Nonzero Mean Curvature Research Articles

Planar Pseudo-geodesics and Totally Umbilic Submanifolds

We study totally umbilic isometric immersions between Riemannian manifolds. First, we provide a novel characterization of the totally umbilic isometric immersions with parallel normalized mean curvature vector, i.e., those having nonzero mean curvature vector and such that the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. Such characterization is based on a family of curves, called planar pseudo-geodesics, representing a natural extrinsic generalization of both geodesics and Riemannian circles: being planar, their Cartan development in the tangent space is planar in the ordinary sense; being pseudo-geodesics, their geodesic and normal curvatures satisfy a linear relation. We study these curves in detail and, in particular, establish their local existence and uniqueness. Moreover, in the case of codimension-one immersions, we prove the following statement: an isometric immersion ι:M↪Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\iota :M \\hookrightarrow Q$$\\end{document} is totally umbilic if and only if the extrinsic shape of every geodesic of M is planar. This extends a well-known result about surfaces in 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}.

ІНФІНІТЕЗИМАЛЬНІ ДЕФОРМАЦІЇ ПОВЕРХОНЬ ІЗ ЗАДАНОЮ ЗМІНОЮ ТЕНЗОРА РІЧЧІ

In three-dimensional Euclidean space, we study the problem of the existence of an infinitesimal first-order deformation of single-connected regular surfaces with a predetermined change in the Ricci tensor. It is shown that for surfaces of nonzero Gaussian curvature, this problem is reduced to the study and solution of a system of seven equations (including differential equations) with respect to seven unknown functions, each solution of which determines a vector field that is a univariate function (with an accuracy of a constant vector) and can be interpreted as a moment-free stress state of equilibrium of a loaded shell. For regular surfaces of non-zero Gaussian and mean curvatures, the problem is reduced to finding solutions to one second-order partial differential equation with respect to two unknown functions. Given one of these functions, the resulting equation will in general be a nonhomogeneous second-order partial differential equation (nonhomogeneous Weingarten differential equation). It is proved that any regular surface of positive Gaussian and non-zero mean curvature admits an infinitesimal first-order deformation with a given change in the Ricci tensor in a sufficiently small region. In this case, the tensor fields will be represented by an arbitrary and predefined regular function. By considering the Neumann problem, it is shown that a single-connected regular surface of elliptic type of positive Gaussian and negative mean curvature with a regular boundary under a certain boundary condition admits, in general, an infinitesimal first-order deformation with a predetermined change in the Ricci tensor. In this case, the tensor fields will be determined uniquely. For surfaces of negative Gaussian and non-zero mean curvature, the resulting inhomogeneous partial differential equation with second-order partial differentials will be of hyperbolic type with known coefficients and right-hand side. The Darboux problem is considered for this equation. It is proved that any regular surface of negative Gaussian and non-zero mean curvature admits an infinitesimal first-order deformation with a given change in the Ricci tensor. Tensor fields are expressed through a given function of two variables and through two arbitrary regular functions of one variable. Keywords: infinitesimal deformation, Ricci tensor, tensor fields, Gaussian curvature, mean curvature.

Canonical Coordinates and Natural Equation for Lorentz Surfaces in R13

We consider Lorentz surfaces in R13 satisfying the condition H2−K≠0, where K and H are the Gaussian curvature and the mean curvature, respectively, and call them Lorentz surfaces of general type. For this class of surfaces, we introduce special isotropic coordinates, which we call canonical, and show that the coefficient F of the first fundamental form and the mean curvature H, expressed in terms of the canonical coordinates, satisfy a special integro-differential equation which we call a natural equation of the Lorentz surfaces of a general type. Using this natural equation, we prove a fundamental theorem of Bonnet type for Lorentz surfaces of a general type. We consider the special cases of Lorentz surfaces of constant non-zero mean curvature and minimal Lorentz surfaces. Finally, we give examples of Lorentz surfaces illustrating the developed theory.

Translation surfaces with non-zero constant mean curvature in Euclidean space

We prove that the only surface in 3-dimensional Euclidean space $${\mathbb {R}}^3$$ with constant and non-zero mean curvature H, constructed by the sum of a planar curve and a space curve, is the circular cylinder of radius $$\frac{1}{2|H|}$$ .

