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We study a variational model for soap films in which the films are represented by sets with fixed small volume rather than surfaces. In this problem, a minimizing sequence of completely “wet" films, or sets of finite perimeter spanning a wire frame, may converge to a film containing both wet regions of positive volume and collapsed (dry) surfaces. When collapsing occurs, these limiting objects lie outside the original minimization class and instead are admissible for a relaxed problem. Here we show that the relaxation and the original formulation are equivalent by approximating the collapsed films in the relaxed class by wet films in the original class. [ABSTRACT FROM AUTHOR]

In this paper, we study conformal solitons for the mean curvature flow in hyperbolic space H n + 1 . Working in the upper half-space model, we focus on horo-expanders, which relate to the conformal field - ∂ 0 . We classify cylindrical and rotationally symmetric examples, finding appropriate analogues of grim-reaper cylinders, bowl and winglike solitons. Moreover, we address the Plateau and the Dirichlet problems at infinity. For the latter, we provide the sharp boundary convexity condition to guarantee its solvability and address the case of non-compact boundaries contained between two parallel hyperplanes of ∂ ∞ H n + 1 . We conclude by proving rigidity results for bowl and grim-reaper cylinders. [ABSTRACT FROM AUTHOR]

The authors generalize the Fenchel theorem for strong spacelike closed curves of index 1 in the 3-dimensional Minkowski space, showing that the total curvature must be less than or equal to 2π. Here the strong spacelike condition means that the tangent vector and the curvature vector span a spacelike 2-plane at each point of the curve γ under consideration. The assumption of index 1 is equivalent to saying that γ winds around some timelike axis with winding number 1. This reversed Fenchel-type inequality is proved by constructing a ruled spacelike surface with the given curve as boundary and applying the Gauss-Bonnet formula. As a by-product, this shows the existence of a maximal surface with γ as the boundary. [ABSTRACT FROM AUTHOR]

We give a fairly complete solution to the asymptotic Plateau Problem for area minimizing surfaces in H2×R. In particular, we identify the collection of Jordan curves in ∂∞(H2×R) which bounds an area minimizing surface in H2×R. Furthermore, we study the similar problem for minimal surfaces, and show that the situation is highly different. [ABSTRACT FROM AUTHOR]

I review the present status of the problem of initial conditions for inflation and describe several ways to solve this problem for many popular inflationary models, including the recent generation of the models with plateau potentials favored by cosmological observations. [ABSTRACT FROM AUTHOR]

Adhesion of bio-inspired microfibre arrays to a rough surface is studied theoretically. The array consists of vertical elastic rods fixed on a rigid backing layer, and the surface is modeled by rigid steps with a normally distributed height. Analytical expressions are obtained for the adhesion forces in both the approach and retraction processes. It is shown that, with the increasing preload, the pull-off force increases at first and then attains a plateau value. The results agree with the previous experiments and are expected helpful in adhesion control of the array in practical applications. [ABSTRACT FROM AUTHOR]

We show the area-minimality property of all homogeneous area-minimizing hypercones in Euclidean spaces (classified by Lawlor) following Lawson's original idea in his 72' Trans. A.M.S. paper 'The equivariant Plateau problem and interior regularity'. Moreover, each of them enjoys (coflat) calibrations singular only at the origin. [ABSTRACT FROM AUTHOR]

We give several regularity results for two-dimensional solutions to the c-Plateau problem in $$\mathbb {R}^{3}:$$ given $$c>0$$ and an integer one-rectifiable current $$\varGamma $$ without boundary in $$\mathbb {R}^{3},$$ we study integer two-rectifiable currents which minimize c-isoperimetric mass $$\mathbf {M}^{c}(T):=\mathbf {M}(T)+c \mathbf {M}(\partial T)^{2},$$ where $$\mathbf {M}$$ is the usual mass on currents, amongst all integer two-rectifiable currents T with boundary of the form $$\partial T = \varGamma + \varSigma $$ where $$\varGamma ,\varSigma $$ have disjoint supports. We show the following three results for $$\mathbf {T}_{c}$$ a solution to the c-Plateau problem with $$\partial \mathbf {T}_{c} = \varGamma +\varSigma _{c}$$ : if $$\varSigma _{c}$$ is a smooth closed embedded curve, then $$\varSigma _{c}$$ parameterized by arc-length must have at some point large third derivative; $$\mathbf {T}_{c}$$ cannot have a tangent cone at a singular point of $$\varSigma _{c}$$ supported in a plane but with constant orientation; $$\varSigma _{c}$$ is regular wherever we can write the support of $$\mathbf {T}_{c}$$ as a finite union of $$C^{1}$$ surfaces-with-boundary. [ABSTRACT FROM AUTHOR]

