本篇論文探討在負曲率的完備黎曼流行上是否存在有正的調和函數的問題, 邱成桐, Greene 和 Wu 都有討論過相關的問題, 邱成桐曾經在1975證明過在非負瑞奇曲率下的完備黎曼流型上不會有常數函數以外的正調和函數, Greene 和 Wu 則也有類似的定理, 不過條件比較複雜, 大概來說, 如果滿足曲率在夠遠的地方不會彎曲的太利害, 則用Liouville定理可以得到說這樣的流型下, 不會有有界的調和函數. 以上都是證明沒有調和函數的定理, 可是對於存在性的定理卻是非常的少, 本篇主要引用Hyeong In Chou 的文章, 討論在於負曲率下, 我們可以證明這樣的調和函數存在

This paper is only a review of chapter 2 in "Lectures on Differential Geometry" written by R. Schoen and S.-T. Yau. We use Perron method to solve Dirichlet problem on the Riemannian manifold with bounded negative sectional
curvature.

References
[1] P. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math.
46(1973), 45-109
[2] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equa-
tions of second order, second edition, Springer-Verlag
[3] Hyeong In Choi, Asymptotic Dirichlet problem for harmonic func-
tions on Riemannian manifolds, AMS, 1984, vol.281, no.2
[4] R. Scheon and S.T. Yau, Lectures on Differential Geometry, Inter-
national Press, 1994
[5] Jeff Cheeger and David G. Ebin, Comparison Theorems in Rie-
mannian Geometry, North-Hooland Publishing Company, 1975
[6] Michael T. Anderson and Richard Schoen, Positive harmonic func-
tions on complete manifolds of negative curvature Annals of Math-
ematics, 1985, 429-461