# Research

My research interests lie in the area of complex geometry and number theory, in particular, the relationship between Nevanlinna theory (the theory of complex hyperbolicities) and Diophantine approximation.

The theory of diophantine geometry has a long rich history dating back all the way to the Greek schools and culminating in the great breakthrough in the 1980s by Faltings in the resolution of the Mordell Conjecture and the proof by Wiles of Fermat's Last Theorem in 1994 concerning the non-existence of integer solutions of the Fermat polynomial x^n+y^n=x^n. Similar questions arise in higher dimension as well, namely, solutions of polynomials of more variables. Analogous questions are also investigated by algebraic geometers, searching for solutions over function fields and by complex geometers, searching for meromorphic solutions. Such theory is called the Nevanlinna theory. Research in this domain has been carried out by many outstanding mathematicians, starting from Nevanlinna's time until recent years. It is perhaps worth noticing that the first Fields Medal was awarded, in 1936, to Lars Ahlfors for his works in this field. Some notable well-known mathematicians like S.S. Chern, P.H. Griffiths, Yum-Tong Siu, J.P. Demailly all have worked in this subject. The research in Diophantine geometry itself also produced at least five Fields medalists (Roth, Baker, Bombieri, Faltings and Wiles).

In 1983, C.H. Osgood was the first one who noticed the striking analogy between the subject of Nevanliina theory in complex geometry and Diophantine approximation in number theory. The link between these two theories, since then, has been deeply investigated, notably by Serge Lang, Paul Vojta, P.M. Wong, and myself etc. The research in exploring the deep relationship betweek these two subjects has spurred great advances in both fields. Lang conjectured that if M is a projective variety defined over a number field K and is hyperbolic (i.e. f every entire map f from the complex plane C to M is constant), then there are only finitely many K-rational points on M. This serves a guideline of our study.

My another research interest is the value distribution properties of the Gauss maps of minimal surfaces in R^n, a program initiated by S.S.Chern and Robert Osserman in early sixties.

