# Professor Ngaiming Mok 莫毅明教授

Research Interests:

Professor Mok’s recent works in Complex Differential Geometry revolve around bounded symmetric domains, holomorphic isometries and measure-preserving maps, rigidity problems and holomorphic geodesic cycles on Hermitian locally symmetric spaces. In Algebraic Geometry his recent works are primarily related to Fano manifolds and more generally uniruled projective varieties. He is on the editorial boards of Mathematische Annalen, Science China Mathematics and Chinese Annals of Mathematics.

Preprints:

• (with J. Pila & J. Tsimerman) Ax-Schanuel for Shimura Varieties (from arXiv:1711.02189v2). (pdf)
• (with S.-T. Chan) Asymptotic total geodesy of local holomorphic curves exiting a bounded symmetric domain and applications to a uniformization problem for algebraic subsets. (pdf)
• Some Recent Results on Holomorphic Isometries of the Complex Unit Ball into Bounded Symmetric Domains and Related Problems. (pdf)
• Zariski Closures of Images of Algebraic Subsets under the Uniformization Map on Finite-Volume Quotients of the Complex Unit Ball. (pdf)
• Full cones swept out by minimal rational curves on irreducible Hermitian symmetric spaces as examples of varieties underlying geometric substructures. (pdf)
• (with S.-C. Ng) On compact splitting complex submanifolds of quotients of bounded symmetric domains. (pdf)
• (with K.-K. Wong) Extension of inverses of Γ-equivariant holomorphic embeddings of bounded symmetric domains of rank ≥ 2 and applications to rigidity problems. (pdf)
Publications:

