Stable branched minimal immersions with prescribed boundary

Abstract

We describe a method for producing smooth 2-valued minimal graphs over the cylindrical region (D \{0})×R n−2 , where D is the disk in R 2 , subject to given continuous 2-valued boundary data on ∂D × R n−2 . Subject to appropriate symmetry assumptions, the construction produces branched minimal immersions in D × R n−2 × R with prescribed boundary and branching at every point of {0} × R n−2 , and we also discuss the nature of the possible singularities along {0} × R n−2 in case of general boundary data.

Regularity of stable minimal hypersurfaces: Recent advances in the theory and applications

“…Simon and the author ([SimWic07]) constructed a large family of such examples as multi-valued codimension 1 graphs over the cylinder B 2 1 (0) × R n−2 with prescribed multi-valued boundary data satisfying a mild symmetry condition. Rosales [Ros10] constructed a different family of two dimensional branched surfaces using a variant of the method introduced in [SimWic07], and more recently, Krummel [Krum1] gave a direct PDE theoretic approach to the results in [SimWic07], also extending the construction to higher codimension under an additional necessary smallness hypothesis on the boundary data.…”

The geometric structure of solutions to the two-valued minimal surface equation

“…This analysis is in fact subtle, and relies on the theory of varifolds (we refer to [7] for an introduction to varifolds). The present work gives a more complete analysis of this asymptotic behavior, and as a consequence various new results are obtained, including the existence of some new classes of stable branched minimal immersions without assuming the symmetries imposed in [8].…”

“…In Sect. 6 we show that G at (0, lim r →0 u 0 ) and (0, lim r →0 u 0 ) has a unique tangent cone consisting of a vertical plane, and we give a description of G near these points as being part of a two-valued graph as in [8] off of each tangent cone.…”

“…In R 3 , this method involves solving an equation, which we call the Two-Valued Minimal Surface Equation, or the 2MSE, over the punctured unit disk D\{0} given boundary data ϕ 0 ∈ C(∂D). It is then shown that either u 0 cannot be extended continuously across the origin or the two-valued graph Most of the work in [8] involves showing that the latter alternative occurs subject to a 2π k rotational symmetry on the boundary data for k > 1 odd, through an investigation of the asymptotic behavior of solutions u 0 that cannot be extended continuously across the origin. This analysis is in fact subtle, and relies on the theory of varifolds (we refer to [7] for an introduction to varifolds).…”

“…In [8] a PDE method is introduced producing a class of stable minimal orientable hypersurfaces each of which has a branching set equal to an (n − 2)-dimensional C 1,α submanifold for some α ∈ (0, 1). In R 3 , this method involves solving an equation, which we call the Two-Valued Minimal Surface Equation, or the 2MSE, over the punctured unit disk D\{0} given boundary data ϕ 0 ∈ C(∂D).…”