方程Δg-aKg=0无正函数解的充分条件

徐海峰

数学学报 ›› 2010, Vol. 53 ›› Issue (5) : 945-952.

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PDF(447 KB)
数学学报 ›› 2010, Vol. 53 ›› Issue (5) : 945-952. DOI: 10.12386/A2010sxxb0105
论文

方程Δg-aKg=0无正函数解的充分条件

    徐海峰
作者信息 +

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

    Hai Feng XU
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文章历史 +

摘要

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.

 

Abstract

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.

 

关键词

Δ-q算子 / 共形度量 / 高斯曲率 / Laplace算子的第一特征值

Key words

Δ-q operator / conformal metric / Gauss curvature / the first eigenvalue of the Laplace operator

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导出引用
徐海峰. 方程Δg-aKg=0无正函数解的充分条件. 数学学报, 2010, 53(5): 945-952 https://doi.org/10.12386/A2010sxxb0105
Hai Feng XU. A Sufficient Condition of Nonexistence of Positive Solution of Equation Δg-aKg=0. Acta Mathematica Sinica, Chinese Series, 2010, 53(5): 945-952 https://doi.org/10.12386/A2010sxxb0105

参考文献


[1] Fischer-Colbrie D., Schoen R., The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Comm. Pure. Appl. Math., 1980, 33: 199--211.

[2] Glazman I. M., Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Jerusalem: Israel Program for Scientific Translations, 1965.

[3] Hobson E. W., Spherical and Ellipsoidal Harmonics, New York: Chelsea, 1955.

[4] Wu H. X., Shen C. L., Yu Y. L., An Introduction to Riemannian Geometry, Beijing: Beijing University Press, 2000 (in Chinese).

[5] Bai Z. G., Shen Y. B., Shui N. X., Guo X. Y., An Introduction to Riemannian Geometry, Beijing: Higher Education Press, 1992 (in Chinese).

[6] Wang P. H., Shen C. L., A global curvature pinching result of the first eigenvalue of Laplacian, Acta Mathematica Sinica, Chinese Series, 2008, 51(1): 115--122.

[7] Gilbarg D., Trudinger N. S., Elliptic Partial Differential Equations of Second Order, New York: Springer-Verlag, 1977.

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