# Publications

1. X. Pang & Min Ru. On the total number of entire deficient functions of entire functions, Chinese Ann. of Math., Vol 6, Ser. A, No. 4(1985), 411-424, (in Chinese).
2. Min Ru. A general theorem on the total number of deficient values for a class of meromorphic functions, Chinese Ann. of Math., Vol 9, Ser. A, No. 2 (1988), 178-187 (in Chinese). MR: 90h: 30077.
3. Min Ru. Some discussion on common Borel directions of meromorphic functions, J. East China Normal Univ. Sci. Ed.,} No. 1 (1989), 39-50, (in Chinese). MR: 90h: 30078.
4. Min Ru & W. Stoll. Courbes holomorphes $\acute e$vitant des hyperplans mobiles, C. R. Acad. Sci. Paris t310, sere I, (1990), 45-48.
5. Min Ru & P.M. Wong. Integral points of P^n - {2n + 1 hyperplanes in general position}, Inventiones Mathematicae, Vol. 106, (1991), 195-216.
6. Min Ru. On the Gauss map of minimal surfaces immersed in R^n, Journal of Differential Geometry, Vol. 34, No. 2, (1991), 411-423.
7. Min Ru & W. Stoll. The second main theorem for moving targets, Journal of Geometric Analysis, Vol. 1, No. 2, (1991), 99-138.
8. Min Ru. On the Gauss map of minimal surfaces with finite total curvature, Bulletin of the Australian Math. Soc., 44(1991), 225-232.
9. Min Ru & W. Stoll. The Cartan conjecture for moving targets, Proceeding of Symposia in Pure Mathematics, AMS, Vol. 52, Part 2, (1991), 477-508.
10. Min Ru & W. Stoll. The Nevanlinna conjecture for moving targets, Research and Lecture notes in Mathematics, Mediterranean Press, 1991, 293-308.
11. Min Ru. Integral points and the hyperbolicity of the complement of hypersurfaces, J. Reine Angew. Math., 442(1993), 163-176.
12. Min Ru. Gauss map of minimal surfaces with ramification, Trans. Amer. Math. Soc., Vol. 339, No. 2, (1993), 751-764.
13. Shanyu Ji & Min Ru. Global Lojasiewicz inequality, defect relation and applications of holomorphic curve theory, Contemp. Math., Amer. Math. Soc., Vol 142, (1993), 49-59.
14. Min Ru. Geometric and arithmetic aspects of $P^n$ minus hyperplanes, American Journal of Mathematics, Vol. 117, No. 2, (1995), 307-321.
15. Min Ru & P. Vojta. Schmidt's subspace theorem with moving targets, Inventiones Mathematicae, Vol. 127 (1997), 51-65.
16. Min Ru. The second main theorem on parabolic manifolds, Indiana Univ. Math. Journal., Vol. 46, No. 1, (1997), 299-318.
17. R. Osserman & Min Ru. An estimate for the Gauss curvature on minimal surfaces in ${\bf R}^m$ whose Gauss map omits a set of hyperplanes, Journal of Differential Geometry, 46(1997), 578-593.
18. Min Ru. On the general form of the second main theorem, Trans. Amer. Math. Soc., 349(1997), 5093-5105.
19. Min Ru. Nevanlinna theory and its relation with Diophantine approximation, Bulletin of the Hong Kong Math. Soc., Vol 1 (1997), 343-349.
20. K. Gyory and Min Ru. Integer solutions of a sequence of decomposable from inequalities, Acta Arithmetica, Vol. 86 (1998), 227-237.
21. Min Ru and J. Tzu-Yueh Wang. Diophantine approximation with algebraic points of bounded degree, Journal of Number Theory, 81(2000), 110-119.
22. Min Ru. Algebroid functions, Wirsings theorem and their relations, Mathematische Zeitschrift, Vol. 233, No. 1 (2000), 137-148.
23. Min Ru. A uniqueness theorem for rational points in projective space, Journal of Number Theory, 85(2000), 85-91.
24. Min Ru. A weak effective Roth's theorem over function fields, Rocky Mountain Journal of Mathematics, Vol. 30, No. 2, (2000), 723-734.
25. Min Ru. A note on p-adic Nevanlinna theory, Proc. Amer. Math. Soc. 129(2001), 1263-1269.
26. Min Ru. The moving target problems in Nevanlinna theory, Complex Variables, 43(2001), 417-431.
27. Min Ru. A uniqueness theorem for moving targets without counting multiplicities, Proc. Amer. Math. Soc., 129(2001), 2701-2707.
28. Min Ru. Uniqueness theorems for p-adic holomorphic curves, Illinois Journal of Mathematics, 45(2001), 487-493.
29. Min Ru and Julie Wang. Truncated second main theorem with moving targets, Trans. Amer. Math. Soc., 356(2004), No. 2, 557-571.
30. Min Ru. A defect relation for holomorphic curves intersecting hypersurfaces, American Journal of Mathematics, 126(2004), 215-226.
31. W. Cherry and Min Ru. Rigid analytic Picard theorems, American Journal of Mathematics, 126(2004), 873-889.
32. Min Ru and Eunjeong Yi. Nevanlinna theory and iteration of rational maps, Mathematische Zeitschrift, 249(2005), 125-138.
33. Zhihua Chen and Min Ru. Decomposable form equations without the finiteness property, Proc. Amer. Math. Soc., 133(2005), 1929-1933.
34. Zhihua Chen and Min Ru. Integer solutions to decomposable form inequalities, Journal of Number Theory, 115(2005), 58-70.
