Stable Minimal Surfaces Research Articles

A note on area estimates for stable H-hypersurfaces and applications

We show some area estimates for stable CMC hypersurfaces immersed in Riemannian manifolds with scalar and sectional curvature bounded from below. In particular, we focus on immersions in three-dimensional Riemannian manifolds. As an application, we derive some upper estimates for the bottom spectrum of these hypersurfaces. This paper generalizes and builds on the techniques developed by Munteanu, Sung and Wang [19] for stable minimal surfaces.

Generalized Henneberg Stable Minimal Surfaces

We generalize the classical Henneberg minimal surface by giving an infinite family of complete, finitely branched, non-orientable, stable minimal surfaces in mathbb {R}^3. These surfaces can be grouped into subfamilies depending on a positive integer (called the complexity), which essentially measures the number of branch points. The classical Henneberg surface H_1 is characterized as the unique example in the subfamily of the simplest complexity m=1, while for mge 2 multiparameter families are given. The isometry group of the most symmetric example H_m with a given complexity min mathbb {N} is either isomorphic to the dihedral isometry group D_{2m+2} (if m is odd) or to D_{m+1}times mathbb {Z}_2 (if m is even). Furthermore, for m even H_m is the unique solution to the Björling problem for a hypocycloid of m+1 cusps (if m is even), while for m odd the conjugate minimal surface H_m^* to H_m is the unique solution to the Björling problem for a hypocycloid of 2m+2 cusps.

Area and Spectrum Estimates for Stable Minimal Surfaces

This paper mainly concerns the area growth and bottom spectrum of complete stable minimal surfaces in a three-dimensional manifold with scalar curvature bounded from below. When the ambient manifold is the Euclidean space, by an elementary argument, it is shown directly from the stability inequality that the area of such minimal surfaces grows exactly as the Euclidean plane. Consequently, such minimal surfaces must be flat, a well-known result due to Fisher-Colbrie and Schoen as well as do Carmo and Peng. In the case of general ambient manifold, explicit area growth estimate is also derived. For the bottom spectrum, a self-contained argument involving positive Green’s function is provided for its upper bound estimates. The argument extends to stable minimal hypersurfaces in a complete manifold of dimension up to six with sectional curvature bounded from below.

Mean-stable surfaces in static Einstein–Maxwell theory

We use the theory of mean-stable surfaces (stable minimal surfaces included) to explore the static Einstein–Maxwell space-time. We first prove that the zero set of the lapse function must be contained in the horizon boundary. Then, we explore some implications of it providing some results of the nonexistence of stable minimal surfaces in the interior of an electrostatic space, subject to certain initial-boundary data. We finish by proving that the ADM mass is bounded from above by the Hawking quasi-local mass with charge.

On the size of stable minimal surfaces in $${\mathbb {R}}^4$$

The Gauss map g of a surface \(\Sigma \) in \({\mathbb {R}}^4\) takes its values in the Grassmannian of oriented 2-planes of \({\mathbb {R}}^4\): \(G^+(2,4) \). We give geometric criteria of stability for minimal surfaces in \({\mathbb {R}}^4\) in terms of g. We show in particular that if the area of the Gauss map \( |g(\Sigma ) | \) of a minimal surface is smaller than \( 2\pi \) then the surface is stable by deformations which fix the boundary of the surface. This answers the question of Barbosa and Do Carmo (Math Z 173:13–28, 1980) in \({\mathbb {R}}^4\).

Curvature estimates for stable minimal surfaces with a common free boundary

The minimal surfaces meeting in triples with equal angles along a common boundary naturally arise from soap films and other physical phenomena. They are also the natural extension of the usual minimal surfaces. In this paper, we consider the multiple junction surface and show the Bernstein’s Theorem still holds for the stable multiple junction surfaces in some special cases. The key part is to derive the \(L^p\) estimates of the curvature for multiple junction surfaces.

On stability of the fibres of Hopf surfaces as harmonic maps and minimal surfaces

We construct a family of Hermitian metrics on the Hopf surface S 3 × S 1 \mathbb {S}^3\times \mathbb {S}^1 , whose fundamental classes represent distinct cohomology classes in the Aeppli cohomology group. These metrics are locally conformally Kähler. Among the toric fibres of π : S 3 × S 1 → C P 1 \pi :\mathbb {S}^3\times \mathbb {S}^1\to \mathbb {C} P^1 two of them are stable minimal surfaces and each of the two has a neighbourhood so that fibres therein are given by stable harmonic maps from 2-torus and outside, far away from the two tori, there are unstable harmonic ones that are also unstable minimal surfaces. A similar result is true for S 2 n − 1 × S 1 \mathbb {S}^{2n-1}\times \mathbb {S}^{1} .

