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# Cheng, Jih-Hsin

 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 at 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. Email : cheng AT math.sinica.edu.tw Phone:+886 2 2368-5999 ext. 631 Fax: +886 2 2368-9771

### Publications :

 1 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 (2017), 23-35 2 An integral formula on the scalar curvature of algebraic manifolds, Proceedings of AMS , 81(1981), 451-454 3 On the second fundamental form of algebraic hypersurfaces, Bulletin of IMAS , 10(1982), 25-34 4 CR structures for a family of homogeneous spaces, draft 5 (with C.H. Chang, H.P. Lee and A.N. Wang) Examples of spiralling chains, draft 6 Graded Lie algebras of the second kind, Transactions of AMS , 302(1987), 467-488 7 On the curvature of CR structures and its covariant derivatives, Math. Zeit. , 196(1987), 203-209 8 Chain-preserving diffeomorphisms and CR equivalence, Proceedings of AMS , 103(1988), 75-80 9 CR umbilicity for certain class of real hypersurfaces in $C^{n+1}$, draft 10 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) 11 (with John M. Lee) The Burns-Epstein invariant and deformation of CR structures, Duke Math. J. , 60(1990), 221-254 12 淺談Jones-Witten理論及Floer同調群., 數學傳播 , 15(1991-03), no. 1, 12-22 13 Curvature functions for the sphere in pseudohermitian geometry, Tokyo J. Math. , 14(1991), 151-163 14 On embeddable CR 3-manifolds, draft (1992) 15 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., (1993), 105-111 16 (with Shuenn-Jyi Sheu) Brownian motion on pseudohermitian manifolds, incomplete draft (1994) 17 Some applications of Cartan's theory on three-dimensional Cauchy-Riemann geometry, Math. Zeit. , 218(1995), 527-548 18 (with John M. Lee) A local slice theorem for 3-dimensional CR structures, Amer. J. Math. , 117(1995), 1249-1298 19 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., (1998), 201-208 20 (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(2000), 301-346 21 Rigidity of automorphisms and spherical CR structures. Proc. Amer. Math. Soc., 128(2000), 1825-1832; math. DG/9806108., Proc. Amer. Math. Soc. , 128(2000), 1825-1832 22 Cauchy-Riemann geometry and contact topology in three dimensions. Proc. Natl. Sci. Counc., ROC(A), 24(2000), 87-94; math. DG/0003211., , 24(2000), 87-94 23 (with 李瑩英, 蔡宜洵) 有朋自遠方來 ---Rick Schoen 訪談紀要., 數學傳播 , 24(2000-12), no. 4, 58-62 24 Rick Schoen, Yamabe 問題與正質量定理., 數學傳播 , 24(2000-12), no. 4, 63-67 25 淺談場量子化與低維拓樸., 數學傳播 , 25(2001-12), no. 4, 4-7 26 (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) 27 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) 28 (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) 29 Deformation of fillable CR structures, preprint; math.DG/0202051., 30 (with Jenn-Fang Hwang) Properly embedded and immersed minimal surfaces in the Heisenberg group, Bull. Aus. Math. Soc. , 70(2004), 507-520 31 (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(2005), no. 1, 129-177 32 The mean curvature equation in pseudohermitian geometry, Contemporary Mathematics , 400(2006), 89-94 33 (with Jenn-Fang Hwang ! Paul Yang) Existence and uniqueness for p-area minimizers in the Heisenberg group, Math. Annalen , 337(2007), 253-293 34 Surfaces of prescribed p-mean curvature in the Heisenberg group, Proceedings of the 4th ICCM , II(2007), 40-58(Dec.17-22, Hangzhou, ed. L. Ji, K. Liu, L. Yang, and S.T. Yau.) 35 (with Shu-Cheng Chang ! Hung-Lin Chiu) A fourth order curvature flow on a CR 3-manifold, Indiana Univ. Math. J. , 56(2007), 1793-1826 36 (with Jenn-Fang Hwang!Paul Yang) Regularity of $C^{1}$ smooth surfaces with prescribed p-mean curvature in the Heisenberg group, Math. Annalen , 344(2009), no. 1, 1-35 37 (with 劉太平, 江孟蓉) 有朋自遠方來--專訪Kenji Fukaya教授, 數學傳播 , 33(2009-09), no. 3, 3-12 38 The prescribed p-mean curvature equation of low regularity in the Heisenberg group, Science in China, Series A , 52(2009), no. 12, 2604-2609(2007 International Conference on Several Complex Variables and Complex Geometry, May 28-31, 2007, Xiamen.) 39 (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(2010), no. 4, 369-412 40 (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(2012), 131-198 41 (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.) 42 (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(2014), 579-597(published online: 29 June 2013) 43 (with Chiu H.-L., Yang P.) Uniformization of spherical CR manifolds, Advances in Mathematics , 255(2014), 182-216 44 Flows and a tangency condition for embeddable CR structures in dimension 3, Methods and Applications of Analysis , 21(2014), no. 3, 337-356 45 (with 劉太平, 蔡忠潤) 有朋自遠方來--專訪Clifford Taubes教授, 數學傳播 , 37(2013-03), no. 1, 3-14 46 (with 夏杼,何南國) 有朋自遠方來--專訪Nigel Hitchin教授, 數學傳播 , 39(2015-12), no. 4, 3-16 47 (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 , 2018(2018-05), no. 738, 203-235 48 (with Malchiodi A., Yang P.) A positive mass theorem in three dimensional Cauchy-Riemann geometry, Advances in Mathematics , 308(2017), 276-347 49 (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(2016-06), no. 3 50 (with Hsiao C.-Y., Tsai I-H.) Heat kernel asymptotics, local index theorem and trace integrals for CR manifolds with $S^1$ action, (preprint)(arXiv:1511.00063.) 51 (with Case J.S., Yang P.) An integral formula for the Q-prime curvature in 3-dimensional CR geometry, Proceedings of AMS (2018-07 accepted)(arXiv:1709.05367v1) 52 (with Yang P., Zhang Y.) Invariant surface area functionals and singular Yamabe problem in 3-dimensional CR geometry, Advances in Mathematics (2018-07 online)(arXiv:1711.04120v1) 53 (with Marugame T., Matveev V.S., Montgomery R.) Chains in CR geometry as geodesics of a Kropina metric, . (preprint)(arXiv:1806.01877) 54 (with Chiu H.-L.) Connected sum of spherical CR manifolds with positive CR Yamabe constant, The Journal of Geometric Analysis (2018-09 accepted)(arXiv:1805.08485v1) 55 (with Chiu H.-L., Hwang J.-F., Yang P.) Strong Maximum principle for mean curvature operator on subriemannian manifolds, Math. Annalen (2018-06 online), 0-0