Dynamical instability of minimal surfaces at flat singular points

Second order contact of minimal surfaces

Flat Lagrangian minimal surfaces in the Lorentzian complex plane \({\mathbb{C}}^{2}_{1}\) are classified by B. Y. Chen and L. Vrancken in [8]. On the other hand, Vrancken proves in [11] that Lagrangian minimal surfaces of constant curvature in \({\mathbb{C}}^{2}_{1}\) are flat surfaces. In this article, we classify all Lagrangian minimal surfaces in \({\mathbb{C}}^{2}_{1}\) which are free from flat points.

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.

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.

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.

Dynamic crack surface instabilities initiated at dopants

A surface of constant mean curvature (CMC) equal to H in a sub-Riemannian 3-manifold is strongly stable if it minimizes the functional area+2Hvolume up to second order. In this paper we obtain some criteria ensuring strong stability of surfaces in Sasakian 3-manifolds. We also produce new examples of C1 complete CMC surfaces with empty singular set in the sub-Riemannian 3-space forms by studying those ones containing a vertical line. As a consequence, we are able to find complete strongly stable non-vertical surfaces with empty singular set in the sub-Riemannian hyperbolic 3-space M(−1). In relation to the Bernstein problem in M(−1) we discover strongly stable C∞ entire minimal graphs in M(−1) different from vertical planes. These examples are in clear contrast with the situation in the first Heisenberg group, where complete strongly stable surfaces with empty singular set are vertical planes. Finally, we analyze the strong stability of CMC surfaces of class C2 and non-empty singular set in the sub-Riemannian 3-space forms. When these surfaces have isolated singular points we deduce their strong stability even for variations moving the singular set.

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.

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} .

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

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.

Surface instabilities in a Mooney–Rivlin body with frictional boundary conditions

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.

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.

The instability of field emission current arises from the atomic changes of the field emitter surface which can be induced by the sputtering of the secondary ions, by surface diffusion of containment atoms, and by the dynamical behavior of the emitter surface itself. There are three major causes of the instability: (1) the ion-bombardment- or ion-sputtering-induced surface changes, (2) contamination and surface diffusion of contaminant-induced emission current fluctuations, and (3) the intrinsic thermal instability of the field emitter surface itself. A few possible means of alleviating these problems are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>