A local optimal diastolic inequality on the two-sphere

TL;DR

This paper proves a local optimal inequality relating diastole and area on the two-sphere, supporting Calabi's conjecture about a specific singular metric's optimality in this geometric context.

Contribution

It introduces a novel approach using a ramified cover to establish a local diastolic inequality near a singular metric on the two-sphere.

Findings

The singular metric made of two equilateral triangles is locally optimal for the diastolic inequality.

The inequality supports Calabi's conjecture about the optimal ratio of area to squared shortest geodesic length.

The method involves a ramified cover of the sphere by a torus to analyze the inequality.

Abstract

Using a ramified cover of the two-sphere by the torus, we prove a local optimal inequality between the diastole and the area on the two-sphere near a singular metric. This singular metric, made of two equilateral triangles glued along their boundary, has been conjectured by E. Calabi to achieve the best ratio area over the square of the length of a shortest closed geodesic. Our diastolic inequality asserts that this conjecture is to some extent locally true.