Let (M, g) be a Riemannian manifold of dimension 2 and K a given function on M. The problem of realizing K as the curvature of a metric g pointwise conformal to g (i.e., g = $e\sp{2u}g$ for some $u\ \in\ C\sp\infty(M$)) is equivalent to the problem of solving the nonlinear equation$$\Delta u - k + Ke\sp{2u} = 0\eqno(*)$$where k and $\Delta$ are the curvature and Laplacian respectively in the given metric g We study the equation ($\*$) in the case that M is a noncompact 2-manifolds of finite topological type with only parabolic ends and K is nonpositive. M. Kalka and D. G. Yang $\lbrack6\rbrack$ have the existence results relating to the Euler characteristic of the surfaces. We discuss the completeness of solutions. The existence of complete solutions is stated in Theorem 3.1. We show that if $-K(r\sb{i},\theta\sb{i}$) goes to 0 at a rate faster than $r\sbsp{i}{\beta i}$ near each parabolic end with $\sum\sbsp{i=1}{n}\beta\sb{i}\ >\ 2\sb\chi(M$) and $\beta\sb{i}\ >$ 0, there exists a complete solution for the equation ($\*$) on (M, g). We also discuss the nonexistence of complete solutions. The result is stated in Theorem 4.2 We discuss harmonic functions on parabolic surfaces in Section 2.4. In chapter 3 and 4 we show how harmonic functions affect the completeness of solutions, especially trigonometric components. The shifting lemma $\lbrack6\rbrack$ and the generalized maximum principle $\lbrack11\rbrack$ are used as our main tools / acase@tulane.edu
| Identifer | oai:union.ndltd.org:TULANE/oai:http://digitallibrary.tulane.edu/:tulane_24123 |
| Date | January 1995 |
| Contributors | Liu, Wenhong (Author), Kalka, Morris (Thesis advisor) |
| Publisher | Tulane University |
| Source Sets | Tulane University |
| Language | English |
| Detected Language | English |
| Rights | Access requires a license to the Dissertations and Theses (ProQuest) database., Copyright is in accordance with U.S. Copyright law |
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