Stable constant-mean-curvature surfaces bounded by a circle

Determine whether planar disks and spherical caps are the only stable compact constant-mean-curvature surfaces bounded by a circle, including surfaces of positive genus.

Background

The paper identifies this as a related open problem posed as Problem 2.9 in the cited survey by López. The corresponding classification is known for disks, but the general problem concerns stable compact constant-mean-curvature surfaces of arbitrary genus whose boundary is a circle.

The authors explain that their investigation of this stability problem involved attempting to bound the genus of a stable surface. The question remains unresolved beyond the disk case, which is why it is included as an explicit open problem rather than as a result of the present paper.