On the validity of Damköhler's first hypothesis in turbulent Bunsen burner flames: A computational analysis

The validity of Damköhler's first hypothesis, which relates the turbulent flame speed to turbulent flame surface area under the condition where the integral length scale of turbulence is greater than the flame thickness, has been assessed using three-dimensional Direct Numerical Simulations (DNS) of turbulent premixed Bunsen burner flames over a range of values of Reynolds number, pressure and turbulence intensity. It has been found for the Bunsen configuration that the proportionality between volume-integrated burning rate and the overall flame surface area is not strictly maintained according to Damköhler's first hypothesis. The discrepancy is found to originate physically from the local stretch rate dependence of displacement speed, and this helps to explain differences observed previously between flames with and without mean curvature. Approximating the local density-weighted flame propagation speed with the unstrained laminar burning velocity is shown to be inaccurate, and can have a significant influence on the prediction of the overall burning rate for flames with non-zero mean curvature. Using a two-dimensional projection of the actual scalar gradient for flame area evaluation is shown to exacerbate the loss of proportionality between volume-integrated burning rate and the overall flame surface area. The current analysis identifies the conditions under which Damköhler's hypothesis remains valid and the necessary correction for non-zero mean flame curvature. Further, it has been demonstrated that surface-weighted stretch effects on displacement speed need to be accounted for in order to ensure the validity of Damköhler's hypothesis under all circumstances. Finally, it has been found that the volume-integrated density-weighted scalar dissipation rate remains proportional to the overall burning rate for all flames considered here irrespective of the value of Reynolds number, pressure and turbulence intensity. However, this proportionality is lost when the scalar dissipation rate is evaluated using the two-dimensional projection of the actual scalar gradient.

Persistently Auxetic Materials: Engineering the Poisson Ratio of 2D Self-Avoiding Membranes under Conditions of Non-Zero Anisotropic Strain.

Entropic surfaces represented by fluctuating two-dimensional (2D) membranes are predicted to have desirable mechanical properties when unstressed, including a negative Poisson's ratio ("auxetic" behavior). Herein, we present calculations of the strain-dependent Poisson ratio of self-avoiding 2D membranes demonstrating desirable auxetic properties over a range of mechanical strain. Finite-size membranes with unclamped boundary conditions have positive Poisson's ratio due to spontaneous non-zero mean curvature, which can be suppressed with an explicit bending rigidity in agreement with prior findings. Applying longitudinal strain along a singular axis to this system suppresses this mean curvature and the entropic out-of-plane fluctuations, resulting in a molecular-scale mechanism for realizing a negative Poisson's ratio above a critical strain, with values significantly more negative than the previously observed zero-strain limit for infinite sheets. We find that auxetic behavior persists over surprisingly high strains of more than 20% for the smallest surfaces, with desirable finite-size scaling producing surfaces with negative Poisson's ratio over a wide range of strains. These results promise the design of surfaces and composite materials with tunable Poisson's ratio by prestressing platelet inclusions or controlling the surface rigidity of a matrix of 2D materials.

Faceted particles formed by the frustrated packing of anisotropic colloids on curved surfaces.

We use computer simulations and simple theoretical models to analyze the morphologies that result when rod-like particles end-attach onto a curved surface, creating a finite-thickness monolayer aligned with the surface normal. This geometry leads to two forms of frustration, one associated with the incompatibility of hexagonal order on surfaces with Gaussian curvature, and the second reflecting the deformation of a layer with finite thickness on a surface with non-zero mean curvature. We show that the latter effect leads to a faceting mechanism. Above threshold values of inter-particle attraction strength and surface mean curvature, the adsorbed layer undergoes a transition from orientational disorder to an ordered state that is demarcated by reproducible patterns of line defects. The number of facets is controlled by the competition between line defect energy and intra-facet strain. Tuning control parameters thus leads to a rich variety of morphologies, including icosahedral particles and irregular polyhedra. In addition to suggesting a new strategy for the synthesis of aspherical particles with tunable symmetries, our results may shed light on recent experiments in which rod-like HIV GAG proteins assemble around nanoscale particles.

On the geometry of complete submanifolds immersed in the hyperbolic space

We deal with $n$-dimensional complete submanifolds immersed with parallel nonzero mean curvature vector ${\bf H}$ in the hyperbolic space $\mathbb{H}^{n+p}$. In this setting, we establish sufficient conditions to guarantee that such a submanifold $M^n$ must be pseudo-umbilical, which means that ${\bf H}$ is an umbilical direction. In particular, we conclude that $M^n$ is a minimal submanifold of a small hypersphere of $\mathbb{H}^{n+p}$.

Constructing constant mean curvature surfaces with fixed-point half dynamic model

In this paper, we present a method to generate a constant mean curvature (CMC) surface with the given boundary. By assuming the velocity to zero in the dynamic model for every step, a fixed-point iteration is carried out. Either the variation of Dirichlet energy or area energy can be chosen to be the approximation of the mean curvature. Experiments and results show that our method can work for minimal surfaces or CMC surfaces with nonzero mean curvature and with fixed or free boundary. Meanwhile it is robust with poor quality initial mesh as well as non-uniformly distributed vertices.

