Minimal Annulus Research Articles

Free Boundary Minimal Annuli Immersed in the Unit Ball

We construct a family of compact free boundary minimal annuli immersed in the unit ball B3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {B}^3$$\\end{document} of R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}, the first such examples other than the critical catenoid. This solves a problem formulated by Nitsche in 1985. These annuli are symmetric with respect to two orthogonal planes and a finite group of rotations around an axis, and are foliated by spherical curvature lines. We show that the only free boundary minimal annulus embedded in B3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {B}^3$$\\end{document} foliated by spherical curvature lines is the critical catenoid; in particular, the minimal annuli that we construct are not embedded. On the other hand, we also construct families of non-rotational compact embedded capillary minimal annuli in B3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {B}^3$$\\end{document}. Their existence solves in the negative a problem proposed by Wente in 1995.

On the Minimal Annulus of Triangles and Parallelograms

Sharp upper and lower bounds for the isoperimetric deficit of triangles or parallelograms with the minimal annulus of radii $R$ and $r$ are given.

A new approach to the Fraser-Li conjecture with the Weierstrass representation formula

In this paper, we provide a sufficient condition for a curve on a surface in R 3 \mathbb {R}^3 to be given by an orthogonal intersection with a sphere. This result makes it possible to express the boundary condition entirely in terms of the Weierstrass data without integration when dealing with free boundary minimal surfaces in a ball B 3 \mathbb {B}^3 . Moreover, we show that the Gauss map of an embedded free boundary minimal annulus is one to one. By using this, the Fraser-Li conjecture can be translated into the problem of determining the Gauss map. On the other hand, we show that the Liouville type boundary value problem in an annulus gives some new insight into the structure of immersed minimal annuli orthogonal to spheres. It also suggests a new PDE theoretic approach to the Fraser-Li conjecture.

Properly embedded minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$

Abstract We prove that every properly embedded minimal annulus in $\mathbb{S}^2\times\mathbb{R}$ is foliated by circles. We show that such minimal annuli are given by periodic harmonic maps $\mathbb{C} \to \mathbb{S}^2$ of finite type. Such harmonic maps are parameterized by spectral data, and we show that continuous deformations of the spectral data preserve the embeddedness of the corresponding annuli. A curvature estimate of Meeks and Rosenberg is used to show that each connected component of spectral data of embedded minimal annuli contains a maximum of the flux of the third coordinate. A classification of these maxima allows us to identify the spectral data of properly embedded minimal annuli with the spectral data of minimal annuli foliated by circles.

Index characterization for free boundary minimal surfaces

In this paper, we compute the Morse index for a free boundary minimal submanifold from data of two simpler problems. The first one is the corresponding problem with fixed boundary condition; and the second is associated with the Dirichlet-to-Neumann map for Jacobi fields. As an application, we show that the Morse index of a free boundary minimal annulus is equal to 4 if and only if it is the critical catenoid.

A characterization of the critical catenoid

We show that an embedded minimal annulus $\Sigma^2 \subset B^3$ which intersects $\partial B^3$ orthogonally and is invariant under reflection through the coordinate planes is the critical catenoid. The proof uses nodal domain arguments and a characterization, due to Fraser and Schoen, of the critical catenoid as the unique free boundary minimal annulus in $B^n$ with lowest Steklov eigenvalue equal to 1. We also give more general criteria which imply that a free boundary minimal surface in $B^3$ invariant under a group of reflections has lowest Steklov eigenvalue 1.

Metrics with nonnegative Ricci curvature on convex three-manifolds

We prove that the space of smooth Riemannian metrics on the three-ball with non-negative Ricci curvature and strictly convex boundary is path connected; and, moreover, that the associated moduli space (i.e., modulo orientation-preserving diffeomorphisms of the three-ball) is contractible. As an application, using results of Maximo, Nunes, and Smith [MNS13], we show the existence of properly embedded free boundary minimal annulus on any three-ball with non-negative Ricci curvature and strictly convex boundary.