The Kirchhoff-Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a soap film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. Adopting a variational approach, we define an energy associated with shape deformations of the system and then derive general equilibrium and (linear) stability conditions by considering the first and second variations of the energy functional. We analyze in detail the transition to instability of flat circular configurations, which are ground states for the system in the absence of surface tension, when the latter is progressively increased. Such a theoretical study is particularly useful here, since the many different perturbations that can lead to instability make it challenging to perform an exhaustive experimental investigation. We generalize previous results, since we allow the filament to possess a curved intrinsic shape and also to display anisotropic flexural properties (as happens when the cross section of the filament is noncircular). This is accomplished by using a rod energy which is familiar from the modeling of DNA filaments. We find that the presence of intrinsic curvature is necessary to obtain a first buckling mode which is not purely tangent to the spanning surface. We also elucidate the role of twisting buckling modes, which become relevant in the presence of flexural anisotropy. [ABSTRACT FROM AUTHOR]

Hopf's well-known conjecture is considered, which states that there exists no metric of strictly positive curvature on the topological product S × S of two 2-spheres. Three theorems are proved. [ABSTRACT FROM AUTHOR]

The Euler-Plateau problem, proposed by Giomi and Mahadevan in Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 468, 1851-1864 (), concerns a soap film spanning a flexible loop. The shapes of the film and the loop are determined by the interactions between the two components. In the present work, the Euler-Plateau problem is reformulated to yield a boundary-value problem for a vector field that parameterizes both the spanning surface and the bounding loop. Using the first and second variations of the relevant free-energy functional, detailed bifurcation and stability analyses are performed. For a spanning surface with energy density σ and a bounding loop with length 2 πR and flexural rigidity a, the first bifurcation, during which the spanning surface remains flat but the bounding loop becomes noncircular, occurs at R σ/ a=3, confirming a result obtained previously via an energy comparison. All other bifurcation solution branches emanating from the flat circular solution branch, including those to nonplanar solution branches, are found to be unstable. [ABSTRACT FROM AUTHOR]

A new species of Tsaidamotherium (Bovidae, Artiodactyla), T. brevirostrum, described here represents only the second discovery of the genus. Tsaidamotherium is a mid-sized late Miocene bovid with an odd-looking plate-like horn-core apparatus so far known only from northwestern China. The holotype of T. brevirostrum is a cranial part of skull from the Liushu Formation (late Miocene) of the Linxia Basin. The new species differs from the type species, T. bohlini, mainly in horn-core morphology, having the frontal horn-cores proper much smaller and less distinctly separated from the plate-shaped posterior portion. Furthermore, the new species has smaller frontal sinuses, but a thicker layer of compact bone beneath the horn-core apparatus as seen in frontal section. A complete skull with a smaller horn-core apparatus is considered to represent a female individual of T. brevirostrum. The facial part of the skull, which is first known for this genus, is very short and high, with a high nasal cavity, strongly retracted nasals, and shortened premaxillae and premolar rows. All this probably suggests that the animal with an enlarged nasal cavity, like the modern Saiga and Budorcas, may have had a special adaptation to the high plateau environment. [ABSTRACT FROM AUTHOR]

Plateau's problem is to show the existence of an area-minimizing surface with a given boundary, a problem posed by Lagrange in 1760. Experiments conducted by Plateau showed that an area-minimizing surface can be obtained in the form of a film of oil stretched on a wire frame, and the problem came to be called Plateau's problem. Special cases have been solved by Douglas, Rado, Besicovitch, Federer and Fleming, and others. Federer and Fleming used the chain complex of integral currents with its continuous boundary operator, a Poincaré Lemma, and good compactness properties to solve Plateau's problem for orientable, embedded surfaces. But integral currents cannot represent surfaces such as the Möbius strip or surfaces with triple junctions. In the class of varifolds, there are no existence theorems for a general Plateau problem. We use the chain complex of differential chains, a geometric Poincaré Lemma, and good compactness properties of the complex to solve Plateau's problem in such generality as to find the first solution which minimizes area taken from a collection of surfaces that includes all previous special cases, as well as all smoothly immersed surfaces of any genus type, orientable or nonorientable, and surfaces with multiple junctions. Our result holds for all dimensions and codimension-one surfaces in ℝ. [ABSTRACT FROM AUTHOR]

