2008/07/07 by David Hoffman, Hoffman, David, Brian White +1
Mathematics · #49Q05 #53A10 #58E12 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Nonlinear Partial Differential Equations #math.DG #msc:49Q05 #msc:53A10 #msc:58E12
paper · pdf · doi:10.48550/arxiv.0807.0933
13 pages Dedicated to Jean Pierre Bourguignon on the occasion of his 60th birthday. One tex 'newcommand' revised because arxiv version had an error. Two illustrations and one proof have been added. May 2009: Abstract, key words, MSC codes added. One typo fixed. Paper has been published in Asterisque
openalex publication_date 2008/07/07 · arxiv created 2009/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We generalize the following result of White: Suppose N is a compact, strictly convex domain in \RR3 with smooth boundary. Let Σ be a compact 2-manifold with boundary. Then a generic smooth curve Γ≅ ∂Σ in ∂ N bounds an odd or even number of embedded minimal surfaces diffeomorphic to Σ according to whether Σ is or is not a union of disks. First, we prove that the parity theorem holds for any compact riemannian 3-manifold N such that N is strictly mean convex, N is homeomorphic to a ball, ∂ N is smooth, and N contains no closed minimal surfaces. We then further relax the hypotheses by allowing N to be weakly mean convex and to have piecewise smooth boundary. We extend the parity theorem yet further by showing that, under an additional hypothesis, it remains true for minimal surfaces with prescribed symmetries. The parity theorems are used in an essential way to prove the existence of embedded genus-g helicoids in \SS2× \RR. We give a very brief outline of this application. (The full argument will appear elsewhere.)