English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

P. Papasoglu asked in [Pap13] whether for any Riemannian 3-disk $M$ withdiameter $d$, 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 tosubdivide $M$ by a disk $D$ into two regions of volume $V/4$ so that the areaof $D$ is bounded by some function $h(d,A,V)$.  In this paper, we answer the questions above in the negative. We furtherprove that given $N>0$ and $c\in(0,1)$, one can construct a metric $g'$ so thatany 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 Riemannian3-sphere $M$, there is a surface that subdivides the disk into two regions ofvolume 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 homologicalfilling function of $M$.

Subdividing Three-Dimensional Riemannian Disks