Subdividing Three-Dimensional Riemannian Disks

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 $M$.