This is the second in a series of two papers discussing the elementary but beautiful and fundamental question (open for some eighty years) of whether or not a minimal surface spanning a sufficiently smooth curve, which is a local minimizer, is immersed up to and including the boundary. We show that C minimizers of energy or area cannot have nonexceptional boundary branch points. [ABSTRACT FROM AUTHOR]

We define the notion of an exceptional branch point for a minimal surface in $${{\mathbb R}^3}$$ and show that it is closely related to the notion of false branch point. We prove that even in the presence of an exceptional branch point we may, under certain conditions, reduce Dirichlet’s energy and therefore also reduce area. [ABSTRACT FROM AUTHOR]

In this paper we study minimal surfaces in M × ℝ, where M is a complete surface. Our main result is a Jenkins-Serrin type theorem which establishes necessary and sufficient conditions for the existence of certain minimal vertical graphs in M × ℝ. We also prove that there exists a unique solution of the Plateau’s problem in M × ℝ whoseboundaryisaNitschegraphandweconstructaScherk-typesurfaceinthisspace. [ABSTRACT FROM AUTHOR]

We prove existence theorems for two-dimensional, noncompact, complete minimal surfaces in ℝn of annular type, which span a given contour and have a finite total curvature end and prescribed asymptotical behavior. For arbitrary rectifiable Jordan curves, we show the existence of such surfaces with a flat end, i.e., within a bounded distance from a 2-plane. For more restricted classes of curves, we prove the existence of minimal surfaces with higher multiplicity flat ends as well as of surfaces with polynomial-type nonflat ends. [ABSTRACT FROM AUTHOR]

Using a standard fact in hyperbolic geometry, we give a simple proof of the uniqueness of PL minimal surfaces, thus filling in a gap in the original proof of Jaco and Rubinstein. Moreover, in order to clarify some ambiguity, we sharpen the definition of PL minimal surfaces, and prove a technical lemma on the Plateau problem in the hyperbolic space. [ABSTRACT FROM AUTHOR]

We construct new examples of complete minimal tori in the three-dimensional Euclidean space with an arbitrary even number n ≥ 6 of planar embedded ends. [ABSTRACT FROM AUTHOR]

We prove existence and almost everywhere regularity of an area minimizing soap film with a bound on energy spanning a given Jordan curve in Euclidean space R 3. The energy of a film is defined to be the sum of its surface area and the length of its singular branched set. The class of surfaces over which area is minimized includes images of disks, integral currents, nonorientable surfaces and soap films as observed by Plateau with a bound on energy. Our area minimizing solution is shown to be a smooth surface away from its branched set which is a union of Lipschitz Jordan curves of finite total length. [ABSTRACT FROM AUTHOR]

In this paper, we continue our study on the heat flow for the minimal surface with Plateau boundary condition in. The aim in introducing the heat flow is to establish the Morse theory, the minimax methods for minimal surfaces spanned by a curve Γ. In contrast with the previous paper, now we are concerned with minimal surfaces in a compact Riemannian manifold N rather than that in the Euclidean space. [ABSTRACT FROM AUTHOR]

The heat flow for the minimal surface under Plateau boundary condition is defined to be a parabolic variational inequality, and then the existence, uniqueness, regularity, continuous dependence on the initial data and the asymptotics are studied. It is applied as a deformation of the level sets in the critical point theory. [ABSTRACT FROM AUTHOR]

Asymptotic solutions are constructed for the problem of the shock‐wave structure in mixtures of gases with disparate molecular masses. The effect of emergence of a plateau on the density profile of the light component and nonmonotonicity of the temperature profile of the heavy component is described. Based on a comparison with calculations of the full model, the range of applicability of asymptotic solutions is determined. [ABSTRACT FROM AUTHOR]

Given a compact, oriented Riemannian manifold M, without boundary, and a codimension-one homology class in H* (M, Z) (or, respectively, in H* (M, Zp) with p an odd prime), we consider the problem of finding a cycle of least area in the given class: this is known as the homological Plateau’s problem. We propose an elliptic regularization of this problem, by constructing suitable fiber bundles ξ (resp. ζ) on M, and one-parameter families of functionals defined on the regular sections of ξ, (resp. ζ), depending on a small parameter ε. As ε → 0, the minimizers of these functionals are shown to converge to some limiting section, whose discontinuity set is exactly the minimal cycle desired. [ABSTRACT FROM AUTHOR]