2020/11/13 by Kusner, Robert, McGrath, Peter · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2011.06884
We develop new methods to compare the span C(Σ) of the coordinate functions on a free boundary minimal submanifold Σ embedded in the unit n-ball \mathbbBn with its first Steklov eigenspace Eσ1(Σ). Using these methods, we show that C(A)=Eσ1(A) for any embedded free boundary minimal annulus A in \mathbbB3 invariant under the antipodal map, and thus prove that A is congruent to the critical catenoid. We also confirm that C=Eσ1 for any free boundary minimal surface embedded in \mathbbB3 with the symmetries of many known or expected examples, including: examples of any positive genus from stacking at least three disks; two infinite families of genus 0 examples with dihedral symmetry, as well as a finite family with the various Platonic symmetries; and examples of any genus by desingularizing several disks that meet at equal angles along a diameter of the ball.