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The Calabi-Yau conjectures for embedded surfaces. (English) Zbl 1142.53012
One of the fundamental questions on complete minimal hypersurfaces in Euclidean space is unboundedness or properness. This question was proposed first by E. Calabi about 1960 and he also asked whether there is a complete minimal surface in \({\mathbb R}^3\) which is a subset of the unit ball. There is an example of a complete nonflat minimally immersed surface between two parallel planes due to L. Jorge and F. Xavier [Ann. Math. (2) 112, 203–206 (1980; Zbl 0455.53004)] and N. Nadirashvili [Invent. Math. 126, 457–465 (1996; Zbl 0881.53053)] constructed a complete immersion of a minimal disk into the unit ball in \({\mathbb R}^3\). Thus the immersed versions of those questions mentioned above turned out to be false. In this paper, the authors prove the Calabi conjectures for embedded surfaces. In fact they prove considerably more containing these conjectures.
The main result in this paper is the so called ‘chord arc bound theorem’ for intrinsic balls which says that there exists a constant \(C>0\) such that if \(\Sigma \subset {\mathbb R}^3\) is an embedded minimal disk, \({\mathfrak B}_{2R} = {\mathfrak B}_{2R}(0)\) is an intrinsic ball in \(\Sigma - \partial \Sigma\) of radius \(2R\), and if \(\sup_{{\mathcal B}_{r_0}} | A| ^2 > r_0^{-2}\), then for \(x \in {\mathfrak B}_R\)
\[ C\cdot\text{dist}_{\Sigma} (x, 0) < | x| + r_0.\tag{1} \] The assumption of a lower bound for the supremum of the square norm of the second fundamental form is a necessary normalization for a chord arc bound and it can be expressed in terms of the curvature because of the Gauss equation. Properness of a complete embedded minimal disk is an immediate consequence of the chord arc bounded theorem by letting the intrinsic distance go to infinity. So, any complete embedded minimal disk in \({\mathbb R}^3\) must be proper and this implies that the first Calabi conjecture is true for embedded minimal disks. Another immediate consequence of the chord arc bound theorem together with the one-sided curvature estimate (Theorem 0.2) in [T. H. Colding and W. P. Minicozzi II, Ann. Math. (2) 160, 573–615 (2004; Zbl 1076.53069)] is that there exists \(\varepsilon >0\) such that if \(\Sigma \subset \{x_3 > 0\} \subset {\mathbb R}^3\) is an embedded minimal disk with intrinsic ball \({\mathfrak B}_{2R} (x) \subset \Sigma - \partial \Sigma\) and \(| x| < R\), then \(\sup_{{\mathfrak B}_R(x)} | A_\Sigma| ^2 \leq R^{-2}\).
As a corollary of this intrinsic one-sided curvature estimate, the authors obtain by letting \(R \to \infty\) that the plane is the only complete embedded minimal disk in \({\mathbb R}^3\) in a half-space and furthermore they prove that the plane is the only complete embedded minimal surface with finite topology in a half-space of \({\mathbb R}^3\). This fact implies that a complete embedded minimal surface with finite topology in \({\mathbb R}^3\) must be proper. Recall that a surface is said to have finite topology if it is homeomorphic to a closed Riemann surface of genus \(g\) with a finite set of punctures. The key ingredient to prove the chord arc bound theorem is the following:
There exists \(\delta >0\) such that if \(\Sigma \subset {\mathbb R}^3\) is an embedded minimal disk, then for all intrinsic balls \({\mathfrak B}_R(x)\) in \(\Sigma \setminus \partial \Sigma\), the component \(\Sigma_{x, \delta R}\) of \(B_{\delta R}(x) \cap \Sigma\) containing \(x\) satisfies \(\displaystyle{\Sigma_{x, \delta R} \subset {\mathfrak B}_{R/2} (x)}\), where \(B\) denotes an extrinsic ball.
Combining this result with a result in [Colding and W. P. Minicozzi II, loc. cit.] one can prove the chord arc bound theorem.
To prove the properness for embedded minimal disks, the authors prove a similar property as above which gives a weak chord arc bound for a compact embedded minimal disk with boundary in the boundary of a ball. To do this the authors show that a compact embedded minimal disk, with boundary in the boundary of an extrinsic ball, is part of a double spiral staircase.

MSC:
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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