Translation surfaces with no convex presentation

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 ∪ E 9 ∪ 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).

Rel leaves of the Arnoux-Yoccoz surfaces

“…For an example in which ∆ is nontrivial and HT x is not a torus, let x be the Escher staircase surface obtained by cyclically gluing 2m squares (see e.g. [LW,Figure 3]). The surface has m parallel cylinders, the torus T is isomorphic to pR{Zq m , and the group ∆ is a cyclic group generated by a homeomorphism of the surface which goes one step up the staircase.…”

Periodic Points in Genus Two: Holomorphic Sections over Hilbert Modular Varieties, Teichmuller Dynamics, and Billiards

“…empty if p is a zero and a singleton containing the image of p under the hyperelliptic involution otherwise. The strategy of this proof, which appears in Section 3, is to use the ideas of Lelièvre, Monteil, and Weiss [LMW16] alluded to above along with constraints on the geometry of eigenforms coming from McMullen's prototype surfaces in [McM05] and work of Lelièvre and Weiss [LW15] on convex representations of eigenforms.…”