Subdividing Three-Dimensional Riemannian Disks

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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 Sₜ contracting the boundary to a point so that the area of Sₜ 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∈(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 3HF₁(2d), where HF₁ is the homological filling function of M.