Structure theorems for singular minimal laminations

Abstract

<jats:title>Abstract</jats:title><jats:p>We apply the local removable singularity theorem for minimal laminations [W. H. Meeks III, J. Pérez and A. Ros, Local removable singularity theorems for minimal laminations, J. Differential Geom. 103 (2016), no. 2, 319–362] and the local picture theorem on the scale of topology [W. H. Meeks III, J. Pérez and A. Ros, The local picture theorem on the scale of topology, J. Differential Geom. 109 (2018), no. 3, 509–565] to obtain two descriptive results for certain possibly <jats:italic>singular minimal laminations</jats:italic> of <jats:inline-formula id="j_crelle-2018-0036_ineq_9999_w2aab3b7e3115b1b6b1aab1c14b1b3Aa"><jats:alternatives><jats:tex-math>{\mathbb{R}^{3}}</jats:tex-math></jats:alternatives></jats:inline-formula>. These two global structure theorems will be applied in [W. H. Meeks III, J. Pérez and A. Ros, Bounds on the topology and index of classical minimal surfaces, preprint 2016] to obtain bounds on the index and the number of ends of complete, embedded minimal surfaces of fixed genus and finite topology in <jats:inline-formula id="j_crelle-2018-0036_ineq_9998_w2aab3b7e3115b1b6b1aab1c14b1b5Aa"><jats:alternatives><jats:tex-math>{\mathbb{R}^{3}}</jats:tex-math></jats:alternatives></jats:inline-formula>, and in [W. H. Meeks III, J. Pérez and A. Ros, The embedded Calabi–Yau conjectures for finite genus, preprint 2018] to prove that a complete, embedded minimal surface in <jats:inline-formula id="j_crelle-2018-0036_ineq_9997_w2aab3b7e3115b1b6b1aab1c14b1b7Aa"><jats:alternatives><jats:tex-math>{\mathbb{R}^{3}}</jats:tex-math></jats:alternatives></jats:inline-formula> with finite genus and a countable number of ends is proper.</jats:p>

Bounds on the topology and index of minimal surfaces

“…A priori, one procedure to obtain surfaces in M with finite genus and infinite topology might be to take limits of sequences of finite total curvature examples in M with a bound on their genus but with a strictly increasing number of ends. Our results in [30,31,32,33] are crucial in understanding that such sequences cannot exist, and they will lead us to a proof of the following main theorem of this manuscript. Theorem 1.4 A properly embedded minimal surface in R 3 with finite topology has a bound on the number of its ends that only depends on its genus.…”

“…The existence of the minimal lamination L of A such that the M n converge to L (after passing to a subsequence) in A−S(L) follows directly from the main statement of Theorem 1.6 in [31]; note that singularities of L are ruled out in our setting by item 7.1 of Theorem 1.6 in [31] because the M n have uniformly bounded genus. The same argument using item 7.1 of Theorem 1.6 in [31] ensures that item 1 of Theorem 2.2 holds. Now assume that S(L) = Ø and we will prove that item 2 holds.…”

“…The proof of Theorem 1.4 depends heavily on results developed in our previous papers [30,31,32,33]. These papers, as well as the present one, rely on a series of deep works by Colding and Minicozzi [3,4,5,6,8] in which they describe the local geometry of a complete, embedded minimal surface in a Riemannian three-manifold, where there is a local bound on the genus of the surface.…”

“…In the proof of our main Theorem 1.4 we will use repeatedly the following statement, which is an application of Theorem 1.6 in [31].…”

“…By construction, the sequence {M k,k } k also converges to L with the same singular set of convergence S(L). However, since the minimal surfaces in the sequence {M k,k } k are compact with genus at most g, then item 7.3 in Theorem 1.6 of [31] implies that item 2 of Theorem 2.2 holds.…”