The zero set of a holomorphic 1-form (phi) on a compact complex surface S is studied. The main result gives, under the assumption that (phi) has a one-dimensional zero set with appropriate self-intersection properties, the existence of a holomorphic map f : S (--->) R onto a Riemann surface. The form (phi) is a pullback via f of a holomorphic 1-form on R and the zero set of (phi) is contained in fibers of f. As a direct consequence of this, any divisor D having the same support as D(,(phi)), the divisor associated to (phi), is shown to satisfy D(.)D (LESSTHEQ) 0 In a different direction, the genus of an irreducible component of the zero set of a holomorphic 1-form is proved to be bounded in terms of the Euler number of S. It is also shown that all curves having sufficiently low genus and zero self-intersection must be contained in the zero set of some holomorphic 1-form on S A structure theorem for elliptic surfaces having a non-vanishing holomorphic 1-form is proved and examples are provided / acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

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

acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

acase@tulane.edu