The minimal lamination closure theorem

Abstract

We prove that the closure of a complete embedded minimal surface M in a Riemannian three-manifold N has the structure of a minimal lamination, when M has positive injectivity radius. When N is R 3 , we prove that such a surface M is properly embedded. Since a complete embedded minimal surface of finite topology in R 3 has positive injectivity radius, the previous theorem implies a recent theorem of Colding and Minicozzi: A complete embedded minimal surface of finite topology in R 3 is proper. More generally, we prove that if M is a complete embedded minimal surface of finite topology and N has nonpositive sectional curvature (or is the Riemannian product of a homogeneously regular Riemannian surface with R), then the closure of M has the structure of a minimal lamination. Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42

Non-properly embedded H-planes in $${\mathbb H}^2\times {\mathbb R}$$ H 2 × R

“…Since the boundaries of the n leave every compact subset of * , for each compact set of * , the norms of the second fundamental forms of the n are uniformly bounded for values n sufficiently large and such a bound does not depend on the chosen compact set. Standard compactness arguments give that, after passing to a subsequence, n converges to a (weak) H -lamination L of * and the leaves of L are complete and have uniformly bounded norm of their second fundamental forms, see for instance [5].…”

“…Abusing the notation, let J P H be the Jacobi function induced by taking the inner product of ∂ θ with the unit normal of P H , then J P H is positive. Finally, since the norm of the second fundamental form of P H is uniformly bounded, standard compactness arguments imply that its closure P H is an H -lamination L of , see for instance [5]. 1 , L 2 ⊂ that form the limit set of P H .…”

“…Based on the techniques in [2], Meeks and Rosenberg [5] then proved that complete minimal surfaces with positive injectivity embedded in R 3 are proper. More recently, Meeks and Tinaglia [7] The first author is partially supported by BAGEP award of the Science Academy, and a Royal Society Newton Mobility Grant.…”

Parabolic minimal surfaces in 𝕄2× ℝ

“…After this, Meeks and Rosenberg proved that if Σ is a complete embedded minimal surface in R 3 which has positive injectivity radius, then Σ is proper, [17]. Finally, Meeks and Rosenberg proved that if f : Σ → R 3 is an injective minimal immersion, with Σ complete and of bounded curvature, then f is proper, [16].…”

Complete Minimal Cylinders Properly Immersed in the Unit Ball

“…An affirmative answer was given by Colding and Minicozzi in the embedded case, namely, they showed that embedded complete minimal surfaces must be unbounded [1]. More results in the embedded case were given in [7,8]. On the other hand, in the immersed case, a negative answer was given by Martin and Morales [5,6].…”