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A MONOTONICITY FORMULA ON COMPLETE KÄHLER MANIFOLDS WITH NONNEGATIVE BISECTIONAL CURVATURE

In this paper, we derive a new monotonicity formula for the plurisuhbarmonic functions on complete Kähler manifolds with nonnegative bisectional curvature. As applications we derive the sharp estimates for the dimension of the spaces of holomorphic functions (sections) with polynomial growth, which in particular, partially solves a conjecture of Yau. §0 Introduction. In [Y], Yau proposed to study the uniformization of complete Kähler manifolds with nonnegative curvature. In particular, one wishes to determine if a complete Kähler manifold M with positive bisectional curvature is biholomorphic to C or not. For this sake, it was further asked in [Y] that if the the ring of the holomorphic functions with polynomial growth, which we denote by OP (M), is finitely generated or not, and if 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 or not. This paper addresses the later question. 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≥0 Od(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 (0.1) dimC(Od(M)) ≤ dimC(O[d](C)). Here [d] is the greatest integer less than or equal to d. Denote Vo(r) the volume of the ball of radius r centered at o. For manifolds with nonnegative Ricci curvature, Vo(r) r2m is monotone decreasing by the Bishop volume comparison theorem. M is called to have the maximum volume growth if limr→∞ Vo(r) r2m > 0. Although we have not quite proved the finite generation for the ring OP (M), we can show that the quotient field generated by OP (M) is finitely generated. In fact, we obtain the following dimension estimate for the general case. Research partially supported by NSF grant DMS-0328624, USA.