Supersqueezed sphere

TL;DR

The paper answers Burago and Petrunin's question in the negative by constructing a genus-0 smooth surface in $\mathbb{R}^3$ with normal curvatures bounded by $1$ that encloses a volume strictly smaller than the unit ball, using a refined variant of Lagunov's fishbowls dubbed the jerrycan. The construction pieces together toric segments, spherical caps, a central cylinder, and flat disks to form a topological sphere with $|k_n|\le1$, achieving a total main-part volume $2a+2b+c<\frac{4}{3}\pi$; after smoothing to strict curvature bounds, the counterexample remains valid. The work situates the result within prior lower-bound studies and discusses conjectures for optimal volume bounds across genera, noting that the torus case and high-genus generalizations present significant challenges. Overall, it establishes a quantitative negative answer and opens avenues for refining lower bounds and understanding genus-dependent extremal shapes.

Abstract

This work pose an example of a smooth closed surface in $\mathbb{R}^3$ which has genus $0$, normal curvatures at most $1$ in absolute value and encloses a volume smaller than the volume of a unit ball. It gives a negative answer to a question asked by Dmitri Burago and Anton Petrunin.

Figures (5)

• Figure 1: The white part of the surface is not important; the volume of the jerrycan near this part can be made arbitrarily small. If we assume that thinness is zero, then (1) red and blue parts are made from pieces of tori obtained by rotating unit circles whose centers are at distances $2$ and $2-\sqrt{3}$ from the axis of rotation, respectively; (2) yellow pieces are cut from the unit sphere; (3) the black part is made from a round cylinder; (4) green semidiscs are flat.
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