Mathematics – Differential Geometry
Scientific paper
2009-10-25
Mathematics
Differential Geometry
Scientific paper
Let X be some Riemann surface, and let omega be a meromorphic quadratic differential form on X, that is, omega can be written in local coordinates as f(z) dz^2, for some meromorphic function f. We say that a curve gamma is part of a horizontal leaf of omega if for each t in I, we have that f(gamma(t)) (gamma'(t))^2 is real and positive. The differential omega is said to be a Strebel differential if its noncompact horizontal leaves form a set of measure 0. Now, most quadratic differential forms on any given Riemann surface are not Strebel (in fact, the space of Strebel differentials on the projective line as a subset of the space of all quadratic differentials on the projective line is a countable union of real lines in the complex plane. Because of this, this paper focuses on writing down explicit examples of Strebel differentials. Most papers that discuss Strebel differentials seem to focus on those that have only second order poles, since they define a cellular decomposition on X. However, we construct an explicit family of examples of holomorphic Strebel differentials on any hyperelliptic curve branched over the points 0, 1, infinity, and 1/2 + ir, for any real r. Later, we look at an example from a previous paper, where omega was an algebraic Strebel differential such that omega had integral residues, on a Riemann surface X defined over the algebraic closure of the rationals that had a transcendental period. We then modify this example to obtain another algebraic Strebel differential omega', defined on an algebraic Riemann surface X', all of whose periods are transcendental.
No associations
LandOfFree
Explicit Examples of Strebel Differentials does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Explicit Examples of Strebel Differentials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Explicit Examples of Strebel Differentials will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-450879