Modified defect relation of Gauss maps on annular ends of minimal surfaces for hypersurfaces of projective varieties in subgeneral position

Abstract

Let A be an annular end of a complete minimal surface S in Rm and let V be a k-dimension projective subvariety of Pn(C)(n=m−1). Let g be the generalized Gauss map of S into V⊂Pn(C). In this paper, we establish a modified defect relation of g on the annular end A for q hypersurfaces {Qi}i=1q of Pn(C) in N-subgeneral position with respect to V. Our result implies that the image g(A) cannot omit all q hypersurfaces Q1,…,Qq if g is nondegenerate over Id(V) and q>(2N−k+1)(M+1)(M+2d)2d(k+1), where M=HV(d)−1 and d is the least of common multiple of deg⁡Q1,…,deg⁡Qq. As our best knowledge, it is the first time the value distribution of the Gauss map on an annular end of a minimal surfaces with hypersurface targets is studied, in particular the product into sum inequality for holomorphic curves on Riemann surfaces with hypersurfaces targets is presented. This our result has been used to study the unicity of the gauss maps in the recent work of C. Lu and X. Chen [14].