The Calabi–Yau conjectures for embedded surfaces

Abstract

In this paper we will prove the Calabi-Yau conjectures for embedded surfaces (i.e., surfaces without self-intersection). In fact, we will prove considerably more. The heart of our argument is very general and should apply to a variety of situations, as will be more apparent once we describe the main steps of the proof later in the introduction.The Calabi-Yau conjectures about surfaces date back to the 1960s. Much work has been done on them over the past four decades. In particular, examples of Jorge-Xavier from 1980 and Nadirashvili from 1996 showed that the immersed versions were false; we will show here that for embedded surfaces, i.e., injective immersions, they are in fact true.Their original form was given in 1965 in [Ca] where E. Calabi made the following two conjectures about minimal surfaces (they were also promoted by S.S. Chern at the same time; see page 212 of [Ch]): Conjecture 0.1. "Prove that a complete minimal hypersurface in R n must be unbounded." Calabi continued: "It is known that there are no compact minimal submanifolds of R n (or of any simply connected complete Riemannian manifold with sectional curvature ≤ 0). A more ambitious conjecture is": Conjecture 0.2. "A complete [non-flat] minimal hypersurface in R n has an unbounded projection in every (n − 2)-dimensional flat subspace."

On curvature estimates for constant mean curvature surfaces

“…In the case of embedded minimal disks such a description was given by Colding and Minicozzi in [7]; see also [32,33] for related results. By rescaling arguments this description can be improved upon once one knows that the helicoid is the unique complete, embedded, non-flat minimal surface in R 3 as explained below; see [17] and also [1] for proofs of the uniqueness of the helicoid which are based in part on results in [6,7,8,9,10].…”

“…Previous important examples of curvature estimates for constant mean curvature surfaces, assuming certain geometric conditions, can be found in the literature; see for instance [2,3,4,9,10,27,28,29,30,34,35].…”

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

“…Recall that η d = cosh −1 ( 2d H+ √ 1−4H 2 +d 2 1−4H 2 Note that α = cosh η d and that as H ∈ (0, 1 2 ), β < 0 < α. Furthermore, 2He −r < 2H < 1 < d. Thus we have,…”

“…In their ground breaking work [2], Colding and Minicozzi proved that complete minimal surfaces embedded in R 3 with finite topology are proper. 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.…”

“…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.…”