Publications since 1997

• (with Y. Zhang) Rigidity of pairs of rational homogeneous spaces of Picard number 1 and analytic continuation of geometric substructures on uniruled projective manifolds, to appear in J. Differential Geom. (pdf)
• (with S.-T. Chan) Holomorphic isometries of $\mathbb B^m$ into bounded symmetric domains arising from linear sections of minimal embeddings of their compact duals, Mathematische Zeitschrift. (pdf)
• Geometric structures and substructures on uniruled projective manifolds, in Foliation Theory in Algebraic Geometry (Simons Symposia), ed. P. Cascini, J. McKernan and J.V. Pereira, Springer-Verlag, Heidelberg-New York-London 2016, pp.103-148. (pdf)
• Holomorphic isometries of the complex unit ball into irreducible bounded symmetric domains, Proc. A.M.S. 144 (2016), 4515-4525. (pdf)
• Local holomorphic curves on a bounded symmetric domain in its Harish-Chandra realization exiting at regular points of the boundary, J. Pure Appl. Math. Quart. 10 (2014), 259-288. (pdf)
• Uniqueness modulo reduction of Bergman meromorphic compactifications of canonically embeddable Bergman manifolds, in Frontiers in Differential Geometry, Partial Differential Equations and Mathematical Physics, volume in memory of Gu Chaohao, World-Scientific, New Jersey-Singapore 2014, pp. 271-290. (pdf)
• (with J. Hong) Characterization of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1, J. Alg. Geom. 22 (2013), 333-362. (pdf)
• Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric, J. Eur. Math. Soc. 14 (2012), 1617-1656. (pdf)
• Projective-algebraicity of minimal compactifications of complex-hyperbolic space forms of finite volume, Perspectives in Analysis, Geometry, Topology, Progr. Math., 296, Birkhäuser/Springer, New York, 2012, pp.331-354. (pdf)
• (with S.-C. Ng) Germs of measure-preserving holomorphic maps from bounded symmetric domains to their Cartesian products, J. Reine Angew. Math. 669, (2012), 47-73.(pdf)
• On singularities of generically immersive holomorphic maps between complex hyperbolic space forms, in Complex and Differential Geometry, Proceedings of a conference at Leibniz Universität, Hannover (2009), Springer-Verlag, Berlin-Heidelberg 2011, pp.323-344. (pdf)
• Geometry of holomorphic isometries and related maps between bounded domains, in Geometry and Analysis, Vol. II, Advanced Lectures in Mathematics 18, Higher Educational Press, Beijing, 2011, pp.225-270. (pdf)
• (with J. Hong) Analytic continuation of holomorphic maps respecting varieties of minimal rational tangents and applications to rational homogeneous manifolds, J. Differential Geom. 86 (2010), 539-567. (pdf)
• (with V. Koziarz) Nonexistence of holomorphic submersions between complex unit balls equivariant with respect to a lattice and their generalizations, Amer. J. Math. 132 (2010), 1347-1363. (pdf)
• On the Zariski closure of a germ of totally geodesic complex submanifold on a subvariety of a complex hyperbolic space form of finite volume, in Complex Analysis, Trends in Mathematics, ed. P. Ebenfelt, etc., Birkhäuser, Basel-Boston 2010, pp. 279-300. (pdf)
• (with S.-C. Ng and Z. Tu) Factorization of proper holomorphic maps on irreducible bounded symmetric domains of rank ³ 2, Science China Mathematics - Special Issue Dedicated to Professor Yang Lo on the Occasion of his 70th Birthday, 53 (2010), 813-826. (pdf)
• (with S.-C. Ng) Second fundamental forms of holomorphic isometries of the Poincaré disk into bounded symmetric domains and their boundary behavior along the unit circle, Science in China Series A: Mathematics - Dedicated to Professor ZHONG TongDe on the occasion of his 80th birthday, 52 (2009), 2628-2646. (pdf)
• On the asymptotic behavior of holomorphic isometries of the Poincaré disk into bounded symmetric domains, Special Volume in honor of Professor Wu Wenjun on the occasion of his 90th birthday, Acta Mathematica Scientia, 29B(4) (2009), 881-902. (pdf)
• (with X. Sun) Remarks on lines and minimal rational curves, Science in China Series A: Mathematics, 52 (2009), 617-630. (pdf)
• Geometric structures on uniruled projective manifolds defined by their varieties of minimal rational tangents, Proceedings of the Conference "Géometrie différentielle, Physique mathématique, Mathématique et Société", Astérisque, 322 (2008), Volume II, 151-205, published by Société Mathématique de France. (pdf)
• 有界全純函數與VMRT幾何理論在剛性問題上的應用，《多複變在中國的研究與發展》，第492-554頁，科技出版社，北京，2009。 (pdf)
• Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents, AMS/IP Studies in Advanced Mathematics, 42 (2008), 41-61. (pdf)
• Nonexistence of proper holomorphic maps between certain classical bounded symmetric domains, Chinese Annals of Mathematics - Series B 29 (2008), 135-146. (pdf)
• Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents, Science in China Series A: Mathematics 51 (2008), 660-684. (pdf)
• Ergodicity, bounded holomorphic functions and geometric structures in rigidity results on bounded symmetric domains, in Proceedings of the International Congress of Chinese Mathematicians (Hangzhou 2007), Volume II, Higher Educational Press, Beijing, 2007, pp.464-505. (pdf)
• Rigidity problems on compact quotients of bounded symmetric domains, AMS/IP Studies in Advanced Mathematics 39 (2007), 201-249. (pdf)
• (with J.-M. Hwang) Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation, Invent. Math. 160 (2005), 591-645.(pdf)
• On holomorphic immersions into Kähler manifolds of constant holomorphic sectional curvature, Science in China Ser. A Mathematics 48 Supp. (2005), 123-145. (pdf)
• Extremal bounded holomorphic functions and an embedding theorem for arithmetic varieties of rank ³ 2, Invent. Math. 158 (2004), 1-31. (pdf)
• (with P. Eyssidieux) On the validity or failure of gap rigidity for certain pairs of bounded symmetric domains, Special Volume in memory of Professor Armand Borel, Asian J. Math. 8 (2004), 773-794. (pdf)
• (with J.-H. Hwang) Birationality of the tangent map for minimal rational curves, Special Volume dedicated to Professor Yum-Tong Siu on his sixtieth birthday, Asian J. Math. 8 (2004), 51-64. (pdf)
• (with J.-M. Hwang) Automorphism groups of the spaces of minimal rational curves on Fano manifolds of Picard Number 1, J. Alg. Geom. 13 (2004), 663-673.
• (with J.-M. Hwang) Deformation rigidity of the 20-dimensional F4-homogeneous space associated to a short root. to appear in Algebraic Transformation Groups and Algebraic Varieties, ed. V.L. Popov, Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin-Heidelberg-New York 2004.
• Negativity of curvature on spaces parametrizing Hodge decompositions of reduced first cohomology groups, Contemp. Math. 332 (2003), 219-234.
• (with J.-M. Hwang) Finite morphisms onto Fano manifolds of Picard number 1 which have curves with trivial normal bundle, J. Alg. Geom. 12 (2003), 627-652.
• Local holomorphic isometric embeddings arising from correspondences in the rank-1 case, in Contemporary Trends in Algebraic Geometry and Algebraic Topology, ed. S.-S. Chern, L. Fu and R. Hain, Nankai Tracts in Mathematics, Vol.5, World Scientific, New Jersey 2002, pp.155-166.
• (with J.-M. Hwang) Deformation rigidity of the rational homogeneous space associated to a long root, Ann. Scient. Ec. Norm. Sup. 35 (2002), 173-184.
• On Fano manifolds with nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents, Trans. A.M.S. 354 (2002), 2639-2658.
• Characterization of certain holomorphic geodesic cycles on quotients of bounded symmetric domains in terms of tangent subspaces, Comp. Math. 132 (2002), 289-309
• (with J.-M. Hwang) Projective manifolds dominated by abelian varieties, Math. Zeit. 238 (2001), 89-150.
• (with J.-M. Hwang) Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1, J. Math. Pure Appl. 80 (2001), 563-575.
• Fibrations of compact Kähler manifolds in terms of cohomological properties of their fundamental groups, Ann. Inst. Fourier 50 (2000), 633-675.
• (with J.-M. Hwang) Varieties of minimal rational tangents on uniruled manifolds, in Several Complex Variables , ed. M. Schneider and Y.-T. Siu, MSRI publications 37, Cambridge University Press, Cambridge 1999, pp.351-389.
• G-structures on irreducible Hermitian symmetric spaces of rank ³ 2 and deformation rigidity, Contemp. Math. 222, 1999, pp.81-107.
• (with J.-M. Hwang) Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds, Invent. Math. 136 (1999), 209-231.
• (with J.-M. Hwang) Characterization and deformation-rigidity of compact irreducible Hermitian symmetric spaces of rank ³ 2 among Fano manifolds, in Lectures in Algebra and Geometry , ed. Kang, M.-C., International Press, Cambridge, Mass., 1998, pp.15-46.
• (with J.-M. Hwang) Uniruled projective manifolds with irreducible reductive G-structures, J. Reine Angew. Math. 490 (1997), 55-64.
• (with J.-M. Hwang) Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation, Invent. Math. 131 (1998), 393-418.
• Steinness of universal coverings of certain compact Kähler manifolds, J. Reine Angew. Math. 498 (1997), 1-14.
• The generalized theorem of Castelnuovo-de Franchis for unitary representations, in Geometry from the Pacific Rim , ed. A.J. Berrick et al., de Gruyter, Berlin-New York 1997, pp.261-284.

