Minimal Lamination Research Articles

On the Teichmüller space of laminations fibering over hyperbolic surfaces

We give for the first time an explicit description of the transversally locally constant Teichmüller space of certain minimal laminations fibering over an infinite type hyperbolic surface.

Erratum to “On minimal laminations of the torus”

Erratum to “On minimal laminations of the torus”

Topology of leaves for minimal laminations by non-simply-connected hyperbolic surfaces

We give the topological obstructions to be a leaf in a minimal lamination by hyperbolic surfaces whose generic leaf is homeomorphic to a Cantor tree. Then, we show that all allowed topological types can be simultaneously embedded in the same lamination. This result, together with results in [arXiv:1906.10029] and [Comment. Math. Helv. 78 (2003), 845–864], completes the panorama of understanding which topological surfaces can be leaves in minimal hyperbolic surface laminations when the topology of the generic leaf is given. In all cases, all possible topologies can be realized simultaneously.

Topology of leaves for minimal laminations by hyperbolic surfaces

We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative version of residual finiteness which may be of independent interest. The main step in establishing this relative version of residual finiteness is to obtain finite covers with control on the second systole of the surface, which is done in the Appendix. In a companion paper, the case of other generic leaves is treated.

An example for nonexistence of minimal foliations

For the high-dimensional Frenkel–Kontorova model on lattices, we have concluded that there are heteroclinic connections between neighboring Birkhoff minimizers which are more periodic. This conclusion is based on the existence of neighboring elements, i.e., the existence of gaps. By adding a large enough oscillation to the local potential, I prove that all minimal foliations can be destroyed into minimal laminations, and hence there always exist gaps.

On the topology of surfaces with the generalised simple lift property

In this paper, we study the geometry of surfaces with the generalised simple lift property. This work generalises previous results by Bernstein and Tinaglia (J Differ Geom 102(1):1–23, 2016) and it is motivated by the fact that leaves of a minimal lamination obtained as a limit of a sequence of property embedded minimal disks satisfy the generalised simple lift property.

Extending an Example by Colding and Minicozzi

Extending an example by Colding and Minicozzi (Trans Am Math Soc 356(1):283–289, 2003), we construct a sequence of properly embedded minimal disks Sigma _i in an infinite Euclidean cylinder around the x_3-axis with curvature blow-up at a single point. The sequence converges to a non-smooth and non-proper minimal lamination in the cylinder. Moreover, we show that the disks Sigma _i are not properly embedded in a sequence of open subsets of mathbb {R}^3 that exhausts mathbb {R}^3.

Structure theorems for singular minimal laminations

Abstract 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 singular minimal laminations of ℝ 3 {\mathbb{R}^{3}} . 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 ℝ 3 {\mathbb{R}^{3}} , 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 ℝ 3 {\mathbb{R}^{3}} with finite genus and a countable number of ends is proper.

Curvature estimates for stable free boundary minimal hypersurfaces

Abstract In this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces satisfying a uniform area bound, which generalize the celebrated Schoen–Simon–Yau interior curvature estimates up to the free boundary. Our curvature estimates imply a smooth compactness theorem which is an essential ingredient in the min-max theory of free boundary minimal hypersurfaces developed by the last two authors. We also prove a monotonicity formula for free boundary minimal submanifolds in Riemannian manifolds for any dimension and codimension. For 3-manifolds with boundary, we prove a stronger curvature estimate for properly embedded stable free boundary minimal surfaces without a-priori area bound. This generalizes Schoen’s interior curvature estimates to the free boundary setting. Our proof uses the theory of minimal laminations developed by Colding and Minicozzi.

Partially hyperbolic sets with a dynamically minimal lamination

We study partially hyperbolic sets of \begin{document}$C^1$\end{document} -diffeomorphisms. For these sets there are defined the strong stable and strong unstable laminations.A lamination is called dynamically minimal when the orbit of each leaf intersects the set densely. We prove that partially hyperbolic sets having a dynamically minimal lamination have empty interior. We also study the Lebesgue measure and the spectral decomposition of these sets. These results can be applied to \begin{document}$C^1$\end{document} -generic/robustly transitive attractors with one-dimensional center bundle.

