A Monotonicity Formula on Complete Kähler Manifolds with Nonnegative Bisectional Curvature

In [Y], Yau proposed to study the uniformization of complete Kähler manifolds with nonnegative curvature. In particular, one wishes to determine whether or not a complete Kähler manifold M with positive bisectional curvature is biholomorphic to C. See also [GW], [Si]. For this sake, it was further asked in [Y] whether or not the ring of the holomorphic functions with polynomial growth, which we denote by OP (M), is finitely generated, and whether or not the dimension of the spaces of holomorphic functions of polynomial growth is bounded from above by the dimension of the corresponding spaces of polynomials on C. This paper addresses the latter questions. We denote by Od(M) the space of holomorphic functions of polynomial growth with degree d. (See Section 3 for the precise definition.) Then OP (M) = ⋃ d≥0Od(M). In this paper, we show that Theorem 0.1. Let M be a complete Kähler manifold with nonnegative holomorphic bisectional curvature. Assume that M is of maximum volume growth. Then