A genus-zero surface with bounded curvature enclosing less volume than the unit sphere

TL;DR

This paper constructs smooth, genus-zero surfaces with bounded principal curvatures in three-dimensional space that enclose less volume than the unit sphere, challenging the assumption that the sphere minimizes volume under curvature constraints.

Contribution

It provides the first explicit family of such surfaces demonstrating they can enclose less volume than the sphere, answering a longstanding folklore question.

Findings

Constructed a family of bounded, smooth genus-zero surfaces with principal curvatures in [-1,1]

Showed these surfaces can enclose less volume than the unit sphere in D

Confirmed the sphere does not minimize volume among such surfaces

Abstract

We produce a family of bodies in $\mathbb R^3$ parameterized by $\varepsilon > 0$, each bounded by a smooth topological sphere with principal curvatures in $[-1, 1]$, and having volume arbitrarily close to $ 16 - 4\sqrt 3 + \left(10 \sqrt 3 - 14\right) \pi - \left(\frac{10}{3} - \sqrt 3\right) \pi^2 \approx 3.70.$ Thus, in contrast to the two-dimensional case, the unit sphere (which bounds a ball of volume $\frac{4 }{ 3} \pi \approx 4.19$) does not enclose the minimal volume among all smooth spheres in $\mathbb R^3$ with principal curvatures in $[-1,1]$. This answers a folklore question of Dmitri Burago and Anton Petrunin.