Translation surfaces with no convex presentation

TL;DR

This paper identifies and classifies translation surfaces that lack convex presentations, providing infinite examples and detailed classifications within specific strata, and establishing the existence of non-lattice examples without strictly convex presentations.

Contribution

It offers the first infinite lists of translation surfaces with no convex presentations and classifies such surfaces in key strata, including the eigenform loci and hyperelliptic strata.

Findings

Surfaces in H(2) with no convex presentations are classified.

All surfaces in certain eigenform loci in H(1,1) lack strictly convex presentations.

Non-lattice surfaces without strictly convex presentations exist in all hyperelliptic strata.

Abstract

We give infinite lists of translations surfaces with no convex presentations. We classify the surfaces in the stratum H(2) which do not have convex presentations, as well as those with no strictly convex presentations. We show that in H(1,1), all surfaces in the eigenform loci E_4, E_9 or E_{16} have no strictly convex presentation, and that the list of surfaces with no convex presentations in H(1,1) - (E_4 union E_9 union E_{16}) is finite and consists of square-tiled surfaces. We prove the existence of non-lattice surfaces without strictly convex presentations in all of the strata H^{(hyp)}(g-1, g-1).