35. Yuancheng Liu and Min Ru. Degeneracy of Holomorphic Curves in Surfaces, Science in China, Series A, Mathematics, 48(2005), 156-167.
36. Yuancheng Liu and Min Ru. A defect relation for meromorphic maps on parabolic manifolds intersecting hypersurfaces, Illinois Journal of Mathematics, 49(2005), 237-257.
37. Min Ru and Julie Wang. A seond main theorem on parabolic manifolds, Asian J. Math. 9(2005), 349-372.
38. Lu Jin and Min Ru. A unicity theorem for moving targets counting multiplicities, Tohoku Math. Journal, 57(2005), 589-595.
39. Zhihua Chen and Min Ru. A uniqueness theorem with moving targets, Houston J. of Math., 32(2006), No. 2, 589-601.
40. Lu Jin and Min Ru. Values of Gauss maps of complete minimal surfaces in ${\Bbb R}^m$ on annular ends, Trans. Amer. Math. Soc., 359(2007), 1527-1546.
41. Lu Jin and Min Ru. Algebraic curves and the Gauss map of algebraic minimal surfaces, Differential Geometry and its Applications, 25(2007), 701-712.
42. Min Ru. The second main theorem with hypersurfaces over function fields, Proceedings of the International Conference on Complex Geometry and Related Fields, AMS/IP Stud. Adv. Math. 39(2007), 251-261.
43. Yan Xu and Min Ru. Uniqueness theorem for algebraic curves on compact Riemann surfaces, Science in China, Series A, 50(2007), 683-688.
44. Min Ru. A fundamental inequality for holomorphic curves into projective varieties, Proceedings ICCM (International Congress of Chinese Mathathematicians), Vol II(2007), 534-544, Higher Educational Press (Beijing, China)/International Press (Somerville, MA, USA).
45. Matt Dulock and Min Ru. A uniqueness theorem for holomorphic curves into encountering hypersurfaces in projective space, Complex Variables & Elliptic Equations, 53(2008), 797-802.
46. Yasheng Ye and Min Ru. A big Picard theorem for holomorphic maps into complex projective space, Canadian Math. Bulletin, 52(2009), 154-160.
47. Min Ru. Holomorphic curves into algebraic varieties, Annals of Mathematics, 169(2009), 255-267.
48. Zhihua Chen, Min Ru and Qiming Yan. The Truncated Second Main Theorem and Uniqueness Theorems, Science in China, 53(2010), 605-616.
49. Matt Dulock and Min Ru. Uniqueness of holomorphic curves into abelian varieties, Trans. Amer. Math. Soc. 363(2011), 131-142.
50. Min Ru. Integer solutions to decomposable and semi-decomposable form inequalities, Publ. Math. Debrecen, 79(2011), 663-673
51. Min Ru and Suraizou Sogome. Non-integrated defect relation for meromorphic maps of complete K$\ddot{A}$hler manifolds into $\mathbb{P}^n(\mathbb{C})$ intersecting hypersurfaces, Trans. Amer. Math. Soc., 364(2012), Number 3, 1145-1162
52. Min Ru and Julie Tzu-Yueh Wang. An effective Schmidt's subspace theorem for Projective Varieties Over Function Fields, IMRN (International Mathematics Research Notices), (2012), 651-684 (first published online March 28, 2011).
53. Gordon Heier and Min Ru. On essentially large divisors, Asian Journal of Mathematics, 16(2012), 387-407
54. Zhihua Chen, Min Ru and Qiming Yan. The Degenerated Second Main Theorem and Schmidt's Subspace Theorem, Science in China, 55(2012), 1367-1380
55. Min Ru and Suraizou Sogome. A uniqueness theorem for meromorphic maps of a complete K$\ddot{a}$hler manifold into ${\bf P}^n({\bf C})$ sharing hypersurfaces, Proc. Amer. Math. Soc., 141(2013), 4229-4239
56. Min Ru. Some Generalizations of the Second Main Theorem Intersecting Hypersurfaces, Methods and Applications of Analysis, 21(2014), 503-526.
57. Hungzen Liao and Min Ru. A note on the Second Main Theorem for holomorphic curves into algebraic varieties, Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 9(2014), No. 4, 671-684
58. Zhihua Chen, Min Ru and Qiming Yan. Schmidt's Subspace Theorem with Moving Hypersurfaces, IMRN (International Mathematics Research Notices), (2015), No. 15, 6305-6329
59. Lei Shi and Min Ru. An improvement of Chen-Ru-Yan's degenerated second main theorem, Science in China, 58(2015), No. 12, 2517-2530
60. Min Ru. A defect relation for holomorphic curves intersecting general divisors on projective varieties, Journal of Geometric Analysis, 26(2016), No. 4, 2751-2776
61. Min Ru .A general diophantine inequality, Funct. Approx. Comment. Math., to appear
62. Jungim Park and Min Ru. Unicity results of Gauss maps of minimal surfaces immersed in ${\bf R}^m$, Journal of Geometry, available as 'Online First' on SpringerLink
63. C. Mills and Min Ru. An improvement defect relations for holomorphic curves in projective varieties. Proceedings of  the Conference Complex Analysis and Dynamical Systems VII, AMS Series Contemporary Mathematics, to appear .

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