Scalar curvature on some open 3-manifolds

This article is a survey of recent results about the existence of Riemannian metrics of positive scalar curvature on some open [Formula: see text]-manifolds. This results culminate in a rigidity theorem obtained using the theory of stable minimal surfaces.

Combinatorial Minimal Surfaces in Pseudomanifolds and Other Complexes

We define combinatorial analogues of stable and unstable minimal surfaces in the setting of weighted pseudomanifolds and their generalizations. We prove that, under mild conditions, such combinatorial minimal surfaces always exist. We use a technique, adapted from work of Johnson and Thompson, called \emph{thin position}. Thin position is defined using orderings of the $n$-dimensional simplices of an $n$-dimensional pseudomanifold. In addition to defining and finding combinatorial minimal surfaces, from thin orderings, we derive invariants of even-dimensional closed simplicial pseudomanifolds called \emph{width} and \emph{trunk}. We study additivity properties of these invariants under connected sum and prove theorems analogous to those in knot theory and 3-manifold theory.

Harmonic 1-forms on immersed hypersurfaces in a Riemannian manifold with weighted Bi-Ricci curvature bounded from below

In this paper, we investigate Lp harmonic 1-forms on a complete noncompact orientable stable hypersurface that is immersed in a Riemannian manifold with weighted Bi-Ricci curvature bounded from below. We obtain several vanishing properties for this class of harmonic 1-forms. Our results can be considered as a generalization and improvement of previous work. Moreover, we consider immersed f-minimal hypersurfaces in a weighted Riemannian manifold and prove that if such a hypersurface is weighted stable then the space of L2 weighted harmonic 1-forms is trivial. As a consequence, any weighted stable minimal surface must have genus zero.

Quantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaces

We establish quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the $s$-fractional perimeter as a particular case. On the one hand, we establish universal $BV$-estimates in every dimension $n \geqslant 2$ for stable sets. Namely, we prove that any stable set in $B_1$ has finite classical perimeter in $B_{1/2}$, with a universal bound. This nonlocal result is new even in the case of $s$-perimeters and its local counterpart (for classical stable minimal surfaces) was known only for simply connected two-dimensional surfaces immersed in $\mathbb{R}^3$. On the other hand, we prove quantitative flatness estimates for minimizers and stable sets in low dimensions $n = 2, 3$. More precisely, we show that a stable set in $B_R$, with $R$ large, is very close in measure to being a half space in $B_1$ – with a quantitative estimate on the measure of the symmetric difference. As a byproduct, we obtain new classification results for stable sets in the whole plane.

The geometry of stable minimal surfaces in metric Lie groups

We study geometric properties of compact stable minimal surfaces with boundary in homogeneous 3-manifolds $X$ that can be expressed as a semidirect product of $\mathbb{R}^2$ with $\mathbb{R}$ endowed with a left invariant metric. For any such compact minimal surface $M$, we provide a priori radius estimate which depends only on the maximum distance of points of the boundary $\partial M$ to a vertical geodesic of $X$. We also give a generalization of the classical Rado's Theorem in $\mathbb{R}^3$ to the context of compact minimal surfaces with graphical boundary over a convex horizontal domain in $X$, and we study the geometry, existence and uniqueness of this type of Plateau problem.

Membrane Structures - Aspects of Form-Finding Process

This paper is concerned with the selected aspects, which are discovered during a design stage of cable and membrane structures, known as “form-finding” process. The aim of this paper is the understanding of the basic principles of the form-finding process and their explanation of the very simple examples. The use of the finding a shape of a tension membrane that is in static equilibrium as an analogy with the search condition of minimal surfaces is explained. The basic principles are demonstrated on simple 2D example, in which the finding a stable minimal surface passes in the finding a stable minimal length.

