scholarly journals Bubbling analysis and geometric convergence results for free boundary minimal surfaces

2019 ◽  
Vol 6 ◽  
pp. 621-664 ◽  
Author(s):  
Lucas Ambrozio ◽  
Reto Buzano ◽  
Alessandro Carlotto ◽  
Ben Sharp
Keyword(s):  
Free Boundary ◽  
Minimal Surfaces ◽  
Convergence Results ◽  
Geometric Convergence

Approximation of minimal surfaces with free boundaries: convergence results

2018 ◽  
Vol 39 (3) ◽  
pp. 1391-1420
Author(s):  
Tristan Jenschke
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Minimal Surfaces ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Free Boundaries ◽  
Convergence Estimate ◽  
Convergence Results ◽  
Discrete Finite Element

Abstract In a previous paper we developed a penalty method to approximate solutions of the free boundary problem for minimal surfaces by solutions of certain variational problems depending on a parameter $\lambda $. There we showed existence and $C^2$-regularity of these solutions as well as convergence to the solution of the free boundary problem for $\lambda \to \infty $. In this paper we develop a fully discrete finite element procedure for approximating solutions of these variational problems and prove a convergence estimate, which includes an order of convergence with respect to the grid size.

scholarly journals Geometric convergence results for closed minimal surfaces via bubbling analysis

2021 ◽  
Vol 61 (1) ◽  
Author(s):  
Lucas Ambrozio ◽  
Reto Buzano ◽  
Alessandro Carlotto ◽  
Ben Sharp
Keyword(s):  
Minimal Surfaces ◽  
Sharp Estimate ◽  
Quantitative Description ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Global Character ◽  
Multiplicity One ◽  
Convergence Results ◽  
Geometric Convergence ◽  
Number Of Ends

AbstractWe present some geometric applications, of global character, of the bubbling analysis developed by Buzano and Sharp for closed minimal surfaces, obtaining smooth multiplicity one convergence results under upper bounds on the Morse index and suitable lower bounds on either the genus or the area. For instance, we show that given any Riemannian metric of positive scalar curvature on the three-dimensional sphere the class of embedded minimal surfaces of index one and genus $$\gamma $$ γ is sequentially compact for any $$\gamma \ge 1$$ γ ≥ 1 . Furthemore, we give a quantitative description of how the genus drops as a sequence of minimal surfaces converges smoothly, with mutiplicity $$m\ge 1$$ m ≥ 1 , away from finitely many points where curvature concentration may happen. This result exploits a sharp estimate on the multiplicity of convergence in terms of the number of ends of the bubbles that appear in the process.

scholarly journals Free boundary minimal surfaces in the unit ball with low cohomogeneity

2016 ◽  
Vol 145 (4) ◽  
pp. 1671-1683 ◽  
Author(s):  
Brian Freidin ◽  
Mamikon Gulian ◽  
Peter McGrath
Keyword(s):  
Free Boundary ◽  
Unit Ball ◽  
Minimal Surfaces

Optimal boundary regularity for minimal surfaces with a free boundary

1981 ◽  
Vol 33 (3-4) ◽  
pp. 357-364 ◽  
Author(s):  
Stefan Hildebrandt ◽  
Johannes C. C. Nitsche
Keyword(s):  
Free Boundary ◽  
Minimal Surfaces ◽  
Boundary Regularity ◽  
Optimal Boundary

scholarly journals Free boundary minimal surfaces in the unit 3-ball

2017 ◽  
Vol 154 (3-4) ◽  
pp. 359-409 ◽  
Author(s):  
Abigail Folha ◽  
Frank Pacard ◽  
Tatiana Zolotareva
Keyword(s):  
Free Boundary ◽  
Minimal Surfaces

scholarly journals Sharp Area Bounds for Free Boundary Minimal Surfaces in Conformally Euclidean Balls

2018 ◽  
Vol 2020 (18) ◽  
pp. 5630-5641 ◽  
Author(s):  
Brian Freidin ◽  
Peter McGrath
Keyword(s):  
Free Boundary ◽  
Minimal Surface ◽  
Minimal Surfaces ◽  
Geodesic Ball ◽  
Euclidean Setting

Abstract We prove that the area of a free boundary minimal surface $\Sigma ^2 \subset B^n$, where $B^n$ is a geodesic ball contained in a round hemisphere $\mathbb{S}^n_+$, is at least as big as that of a geodesic disk with the same radius as $B^n$; equality is attained only if $\Sigma $ coincides with such a disk. More generally, we prove analogous results for a class of conformally euclidean ambient spaces. This follows works of Brendle and Fraser–Schoen in the euclidean setting.

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