Three Circle Theorems on K\"ahler manifolds and applications

Abstract

The classical Hadamard three circle theorem is generalized to complete K\"ahler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem. As corollaries, two sharp monotonicity formulae for holomorphic functions are derived. Among applications, we derive sharp dimension estimates (with rigidity) of holomorphic functions with polynomial growth when the holomorphic sectional curvature is nonnegative. When the bisectional curvature is nonnegative, this was due to Ni. Also we study holomorphic functions with polynomial growth near infinity. On a complete noncompact K\"ahler manifold with nonnegative bisectional curvature, we prove any holomorphic function with polynomial growth is homogenous at infinity. This result is closely related with Yau's conjecture on the finite generation of holomorphic functions (see page $3-4$ for detailed explanation). We also generalize the three circle theorem to K\"ahler manifolds with holomorphic sectional curvature lower bound. These inequalities are sharp in general. As applications, we establish sharp dimension estimates for holomorphic functions with certain growth conditions when the holomorphic sectional curvature is asymptotically nonnegative. Finally observe that Hitchin constructed some complex manifolds which admit K\"ahler metric with positive holomorphic sectional curvature, yet do not even admit K\"ahler metrics with nonnegative Ricci curvature. Thus the set of complete K\"ahler manifolds with nonnegative holomorphic sectional curvature is strictly wider than those with nonnegative bisectional curvature.