Minimal surfaces in Flat Three-Dimensional Spaces

Abstract

In recent years there has been major progress in the classical theory of minimal surfaces. We develop here some of the basic geometric theory and technical tools that have been useful in studying properly embedded minimal surfaces in flat 3-manifolds. A properly embedded surface M in a flat 3-manifold ℝ³/Γ, where Γ⊂Iso(ℝ³) acts freely and properly discontinuously on ℝ³, lifts to a Γ-periodic surface $\widetilde M$ in ℝ³. Up to taking finite index subgroups of Γ, we may assume that either Γ is a lattice in ℝ³ (triply-periodic), Γ is a lattice in ℝ²⊂ℝ³ (doubly-periodic), or Γ is the cyclic group generated by a screw motion symmetry S_θ of ℝ³ with axis the x₃-axis and rotation angle θ (singlyperiodic).

In section 1 we discuss the maximum principle at infinity for proper minimal surfaces and applications. In section 2 we briefly go over the Geometric Dehn’s Lemma and related barrier constructions. In sections 3, 4, and 5 we cover the basic theories of triply, doubly, and singly periodic minimal surfaces. In section 6 we go over some of the very recent work on the topology and asymptotic geometry of properly embedded minimal surfaces in ℝ³.