Miscellaneous:

• Erratum to "Local holomorphic isometric embeddings arising from correspondences in the rank-1 case", 2005. (pdf)

List of Representative Publications (1986-2012):

1. Metric rigidity theorems on locally symmetric Hermitian spaces, Proc. Natl. Acad. Sci. U.S.A. 83 (1986), 2288-2290.
2. Uniqueness theorems of Hermitian metrics of seminegative curvature on locally symmetric spaces of negative Ricci curvature, Ann. Math. 125 (1987), 105-152.
3. The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom. 27 (1988), 179-214.
4. Compactification of complete Kähler surfaces of finite volume satisfying certain curvature conditions, Ann. Math. 129 (1989), 383-425.
5. (with J.-Q. Zhong) Compactifying complete Kähler-Einstein manifolds of finite topological type and bounded curvature, Ann. Math. 129 (1989), 427-470.
6. (with H.-D. Cao) Holomorphic immersions between compact hyperbolic space forms, Invent. Math. 100 (1990), 49-61.
7. Factorization of semisimple discrete representation of Kähler groups, Invent. Math. 110 (1992), 557-614.
8. (with Y.-T. Siu and S.-K. Yeung) Geometric superrigidity, Invent. Math. 113 (1993), 57-83.
9. (with J.-M. Hwang) Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation, Invent. Math. 131 (1998), 393-418.
10. (with J.-M. Hwang) Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds, Invent. Math. 136 (1999), 209-231.
11. Extremal bounded holomorphic functions and an embedding theorem for arithmetic varieties of rank ³ 2, Invent. Math. 158 (2004), 1-31.
12. (with J.-M. Hwang) Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation, Invent. Math. 160 (2005), 591-645.
13. Geometric structures on uniruled projective manifolds defined by their varieties of minimal rational tangents, Proceedings of the Conference "Géometrie différentielle, Physique mathématique, Mathématique et Société", Astérisque 322 (2008), Volume II, 151-205, published by Société Mathématique de France.
14. (with J. Hong) Analytic continuation of holomorphic maps respecting varieties of minimal rational tangents and applications to rational homogeneous manifolds, J. Differential Geom. 86 (2010), 539-567.
15. (with S.-C. Ng) Germs of measure-preserving holomorphic maps from bounded symmetric domains to their Cartesian products, J. Reine Angew. Math.669, (2012), 47-73.
16. Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric, J. Eur. Math. Soc. 14 (2012), 1617-1656.
Lecture Notes:
• Ergodicity, bounded holomorphic functions and geometric structures in rigidity results on bounded symmetric domains, 2007. (pdf)
• Complex Geometry on Bounded Symmetric Domains, 2006.
• I. Metric rigidity(pdf)
• II. Geometric structures and III. Proper holomorphic maps and related problems(pdf)
• Characterization of Holomorphic Geodesic Cycles on Quotients of Bounded Symmetric Domains, 2005. (pdf)
• From Rational Curves to Complex Structures on Fano Manifolds, 2005. (pdf)