Concentration of total curvature of minimal surfaces in $$\mathbb {H}^2\times \mathbb R$$ H 2 × R

We prove a phenomenon of concentration of total curvature for stable minimal surfaces in the product space $$\mathbb H^2 \times \mathbb R,$$ where $$\mathbb H^2$$ is the hyperbolic plane. Under some geometric conditions on its asymptotic boundary, an oriented stable minimal surface immersed in $$\mathbb H^2 \times \mathbb R$$ has infinite total curvature. We exhibit an example of a minimal graph such that in a domain whose asymptotic boundary is a vertical segment the total curvature is finite, but the total curvature of the graph is infinite, by the theorem cited before. We also present some simple and peculiar examples of infinite total curvature minimal surfaces in $$\mathbb H^2 \times \mathbb R$$ and their asymptotic boundaries.

Effective versions of the positive mass theorem

The study of stable minimal surfaces in Riemannian $3$-manifolds $(M, g)$ with non-negative scalar curvature has a rich history. In this paper, we prove rigidity of such surfaces when $(M, g)$ is asymptotically flat and has horizon boundary. As a consequence, we obtain an effective version of the positive mass theorem in terms of isoperimetric or, more generally, closed volume-preserving stable CMC surfaces that is appealing from both a physical and a purely geometric point of view. We also include a proof of the following conjecture of R. Schoen: An asymptotically flat Riemannian $3$-manifold with non-negative scalar curvature that contains an unbounded area-minimizing surface is isometric to flat $\mathbb{R}^3$.

Rigidity of stable minimal hypersurfaces in asymptotically flat spaces

We prove that if an asymptotically Schwarzschildean 3-manifold (M, g) contains a properly embedded stable minimal surface, then it is isometric to the Euclidean space. This implies, for instance, that in presence of a positive ADM mass any sequence of solutions to the Plateau problem with diverging boundaries can never have uniform height bounds, even at a single point. An analogous result holds true up to ambient dimension seven provided polynomial volume growth on the hypersurface is assumed.

Stable and unstable minimal surfaces and Hopf’s problem

Hopf’s well-known conjecture is considered, which states that there exists no metric of strictly positive curvature on the topological product S2 × S2 of two 2-spheres. Three theorems are proved.

Stable Minimal Surfaces in Riemannian Warped Products

We obtain several stability results for minimal two-sided surfaces immersed in a wide class of 3-dimensional Riemannian warped products, which includes the class of Riemannian products. As a consequence, some Bernstein-type results are provided. Moreover, a non-existence theorem for a certain non-linear elliptic equation is obtained.

Topological censorship from the initial data point of view

We introduce a natural generalization of marginally outer trapped surfaces, called immersed marginally outer trapped surfaces, and prove that three-dimensional asymptotically flat initial data sets either contain such surfaces or are diffeomorphic to $\mathbb{R}^3$. We establish a generalization of the Penrose singularity theorem which shows that the presence of an immersed marginally outer trapped surface generically implies the null geodesic incompleteness of any spacetime that satisfies the null energy condition and which admits a non-compact Cauchy surface. Taken together, these results can be viewed as an initial data version of the Gannon-Lee singularity theorem. The first result is a non-time-symmetric version of a theorem of Meeks-Simon-Yau which implies that every asymptotically flat Riemannian 3-manifold that is not diffeomorphic to $\mathbb{R}^3$ contains an embedded stable minimal surface. We also obtain an initial data version of the spacetime principle of topological censorship. Under physically natural assumptions, a 3-dimensional asymptotically flat initial data set with marginally outer trapped boundary and no immersed marginally outer trapped surfaces in its interior is diffeomorphic to $\mathbb{R}^3$ minus a finite number of open balls. An extension to higher dimensions is also discussed.

Area minimizing surfaces in mean convex 3-manifolds

Abstract In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by a simple closed curve in ∂M. Our main result is that for any g ≥ 0, the space of simple closed curves in ∂M where all the absolutely area minimizing surfaces they bound in M has genus ≥ g is open and dense in the space 𝒜 of nullhomologous simple closed curves in ∂M. For showing this we prove a bridge principle for absolutely area minimizing surfaces. Moreover, we show that for any g ≥ 0, there exists a curve γ g in 𝒜 such that the minimum genus of the absolutely area minimizing surfaces γ g bounds is exactly g. As an application of these results, we further prove that the simple closed curves in ∂M bounding more than one minimal surface in M is an open and dense subset of 𝒜. We also show that there are disjoint simple closed curves in ∂M bounding minimal surfaces in M which are not disjoint. This allows us to answer a question of Meeks, by showing that for any strictly mean convex 3-manifold M, there exists a simple closed curve Γ in ∂M which bounds a stable minimal surface which is not embedded. We also gave some applications of these results to the simple closed curves in ℝ3.