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

Hongda Qiu

TL;DR

The paper investigates whether a topological sphere $S$ smoothly embedded in $\mathbb{R}^3$ with normal curvatures bounded by $1$, contained in a ball of radius $2$, must enclose a region containing the unit ball of radius $1$; this addresses a Burago–Petrunin conjecture about minimal enclosed volume $\frac{4}{3}\pi$. It proves a stronger radius-$2$ result by showing the enclosed region is star-shaped via a short radial projection from the bounding sphere and leveraging a curvature-based distance estimate to guarantee a contained unit ball. The work clarifies the landscape of the Burago–Petrunin problem by isolating a finite-radius case, discusses the tightness of the radius bound, and presents a genus-$2$ counterexample construction in the appendix to illustrate potential obstacles. Overall, the results contribute a concrete geometric criterion linking normal curvature bounds and ambient containment to the presence of a unit ball inside the enclosed region, with implications for potential positive or negative resolutions of the conjecture.

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 $\frac{4}{3}π$. The appendix presents an example illustrating an alternative aspect for this problem.

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

TL;DR

The paper investigates whether a topological sphere smoothly embedded in with normal curvatures bounded by , contained in a ball of radius , must enclose a region containing the unit ball of radius ; this addresses a Burago–Petrunin conjecture about minimal enclosed volume . It proves a stronger radius- result by showing the enclosed region is star-shaped via a short radial projection from the bounding sphere and leveraging a curvature-based distance estimate to guarantee a contained unit ball. The work clarifies the landscape of the Burago–Petrunin problem by isolating a finite-radius case, discusses the tightness of the radius bound, and presents a genus- counterexample construction in the appendix to illustrate potential obstacles. Overall, the results contribute a concrete geometric criterion linking normal curvature bounds and ambient containment to the presence of a unit ball inside the enclosed region, with implications for potential positive or negative resolutions of the conjecture.

Abstract

We prove that if a topological sphere smoothly embedded into with normal curvatures absolutely bounded by is contained in an open ball of radius , 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 with normal curvatures absolutely bounded by encloses a volume of at least . The appendix presents an example illustrating an alternative aspect for this problem.

Paper Structure

This paper contains 3 sections, 4 theorems, 3 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['theorem enclosed unit ball']}
  3. Appendix: Petrunin's Fishbowl

Key Result

Theorem 1

If the surface $S$, as defined above, is additionally contained by an open ball of radius $2$, then the region it bounds must contain a unit ball.

Figures (3)

  • Figure 1: An illustration for Inequality \ref{['inequality short']} when $v$ lies in the plane spanned by $x$ and $\nu(x)$.
  • Figure 2: An illustration for the assumption used in the proof of Lemma \ref{['lemma enclosed ball']}.
  • Figure :

Theorems & Definitions (8)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Proof
  • Proof : Proof of Lemma \ref{['lemma radial projection']}
  • Lemma 3
  • Proof : Proof of Lemma \ref{['lemma enclosed ball']}
  • Proof : Proof of Theorem \ref{['theorem enclosed unit ball']}