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一类拟非线性算子P在完全黎曼流形（M, g）上的（次）解的渐近行为。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。本文研究了该算子解的极大值原理。我们可以给出这样一个算子P的广义极大原理，而不需要（M, g）的任何Ricci曲率条件。我们的方法只依赖于流形的体积增长条件。从这个原理我们可以归纳出几个有趣的结果，涉及到(1)标量曲率方程解的唯一性，(2)调和映射的Liouville型定理，(3)保标量曲率的共形变换的等距性质，以及(4)完全流形的最小浸没值分布，这些结果包含了迄今为止黎曼几何中几乎所有已知的结果。进一步研究了（M, g）上测地线球上次调和函数的L^p积分的增长性质，得到了这些积分的最优增长估计。这一结果也与完全流形上的极大原理有关。从这个估计我们可以得到（M, g）是抛物线的一个非常简单的函数理论证明，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).