Volume bounds of conic 2-spheres

TL;DR

This paper establishes sharp volume bounds for conic 2-spheres based on Gaussian curvature constraints, identifying extremal models and computing minimal volumes under specific curvature bounds.

Contribution

It introduces new analytical tools for regions with zero curvature and determines minimal volumes of conic spheres with bounded curvature, extending previous geometric analysis methods.

Findings

Sharp volume bounds for conic 2-spheres based on curvature

Identification of geometric models achieving extremal volume

Calculation of minimal volume when curvature is bounded by 1

Abstract

We obtain sharp volume bound for a conic 2-sphere in terms of its Gaussian curvature bound. We also give the geometric models realizing the extremal volume. In particular, when the curvature is bounded in absolute value by $1$, we compute the minimal volume of a conic sphere in the sense of Gromov. In order to apply the level set analysis and iso-perimetric inequality as in our previous works, we develop some new analytical tools to treat regions with vanishing curvature.