Free boundary minimal hypersurfaces with spherical boundary

We first prove that a compact embedded minimal annulus in meeting two concentric spheres perpendicularly along the boundaries is part of a plane. We also prove that an embedded minimal hypersurface lying outside of a ball is part of either a catenoid or a hyperplane if it is perpendicular to the sphere along the boundary and has only one end that is asymptotic to either a catenoid if , or a hyperplane if .

Finite type annular ends for harmonic functions

In this paper we describe the notion of an annular end of a Riemann surface being of finite type with respect to some harmonic function and prove some theoretical results relating the conformal structure of such an annular end to the level sets of the harmonic function. As an application of these results, we obtain important information on the conformal type of any properly immersed minimal surface M in $$\mathbb {R}^3$$ with compact boundary and which intersects some plane transversely in a finite number of arcs; in particular, such an M is a parabolic Riemann surface. This information is applied by Meeks III and Perez (Embedded minimal surfaces of finite topology, 2015) to classify the asymptotic behavior of annular ends of a complete embedded minimal annulus with compact boundary in terms of the flux vector along its boundary. In the present paper, we apply this information to understand and characterize properly immersed minimal surfaces in $$\mathbb {R}^3$$ of finite total curvature, in terms of their intersections with two nonparallel planes.

THE GREEN–OSHER INEQUALITY IN RELATIVE GEOMETRY

In this paper we give a proof of the Green–Osher inequality in relative geometry using the minimal convex annulus, including the necessary and sufficient condition for the case of equality.

Aortic annulus eccentricity before and after transcatheter aortic valve implantation: comparison of balloon-expandable and self-expandable prostheses

Purpose: A non-circular geometry of the aortic annulus and implanted prosthesis after transcatheter aortic valve implantation (TAVI) might be relevant for short- and long-term valve function. We investigated the influence of valve type and aortic valve calcification on post-implant geometry of the catheter-based aortic valve prostheses. Methods: Eighty consecutive patients with severe aortic valve stenosis (mean age 82±6 years, 43 (53.7%) male, mean logistic Euroscore 27±14) underwent dual source computed tomography (2x64x0.6 mm or 2x128x0.6 mm) before and after TAVI. The minimal and maximal aortic annulus diameters were determined in multiplanar reconstructions exactly aligned with the annulus plane. Eccentricity of the aortic annulus or prosthesis was defined as 1-(minimal/maximal diameter). Influence of prosthesis type and degree of aortic valve calcification on post-implant eccentricity was analysed. Results: Mean aortic annulus dimensions were 21.4±2.2 mm (short diameter) by 27.0±2.4 mm (long-diameter) prior to valve implantation. Following TAVI, mean diameters were 19.9±2.0 mm by 21.6±2.0 mm. The aortic annulus showed a significantly higher eccentricity in all patients before than after TAVI (0.21±0.06 vs. 0.08±0.06, p<0.0001). Post-TAVI eccentricity was significantly lower in 65 (81.2%) patients following implantation of a balloon-expandable Edwards Sapien prosthesis as compared to 15 (18.8%) patients after a self-expandable Medtronic CoreValve prosthesis (0.06±0.05 vs. 0.15±0.07, p<0.0001). There was no significant difference regarding the extent of aortic valve calcification in patients in whom balloon-expandable or self-expandable aortic valve prostheses were implanted (mean calcium volume of 583.4±421.1 mm3 vs. 509.7±362.6 mm3, p=0.66). After TAVI, patients with a higher calcium burden (mean 1048.1±323.7mm3) showed a significantly higher eccentricity compared to patients with a lower calcium burden (mean 187.5±82.0 mm3; Eccentricity of 0.08±0.06 vs. 0.05±0.03, p=0.02). Conclusions: Catheter-based implantation of aortic valve prosthesis reduces eccentricity of the aortic annulus. Patients undergoing TAVI with a balloon-expandable prosthesis are more likely to demonstrate a circular shape of the implanted prosthesis as compared to patients with a self-expandable prosthesis. Eccentricity of the deployed prosthesis is affected by the extent of aortic valve calcification. Further studies need to assess the influence of prosthesis eccentricity on long-term outcome.

