Subdividing Three-Dimensional Riemannian Disks

TL;DR

This paper investigates subdivision and homotopy properties of Riemannian 3-disks, providing negative answers to certain geometric questions and establishing bounds related to volume and boundary area.

Contribution

It demonstrates that certain subdivision and homotopy bounds do not exist universally and introduces constructions with arbitrarily large boundary areas for subdivisions.

Findings

Negative answer to subdivision and homotopy bounding questions.

Existence of metrics with arbitrarily large boundary area for subdivisions.

Bound on subdividing surface area in Riemannian 3-spheres using homological filling functions.

Abstract

P. Papasoglu asked in [Pap13] whether for any Riemannian 3-disk $M$ with diameter $d$, boundary area $A$ and volume $V$, there exists a homotopy $S_t$ contracting the boundary to a point so that the area of $S_t$ is bounded by $f(d,A,V)$ for some function $f$. He further asks whether it is possible to subdivide $M$ by a disk $D$ into two regions of volume $V/4$ so that the area of $D$ is bounded by some function $h(d,A,V)$. In this paper, we answer the questions above in the negative. We further prove that given $N>0$ and $c\in(0,1)$, one can construct a metric $g'$ so that any 2-disk $D$ subdividing $(M,g')$ into two regions of volume at least $cV$, the area of $D$ is greater than $N$. We also prove that for any Riemannian 3-sphere $M$, there is a surface that subdivides the disk into two regions of volume no less than $V/6$, and the area of this surface is bounded by $3\operatorname{HF}_1(2d)$, where $\operatorname{HF}_1$ is the homological filling function of $M$.