2026/07/26 by James Tissot
#math.DG #math.AP
In 1995, Brakke proposed a formulation of the Plateau problem for a given boundary Γ that allows for triple junctions and tetrahedral singularities. Let Γ be a smooth closed curve and let G=π1(ℝ3∖ Γ). For each proper finite index subgroup N of G, Brakke constructs a (M, 0, ∞)-minimal surface ΣN, which is obtained as the projection of the boundary of a perimeter-minimising fundamental domain in the covering space associated to N. We advance the theory in two ways. Firstly, we extend Brakke's construction to include all normal subgroups N\triangleleft G, i.e. possibly with infinite index. Secondly, we prove a compactness result which implies that there exists a proper normal subgroup N0\triangleleft G such that Area(ΣN0)=inf_\N\triangleleft G, N ≠ G\ Area(ΣN). A similar result holds when Γ has many connected components. Furthermore, we study the spanning and minimising properties of the ΣNs and ΣN0.