Asymptotic behavior of solutions of a certain quasi non-linear operator and its application to geometric function theory

本课题的目的是研究完备黎曼流形(M，g)上一类拟非线性算子P的(次)解的渐近性。这里P要么是拉普拉斯算子，要么是平均曲率算子，这是最有趣的情况。研究了与该算子解的极大值原理有关的几个问题。在没有(M，g)的任何Ricci曲率条件的情况下，我们可以证明这类算子P的广义极大值原理。我们的方法仅依赖于该流形的某些体积增长条件。从这个原理我们可以得到几个有趣的结果：(1)标量曲率方程解的唯一性，(2)调和映射的Liouvile型定理，(3)保持标量曲率的共形变换的等距性质，(4)完备流形的极小浸入的值分布，这些结果几乎包含了迄今为止黎曼几何中已知的结果。进一步，我们研究了(M，g)上测地球面上次调和函数的L p-积分的增长性，得到了这些积分的最优增长性估计。这一结果也与完备流形上的最大值原理有关。根据这个估计，我们可以给出(M，g)是抛物线的一个非常简单的函数论证明，并得到与问题(1)~(4)有关的几个结果。

The purpose of this project is to study asymptotic behaviour of (sub-) solutions of a certain quasi non-linear operator P on a complete Riemannian manifold (M, g). Here P is either the Laplacian or the mean curvature operator which is the most interesting case. Several topics related to maximum principle for solutions of that operator have been studied. We could show the generalized maximum principle for such an operator P without any Ricci curvature condition of (M, g). Our method depends only on some volume growth condition of that manifold. From the principle we can induce several interesting results related to (1) uniqueness of solutions of the scaler curvature equation, (2) Liouville type theorem for harmonic maps, (3) isometric property of conformal transformations preserving scaler curvature and (4) value distribution of minimal immersions of complete manifolds, which contain almost all known results up to now in Riemannian geometry. Furthermore we studied a growth property of L^p-integrals of subharmonic functions on geodesic spheres on (M, g), and obtained an optimal growth estimate of those integrals. This result is also related to the maximum principle on complete manifolds. From this estimate we can yield a very simple and function theoretic proof for (M, g) to be parabolic, and get several results related to the problem (1)〜(4).