Finite type minimal annuli in $\mathbb{S}^{2}\times\mathbb{R}$

We study minimal annuli in $\mathbb{S}^{2}\times\mathbb{R}$ of finite type by relating them to harmonic maps $\mathbb{C} \to\mathbb{S} ^{2}$ of finite type. We rephrase an iteration by Pinkall–Sterling in terms of polynomial Killing fields. We discuss spectral curves, spectral data and the geometry of the isospectral set. We consider polynomial Killing fields with zeroes and the corresponding singular spectral curves, bubbletons and simple factors. We investigate the differentiable structure on the isospectral set of any finite type minimal annulus. We apply the theory to a 2-parameter family of embedded minimal annuli foliated by horizontal circles.

Remarks on minimal annuli in a slab

We show that the two closed boundary curves of a minimal annulus in a slab are both convex if one of them is convex and along the other curve the surface meets the plane at a constant angle. And therefore, under the same condition, the minimal annulus is foliated by convex planar curves all of which are parallel to the boundary. In particular, if the convex curve is a circle, then the annulus is part of a catenoid.

Relating the minimal annulus with the circumradius of a convex set

In this paper we relate the minimal annulus of a planar convex body K with its circumradius, obtaining all the upper and lower bounds, in terms of these quantities, for some of the classic geometric measures associated with the set: the diameter, the minimal width and the inradius. We prove the optimal inequalities for each one of those problems, determining also its corresponding extremal sets. Mathematics subject classification (2010): 52A40, 52A10, 52A38.

Minimal annuli with constant contact angle along the planar boundaries

We show that an immersed minimal annulus, with two planar boundary curves along which the surface meets these planes with constant contact angle, is part of the catenoid.

Optimizing geometric measures for fixed minimal annulus and inradius

In this paper we relate the minimal annulus of a planar convex body K with its inradius, obtaining all the upper and lower bounds, in terms of these quantities, for the classic geometric measures associated with the set: area, perimeter, diameter, minimal width and circumradius. We prove the optimal inequalities for each one of those problems, determining also its corresponding extremal sets.

Some optimization problems for the minimal annulus of a convex set

In this paper we relate the minimal annulus of a planar convex body K with the six classic geometric measures associated with it. First, we obtain all the possible bounds (upper and lower bounds) for the measures A,p,D,ω,RK and rK of a convex body K with given minimal annulus. Then, we solve the problem of maximizing and minimizing the area and the perimeter of convex bodies with given circumradius and minimal annulus. We prove the optimal inequalities for each of those problems, determining also its corresponding extremal sets.

NON-COMPACT DOUGLAS-PLATEAU PROBLEM BOUNDED BY A LINE AND A JORDAN CURVE

In this article, we prove the existence of a minimal annulus bounded by a Jordan curve and a straight line.

PROPERLY EMBEDDED MINIMAL ANNULI BOUNDED BY A CONVEX CURVE

We prove that given a convex Jordan curve , the space of properly embedded minimal annuli in the half-space . Moreover, for a fixed positive number , the exterior Plateau problem that consists of finding a properly embedded minimal annulus in the upper half-space, with finite total curvature, boundary has exactly zero, one or two solutions, each one with a different stability character for the Jacobi operator.AMS 2000 Mathematics subject classification: Primary 53A10. Secondary 49Q05; 53C42

PROPERLY EMBEDDED MINIMAL ANNULI BOUNDED BY A CONVEX CURVE

We prove that given a convex Jordan curve , the space of properly embedded minimal annuli in the half-space . Moreover, for a fixed positive number , the exterior Plateau problem that consists of finding a properly embedded minimal annulus in the upper half-space, with finite total curvature, boundary has exactly zero, one or two solutions, each one with a different stability character for the Jacobi operator.AMS 2000 Mathematics subject classification: Primary 53A10. Secondary 49Q05; 53C42