scholarly journals Indecomposable $F_N$-trees and minimal laminations

2015 ◽  
Vol 9 (2) ◽  
pp. 567-597 ◽  
Author(s):  
Thierry Coulbois ◽  
Arnaud Hilion ◽  
Patrick Reynolds
Keyword(s):  
Minimal Laminations

scholarly journals Curvature estimates for stable free boundary minimal hypersurfaces

2020 ◽  
Vol 2020 (759) ◽  
pp. 245-264 ◽  
Author(s):  
Qiang Guang ◽  
Martin Man-chun Li ◽  
Xin Zhou
Keyword(s):  
Free Boundary ◽  
Riemannian Manifolds ◽  
A Priori ◽  
Minimal Submanifolds ◽  
Manifolds With Boundary ◽  
Curvature Estimate ◽  
Minimal Hypersurfaces ◽  
Curvature Estimates ◽  
Minimal Laminations ◽  
Uniform Area

AbstractIn 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.

scholarly journals Structure theorems for singular minimal laminations

2020 ◽  
Vol 2020 (763) ◽  
pp. 271-312
Author(s):  
William H. Meeks III ◽  
Joaquín Pérez ◽  
Antonio Ros
Keyword(s):  
Minimal Surfaces ◽  
Removable Singularity ◽  
Countable Number ◽  
Singularity Theorems ◽  
Structure Theorems ◽  
Number Of Ends ◽  
Minimal Laminations ◽  
Fixed Genus ◽  
Finite Genus ◽  
Differential Geom

AbstractWe 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 {\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 {\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 {\mathbb{R}^{3}} with finite genus and a countable number of ends is proper.

scholarly journals Examples of minimal laminations and associated currents

2010 ◽  
Vol 269 (1-2) ◽  
pp. 495-520 ◽  
Author(s):  
John Erik Fornæss ◽  
Nessim Sibony ◽  
Erlend Fornæss Wold
Keyword(s):  
Minimal Laminations

scholarly journals An example for nonexistence of minimal foliations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Xue-Qing Miao
Keyword(s):  
Local Potential ◽  
High Dimensional ◽  
Minimal Foliations ◽  
Minimal Laminations ◽  
Kontorova Model

AbstractFor 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.

scholarly journals Local removable singularity theorems for minimal laminations

2016 ◽  
Vol 103 (2) ◽  
pp. 319-362 ◽  
Author(s):  
William H. Meeks ◽  
Joaquín Pérez ◽  
Antonio Ros
Keyword(s):  
Removable Singularity ◽  
Singularity Theorems ◽  
Minimal Laminations
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