Oblivious points on translation surfaces

TL;DR

This paper explores the existence and construction of translation surfaces with multiple oblivious points, expanding known examples and demonstrating their presence across all genera greater than or equal to three.

Contribution

It introduces new families of translation surfaces with arbitrarily many oblivious points and proves their existence in every genus at least three.

Findings

Constructed families with arbitrarily many oblivious points.

Proved existence of oblivious points in every genus ≥ 3.

Extended previous finiteness results to new surface families.

Abstract

An oblivious point on a translation surface is a point with no closed geodesic passing through it. Nguyen, Pan, and Su (2017) showed that there are at most finitely many oblivious points on any given translation surface and constructed a family of surfaces with exactly one oblivious point. We construct new families of translation surfaces with arbitrarily many oblivious points and prove that there is a translation surface in every genus $\ge3$ with an oblivious point.