I’ll present a metric construction over compact surfaces called Abelian
Differentials or Flat Structures (flat surfaces). The goal of this
talk will be to present my results regarding tight bounds on the
number of invariant components on a flat surface.
Let S be a compact surface with a complex structure. A flat structure on S
is an atlas of charts defined on S minus a finite set of points, such that
the transition functions are translations. This results in a locally
Euclidean surface with a directional field (vector field) globally
defined.
I’ll also define the moduli space of flat structures and the problem -
finding the best upper bounds in every stratum on the number of invariant
components (invariant to the vertical flow one can define on such
surfaces).
I’ll present the results, and, time allows, will prove some of them. the
talk and proofs will be geometric in nature.