Journal of Geometric Analysis*, 34 ( 136 ), 2024 - (with A. Malchiodi and P. Yang) "On the Sobolev quotient of three-dimensional CR manifolds",
*Revista Matematica Iberoamericana*, 39 ( 6 ), 2023 - (with C.-H. Chang, I-H. Tsai) "Correction to : Theta functions and adiabatic curvature on an elliptic curve",
*The Journal of Geometric Analysis*, 32 (7), 2022-07 - (with Chang C.-H., Tsai I-H.) "Theta functions and adiabatic curvature on an elliptic curve",
*The Journal of Geometric Analysis*, 32 (3), 2022-03 - (with Hung-Lin Chiu) "Positive mass theorem and the CR Yamabe equation on 5-dimensional contact spin manifolds",
*Advances in Mathematics*, 404 (108446), 50-, 2022 - (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 - "微分幾何學--重力研究的好幫手",
*數學傳播*, 44 (4), 27-33, 2020 - "Submanifolds in Cauchy Riemann Geometry",
*Journal of Mathematical Study*, 53 (4), 436-470, 2020 - (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 - (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 - (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 - (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 - (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 - (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*, 203-235, 2018 - (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 - "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 - (with Malchiodi A., Yang P. ) "A positive mass theorem in three dimensional Cauchy-Riemann geometry",
*Advances in Mathematics*, 308, 276-347, 2017 - (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 - (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 - "Flows and a tangency condition for embeddable CR structures in dimension 3",
*Methods and Applications of Analysis*, 21 (3), 337-356, 2014 - (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 - (with Chiu H.-L., Yang P.) "Uniformization of spherical CR manifolds",
*Advances in Mathematics*, 255, 182-216, 2014 - (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 - (with Malchiodi A., Yang P.) "A positive mass theorem in three dimensional Cauchy-Riemann geometry",
*Extended abstract Fall 2013, Research Perspectives CRM, Barcelona*, 2013 - (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 - (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 - (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 - "The prescribed p-mean curvature equation of low regularity in the Heisenberg group",
*Science in China, Series A*, 52 (12), 2604-2609, 2009 - (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 - "Surfaces of prescribed p-mean curvature in the Heisenberg group",
*Proceedings of the 4th ICCM*, 0, 40-58, 2007 - (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 - (with Jenn-Fang Hwang ! Paul Yang) "Existence and uniqueness for p-area minimizers in the Heisenberg group",
*Math. Annalen*, 337, 253-293, 2007 - (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 - "The mean curvature equation in pseudohermitian geometry",
*Contemporary Mathematics*, 400, 89-94, 2006 - (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 - (with Jenn-Fang Hwang) "Properly embedded and immersed minimal surfaces in the Heisenberg group",
*Bull. Aus. Math. Soc.*, 70, 507-520, 2004 - (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
- "Deformation of fillable CR structures, preprint; math.DG/0202051. ", 2002
- (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
- "淺談場量子化與低維拓樸. ",
*數學傳播*, 25 (4), 4-7, 2001-12 - "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
- "Rick Schoen, Yamabe 問題與正質量定理.",
*數學傳播*, 24 (4), 63-67, 2000-12 - (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 - "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
- "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 - (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 - "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
- (with John M. Lee) "A local slice theorem for 3-dimensional CR structures",
*Amer. J. Math.*, 117, 1249-1298, 1995 - "Some applications of Cartan's theory on three-dimensional Cauchy-Riemann geometry",
*Math. Zeit.*, 218, 527-548, 1995 - (with Shuenn-Jyi Sheu) "Brownian motion on pseudohermitian manifolds",
*incomplete draft*, 1994 - "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
- "On embeddable CR 3-manifolds",
*draft*, 1992 - "淺談Jones-Witten理論及Floer同調群. ",
*數學傳播*, 15 (1), 12-22, 1991-03 - "Curvature functions for the sphere in pseudohermitian geometry",
*Tokyo J. Math.*, 14, 151-163, 1991 - (with John M. Lee) "The Burns-Epstein invariant and deformation of CR structures",
*Duke Math. J.*, 60, 221-254, 1990 - "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
- "Chain-preserving diffeomorphisms and CR equivalence",
*Proceedings of AMS*, 103, 75-80, 1988 - "On the curvature of CR structures and its covariant derivatives",
*Math. Zeit.*, 196, 203-209, 1987 - "Graded Lie algebras of the second kind",
*Transactions of AMS*, 302, 467-488, 1987 - "On the second fundamental form of algebraic hypersurfaces",
*Bulletin of IMAS*, 10, 25-34, 1982 - "An integral formula on the scalar curvature of algebraic manifolds",
*Proceedings of AMS*, 81, 451-454, 1981 - "CR umbilicity for certain class of real hypersurfaces in $C^{n+1}$",
*draft* - (with C.H. Chang, H.P. Lee and A.N. Wang) "Examples of spiralling chains",
*draft* - "CR structures for a family of homogeneous spaces",
*draft*