Finite topology minimal surfaces in homogeneous three-manifolds

We prove that any complete, embedded minimal surface M with finite topology in a homogeneous three-manifold N has positive injectivity radius. When one relaxes the condition that N be homogeneous to that of being locally homogeneous, then we show that the closure of M has the structure of a minimal lamination of N. As an application of this general result we prove that any complete, embedded minimal surface with finite genus and a countable number of ends is compact when the ambient space is S3 equipped with a homogeneous metric of nonnegative scalar curvature.

Local removable singularity theorems for minimal laminations

In this paper we prove a local removable singularity theorem for certain minimal laminations with isolated singularities in a Riemannian three-manifold. This removable singularity theorem is the key result used in our proof that a complete, embedded minimal surface in $\mathbb{R}^3$ with quadratic decay of curvature has finite total curvature.

Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem

We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(\hat M, T^1\mathfrak{F})$ of a compact minimal lamination $(M,\mathfrak{F})$ by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal and examples where this action is not minimal. In the first case, we prove that if $\mathfrak{F}$ has a leaf which is not simply connected, the horocyle flow is topologically transitive.

Topological type of limit laminations of embedded minimal disks

We consider two natural classes of minimal laminations in threemanifolds. Both classes may be thought of as limits—in different senses—of embedded minimal disks. In both cases, we prove that, under a natural geometric assumption on the three-manifold, the leaves of these laminations have genus zero. This answers a question posed by Hoffman and White.

The Classification of CMC foliations of {\Bbb R}^3 and {\Bbb S}^3 with countably many singularities

In this paper we generalize our previous Local Removable Singularity Theorem for minimal laminations to the case of weak $H$-laminations (with $H\in{\Bbb R}$ constant) in a punctured ball of a Riemannian three-manifold. We also obtain a curvature estimate for any weak CMC foliation (with possibly varying constant mean curvature from leaf to leaf) of a compact Riemannian three-manifold $N$ with boundary solely in terms of a bound of the absolute sectional curvature of $N$ and of the distance to the boundary of $N$. We then apply these results to classify weak CMC foliations of ${\Bbb R}^3$ and ${\Bbb S}^3$ with a closed countable set of singularities.

Curvatures of embedded minimal disks blow up on subsets of $C^1$ curves

Any sequence of properly embedded minimal disks in an open subset U of Euclidean 3-space has a subsequence such that the curvatures blow up on a relatively closed subset K of U and such that the disks converge in the complement of K to a minimal lamination of U\K. Assuming results of Colding-Minicozzi and an extension due to Meeks, we prove that such a blow-up set K must be contained in a C^1 embedded curve.

A Minimal Lamination of the Interior of a Positive Cone with Quadratic Curvature Blowup

In this note we construct a minimal lamination of the interior of a positive cone with quadratic curvature blowup.

Forms representing forms and linear spaces on hypersurfaces

Monotone variational recurrence relations such as the Frenkel-Kontorova lattice, arise in solid state physics, conservative lattice dynamics and as Hamiltonian twist maps. For such recurrence relations, Aubry-Mather theory guarantees the existence of solutions of every rotation number. They are the action minimizers that constitute the Aubry-Mather set. When the rotation number is irrational, the Aubry-Mather set is either connected or a Cantor set. A connected Aubry-Mather set is called a minimal foliation. In the case of twist maps, it describes an invariant circle, while in solid state physics it corresponds to a continuum of ground states. A Cantor Aubry-Mather set is called a minimal lamination. In this paper we prove that when the rotation number of a minimal foliation is either rational or easy to approximate by rational numbers, then the foliation can be destroyed into a lamination by an arbitrarily small smooth perturbation of the recurrence relation. This generalizes a theorem of Mather for twist maps to general recurrence relations.

A minimal lamination with Cantor set-like singularities

Given a compact closed subset $M$ of a line segment in $\mathbb{R}^3$, we construct a sequence of minimal surfaces $\Sigma_k$ embedded in a neighborhood $C$ of the line segment that converge smoothly to a limit lamination of $C$ away from $M$. Moreover, the curvature of this sequence blows up precisely on $M$, and the limit lamination has non-removable singularities precisely on the boundary of $M$.

The unique ergodicity of equicontinuous laminations

We prove that a transversely equicontinuous minimal lamination on a locally compact metric space $Z$ has a transversely invariant nontrivial Radon measure. Moreover if the space $Z$ is compact, then the tranversely invariant Radon measure is shown to be unique up to a scaling.