Pinching theorems for compact spacelike submanifolds in semi-Riemannian space forms

Let Mn be a compact spacelike submanifold of a semi-Riemannian manifold Mpn+p(c) of constant curvature c, whose index is p, with nonzero mean curvature H and parallel mean curvature vector. We prove that there exists a positive constant C=C(n,H) such that if ‖S‖n2<C, where S is the squared norm of second fundamental form of Mn, then Mn is totally umbilical.

The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps

The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant negative Gauss curvature surfaces (pseudospherical surfaces) in Euclidean 3-space, and for timelike constant non-zero mean curvature (CMC) surfaces in the Lorentz-Minkowski 3- space. We prove that there is a unique solution if the prescribed curve is non-characteristic, and for characteristic initial curves (asymptotic curves for pseudospherical surfaces and null curves for timelike CMC) it is nec- essary and sufficient for similar data to be prescribed along an additional characteristic curve that intersects the first. The proofs also give a means of constructing all solutions using loop group techniques. The method used is the infinite dimensional d'Alembert type representation for surfaces as- sociated with Lorentzian harmonic maps (1-1 wave maps) into symmetric spaces, developed since the 1990's. Explicit formulae for the potentials in terms of the prescribed data are given, and some applications are consid- ered.

Gauss curvature-driven image inpainting for image reconstruction

In this paper, we propose a third-order Gauss curvature-driven geometric diffusion Partial Differential Equation for inpainting and reconstructing images. In Gauss curvature-driven diffusion processes, the rate of diffusion is directly proportional to the Gauss curvature value of the level curve. Since the Gauss curvature is the product of principal curvatures, its value become zero when even one of the principal curvatures is zero. Therefore, when Gauss curvature is used as a driving function for diffusion, the evolution preserves some of the meaningful structures with nonzero mean curvature values (viz. curvy edges, corners, etc.). However, the noise features always have nonzero Gauss curvature value and hence the diffusion process effectively removes them. The inpainting property of geometric PDE based on the Gauss curvature is being used in this work for reconstructing lost or degraded information. A filter is proposed to reconstruct the original images from the observed blurred and noisy images along with inpainting the desired image domain.

Singularities of spacelike constant mean curvature surfaces in Lorentz–Minkowski space

AbstractWe study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space L3. We show how to solve the singular Björling problem for such surfaces, which is stated as follows: given a real analytic null-curve f0(x), and a real analytic null vector field v(x) parallel to the tangent field of f0, find a conformally parameterized (generalized) CMC H surface in L3 which contains this curve as a singular set and such that the partial derivatives fx and fy are given by df0/dx and v along the curve. Within the class of generalized surfaces considered, the solution is unique and we give a formula for the generalized Weierstrass data for this surface. This gives a framework for studying the singularities of non-maximal CMC surfaces in L3. We use this to find the Björling data – and holomorphic potentials – which characterize cuspidal edge, swallowtail and cuspidal cross cap singularities.

Motions of curves in the complex hyperbola and the Burgers hierarchy

We study theory of curves in the complex hyperbola and show that special motions of curves are linked with the Burgers hierarchy, which also leads to a Hamiltonian formulation of the hierarchy and to the difference Burgers equation via their discretization. 1. Introduction It is widely recognized that a lot of differential equations in soliton theory arise from differential geometry especially theory of curves or surfaces ([1, 2, 3]). For example surfaces in the Euclidean 3-space with constant negative Gaussian or non-zero mean curvature are described by the sine or sinh Gordon equation respectively and proper affine spheres are described by the Tzitzeica equatio n. If we refer to theory of curves, the curvature of curves in the Euclidean 2-space evolves according to the mKdV equation under special motions. Pinkall ([4]) showed that the space of closed centroaffine curves in the centroaffine plane possesses a nat ural symplectic structure and the centroaffine curvature evolves according to the KdV e quation when the flow is generated by a Hamiltonian given by the total centroaffine curvature. Chou and Qu ([5, 6]) showed that many soliton equations arise from special motions of plane or space curves. Moreover, Hoffmann and Kutz ([7]) showed that special motions of curves in the complex 1 or 2-space or the complex projective line whose curvature evolves according to the mKdV or KdV equation can be discretized by use of cross ratios. In this paper we study theory of curves in the complex hyperbola which are determined by a certain curvature up to some symmetry. We shall show that special motions of curves are linked with the Burgers hierarchy, which can be formulated as a Hamiltonian system and also leads to the difference Burgers equation ([8]) via their discretization.

