Ratio of Intrinsic Metric to Extrinsic Metric and Volume

TL;DR

This paper investigates the relationship between intrinsic and extrinsic metrics on surfaces within the unit ball in R3, establishing bounds related to surface area and demonstrating the absence of universal lower bounds through examples.

Contribution

It provides lower bounds on the ratio of intrinsic to extrinsic metrics based on surface area and shows that global lower bounds do not always exist.

Findings

Lower bounds on metric ratios in terms of surface area

Examples demonstrating non-existence of universal bounds

Insights into geometric properties of surfaces in R3

Abstract

We study the relationship between the ratio of intrinsic to extrinsic metrics and area. For certain surfaces inside unit ball in R3 we give lower bound on the maximum of ratio in terms of its area. We also give examples to show non-existence of global lower bounds.