方程△g-aKg=0无正函数解的充分条件

Fischer-Colbrie和Schoen曾在1980年研究过复平面中单位圆盘当赋予某个完备度量时,方程Δg-aKg=0在其上无正函数解的充分条件,并将其结果应用到三维非负数量曲率流形中完备稳定的极小曲面上.这里Δ是Laplace算子,K为高斯曲率,a是常数,g是所讨论的单位圆盘上的函数.本文给出了此方程在该圆盘上无正函数解的一个更弱的充分条件.

A Sufficient Condition of Nonexistence of Positive Solution of Equation Δg-aKg=0

Fischer-Colbrie and Schoen have studied the equation Δg-aKg=0 on unit disc in complex plane in 1980. The disc is endowed with a complete conformal metric. They got a sufficient condition for the nonexistence of positive solution on such disc and applied this result to the study of complete stable minimal surfaces in 3-dimensional manifolds of non-negative scalar curvature. Here Δ is Laplace operator, K is Gauss curvature, a is a constant, and g is a function defined on the unit disc. In this paper, we obtain a more weaker sufficient condition which also ensures the nonexistence of positive solution on such unit disc.