Spacelike surfaces in L4 with prescribed Gauss map and nonzero mean curvature.

Spacelike surfaces in L4 with prescribed Gauss map and nonzero mean curvature.

Reduction of cosmological data for the detection of time-varying dark energydensity

We present a method for reducing cosmological data to constraints on the amplitudes ofmodes of the dark energy density as a function of redshift. The modes are chosenso that (1) one of them has constant density and (2) the others are non-zeroonly if there is time-variation in the dark energy density and (3) the amplitudeerrors for the time-varying modes are uncorrelated with each other. We applyour method to various combinations of three-year WMAP data, baryon acousticoscillation data, the ‘Gold’ supernova data set, and the Supernova Legacy Survey dataset. We find no significant evidence for a time-varying dark energy density orfor non-zero mean curvature. Although by some measure the limits on four ofthe time-varying mode amplitudes are quite tight, they are consistent with theexpectation that the dark energy density does not vary on timescales shorter than aHubble time. Since we do not expect detectable time variation in these modes, ourresults should be viewed as a systematic error test which the data have passed.We discuss a procedure to identify modes with maximal signal-to-noise ratio.

Compact hypersurfaces in a unit sphere

We study curvature structures of compact hypersurfaces in the unit sphere Sn+1(1) with two distinct principal curvatures. First of all, we prove that the Riemannian product is the only compact hypersurface in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies where n(n − 1)r is the scalar curvature of hypersurfaces and c2 = (n − 2)/nr. This generalized the result of Cheng, where the scalar curvature is constant is assumed. Secondly, we prove that the Riemannian product is the only compact hypersurface with non-zero mean curvature in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies This gives a partial answer for the problem proposed by Cheng.

Spherical rigidities of submanifolds in Euclidean spaces

In this paper, we study n-dimensional complete immersed submanifolds in a Euclidean space En+p. We prove that if Mn is an n-dimensional compact connected immersed submanifold with nonzero mean curvature H in En+p and satisfies either: (1)S≤n2H2n-1, or (2)n2H2≤(n-1)Rn-2, then Mn is diffeomorphic to a standard n-sphere, where S and R denote the squared norm of the second fundamental form of Mn and the scalar curvature of Mn, respectively. On the other hand, in the case of constant mean curvature, we generalized results of Klotz and Osserman [11] to arbitrary dimensions and codimensions; that is, we proved that the totally umbilical sphere Sn(c), the totally geodesic Euclidean space En, and the generalized cylinder Sn-1(c)×E1 are only n-dimensional (n>2) complete connected submanifolds Mn with constant mean curvature H in En+p if S≤n2H2/(n-1) holds.

Morphological changes during the order-disorder transition in the two- and three-dimensional systems of scalar nonconserved order parameters.

The order-disorder transition is studied in a system of a scalar nonconserved order parameter. We use this well studied system to show that the application of the methods of topology and geometry reveals that our knowledge of the kinetic pathways by which the order-disorder transition proceeds is far from being complete. We show that in two-dimensional (2D) and 3D systems there are three dynamical regimes in the evolution of the system: early, intermediate, and late. In the intermediate regime two length scales govern the behavior of the system, whereas in the early and intermediate regime there is only one length scale. The size distribution of the domain area indicates the pathway by which the domains change their size. There are only two types of domains in a 2D system: circular and elongated with well defined characteristics (scaling of the area with the contour length) which in the late regime do not depend on time after rescaling by the average area and contour in the system. The elongated domains continuously change into circular domains reducing in this way the overall dissipation in the system. In order to reach a Lifshitz-Cahn-Allen (LCA) late stage regime the number of elongated domains must be strongly reduced. In the intermediate regime the number of elongated domains is large and simple LCA scaling does not hold. In a 3D symmetric system we always have a bicontinuous structure that evolves by cutting small connections. The late stage regime seems to be associated with the appearance of the preferred nonzero mean curvature. The early-intermediate regime crossover is associated with the saturation of the order parameter inside the domains, while the intermediate-late stage regime crossover is related to the global breaking of the +/- order parameter symmetry (marked by the appearance of the nonzero mean curvature but still zero average magnetization). The times for the occurrence of these crossovers do not depend on the size of the system.

Surfaces of Constant Mean Curvature in Euclidean 3-space Orthogonal to a Plane along its Boundary

We consider compact surfaces with constant nonzero mean curvature whose boundary is a convex planar Jordan curve. We prove that if such a surface is orthogonal to the plane of the boundary, then it is a hemisphere.