Generalized Bernstein Theorem for stable minimal plateau 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} .

This paper gives an upper bound for the first eigenvalue of the universal cover of a complete, stable minimal surface in hyperbolic space, and a sharper one for least area disks.

Correction to: Stable minimal surfaces and spatial topology in general relativity

We study Hamiltonian stable minimal Lagrangian closed submanifolds in the standard complex projective n-space CP n .It is shown that when n = 2such a surface Σis either totally geodesic or flat if the multiplicity of the Laplacian acting on C∞(Σ)is less than or equal to 6.

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.

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.

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 \({\mathbb {R}}^4\).

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.

Equilibrium surfaces originate from the mechanics of liquids and gases as the interface between two media that are in equilibrium. The equilibrium condition arises from the condition of minimum potential energy of the corresponding mechanical system. Equilibrium surfaces include the classes of minimal surfaces, surfaces of constant mean curvature and equilibrium capillary surfaces. The study of the stability of equilibrium surfaces is closely related to the questions of the existence of a solution to the variational multidimensional problem for the minimum of the potential energy functional. In particular, unstable solutions of the corresponding differential equations are not realizable in nature. Stability is characterized by the positivity of the form of the second variation of the corresponding functional (for example, the area functional for minimal surfaces). In most cases, this property means a lower bound for a quantity similar to the fundamental frequency of a region on a surface. In this article, I follow the approach of Sh.T. Yau obtained lower bounds for the quantity that generalizes the fundamental frequency of the region. Based on these estimates, the stability conditions for minimal surfaces and surfaces of constant mean curvature are proved.

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.

In this paper, we give a lower bound for the spectrum of the Laplacian on\nminimal hypersurfaces immersed into $H^m \\times R$. As an application, in\ndimension 2, we prove that a complete minimal surface with finite total\nextrinsic curvature has finite index. On the other hand, for stable, minimal\nsurfaces in $H^3$ or in $H^2 \\times \\R$, we give an upper bound on the infimum\nof the spectrum of the Laplacian and on the volume growth.\n

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.

International Journal of MathematicsVol. 10, No. 06, pp. 697-706 (1999) No AccessON THE TORELLI THEOREM FOR MINIMAL IMMERSIONSCLAUDIO AREZZOCLAUDIO AREZZOMathematics Institute, University of Warwick, Coventry CV4 7AL, UKThe author was supported by European Community Grant TRM N. ERBFMBICT 950079.https://doi.org/10.1142/S0129167X99000288Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail FiguresReferencesRelatedDetails Recommended Vol. 10, No. 06 Metrics History Received 5 August 1998 PDF download

The use of surface-enhanced Raman scattering (SERS) as an analytical technique continues to grow, but questions about its viability remain. This paper addresses the concern about the long-term stability of the surfaces used with the SERS technique. We have studied the long-term stability of Ag surfaces coated with three different thiol compounds. For this study, we have used 1-propanethiol, 1-do-decanethiol, and p -cresolthiol to coat the Ag surfaces. These surfaces showed long-term stabilities of over a month with minimal surface degradation. In order to mimic a realistic application of SERS, we stored the coated surfaces in a solution of water. Sodium dodecylsulfate (SDS) was added to maintain consistent surface wetting conditions so that reproducible results were obtainable on a day-to-day basis. Spatial heterogeneity of the surfaces and coating was analyzed with a fiber-optic Raman system. The spatial heterogeneity of the surfaces explains the day-to-day variations observed in the detection when the same portion of the surface used for analysis varies from day to day.

The minimal surfaces bounded by a frame formed of a double helix and two horizontal rods are studied. The vibration equation shows that the helicoid is the stable surface when its winding number is small. The catenoid is locally isometric to the helicoid so that their vibration spectra are strongly related. While the catenoid is known to undergo a discontinuous transition to two disks, the helicoid is shown to become unstable through a continuous transition to a ribbon-shaped surface obtained experimentally, numerically, and analytically in the limit of infinite height. The normal forms of the bifurcations confirm the analysis.