On the curvature bounded sphere problem in $\mathbb{R}^3$

TL;DR

This paper proves that a smoothly embedded topological sphere in  with bounded normal curvatures contained in a radius 2 ball must enclose a region containing a unit ball, advancing the understanding of curvature bounds and volume in geometric analysis.

Contribution

It establishes a new curvature-volume relation for smoothly embedded spheres in , providing partial progress on a conjecture by Burago and Petrunin about volume bounds.

Findings

Embedded sphere with curvature  in  and contained in radius 2 ball encloses a volume containing a unit ball.

Supports a conjecture relating curvature bounds to volume in geometric topology.

Includes an example illustrating an alternative aspect of the problem.

Abstract

We prove that if a topological sphere smoothly embedded into $\mathbb{R}^3$ with normal curvatures absolutely bounded by $1$ is contained in an open ball of radius $2$, then the region it bounds must contain a unit ball. This result suggests a potential direction for a problem formulated by D.Burago and A.Petrunin asking whether a topological sphere smoothly embedded in $\mathbb{R}^3$ with normal curvatures absolutely bounded by $1$ encloses a volume of at least $\frac{4}{3}\pi$. The appendix presents an example illustrating an alternative aspect for this problem.