An example of a Teichmuller disk in genus 4 with degenerate Kontsevich-Zorich spectrum

TL;DR

This paper constructs a specific example of a genus 4 Riemann surface with a holomorphic quadratic differential, exhibiting a closed SL(2,R)-orbit and a highly degenerate Kontsevich-Zorich spectrum, extending previous genus 3 work.

Contribution

It provides a new explicit example of a genus 4 Teichmuller disk with degenerate spectrum, advancing understanding of the spectrum's degeneracy in higher genus.

Findings

Constructed a genus 4 quadratic differential with closed SL(2,R)-orbit

Demonstrated a highly degenerate Kontsevich-Zorich spectrum in this example

Extended previous genus 3 constructions to genus 4

Abstract

We construct an orientable holomorphic quadratic differential on a Riemann surface of genus 4 whose SL(2,R)-orbit is closed and has a highly degenerate Kontsevich - Zorich spectrum. This example is related to a previous similar construction in genus 3 by the first author.