Study on global properties of local martingales and martingales on manifolds.

首席调查员A.Atsuji获得了以下三个主要结果。对于一个不能完全浸入的最小曲面，如果它的投影体积是有限的，那么它的总曲率是有限的。在这个结果中，我们在表面上使用了布朗运动的一些性质，这些性质在周围空间上是鞅的。利用一维局部鞅的性质证明了δ-次调和函数的对数导数引理的有效性。抛物线性用δ-次谐波函数表示。利用这个性质，我们给出了从抛物流形到Hadamard流形的有限能量调和映射的一些Liouville型定理。我们利用调和映射作为布朗运动图象给出的鞅的一些性质。给出了C^n中复子流形上亚纯函数的Nevanlinna理论。S.Kotani给出了一维扩散为真鞅的充分必要条件。takgoshi给出了完全黎曼流形为抛物型的若干判据，调和映射的Liouville型定理，以及正标量曲率共形度量等距的若干结果。铃木裕。具有单侧布朗势的随机环境中扩散过程的长时间行为的相同结果。

Head investigator A.Atsuji obtained the following main three results.1. For a minimal surface which may not be properly immersed, its total curvature is finite if its projective volume is finite. In this result we used some properties of Brownian motion on the surfaces which is martingale on the ambient spaces. Lemma for logarithmic derivative of δ-subharmonic functions which is proved by using properties of 1-dimensional local martingale works well.2. Parabolicity is characterized by δ-subharmonic functions. Using this characterization we showed some Liouville type theorems for harmonic maps of finite energy from parabolic manifolds to Hadamard manifolds. We used some properties of martingales given as images of Brownian motion by harmonic maps.3. A Nevanlinna theory for meromorphic functions on complex submanifolds in C^n is obtained. S.Kotani gave a necessarily and sufficient condition for 1 dimensional diffusions to be true martingales. K.Takegoshi gave some criteria for complete Riemannian manifolds to be parabolic, Liouville type theorems for harmonic maps and some results on isometry of conformal metric of positive scalar curvature. Y.Suzuki same results on long time behavior of diffusion processes in random environment with one-sided Brownian potential.