Surfaces of Bounded Mean Curvature In Riemannian Manifolds

Consider a sequence of closed, orientable surfaces of fixed genus g in a Riemannian manifold M with uniform upper bounds on the norm of mean curvature and area. We show that on passing to a subsequence, we can choose parametrisations of the surfaces by inclusion maps from a fixed surface of the same genus so that the distance functions corresponding to the pullback metrics converge to a pseudo-metric and the inclusion maps converge to a Lipschitz map. We show further that the limiting pseudo-metric has fractal dimension two. As a corollary, we obtain a purely geometric result. Namely, we show that bounds on the mean curvature, area and genus of a surface F subset of M, together with bounds on the geometry of M, give an upper bound on the diameter of F. Our proof is modelled on Gromov's compactness theorem for J-holomorphic curves.

Effect of longitudinal surface curvature on boundary layers

We consider here the higher order effect of moderate longitudinal surface curvature on steady, two-dimensional, incompressible laminar boundary layers. The basic partial differential equations for the problem, derived by the method of matched asymptotic expansions, are found to possess similarity solutions for a family of surface curvatures and pressure gradients. The similarity equations obtained by this anaylsis have been solved numerically on a computer, and show a definite decrease in skin friction when the surface has convex curvature in all cases including zero pressure gradient. Typical velocity profiles and some relevant boundary-layer characteristics are tabulated, and a critical comparison with previous work is given.

The curvature of the liquid-vapor reduced pressure curve and its relation with the critical region

For most fluids, there exist a maximum and a minimum in the curvature of the reduced vapor pressure curve, p(r) = p(r)(T-r) (with p(r) = p/p(c) and T-r = T/T-c, p(c) and T-c being the pressure and temperature at the critical point). By analyzing National Institute of Standards and Technology (NIST) data on the liquid-vapor coexistence curve for 105 fluids, we find that the maximum occurs in the reduced temperature range 0.5 <= T-r <= 0.8 while the minimum occurs in the reduced temperature range 0.980 <= T-r <= 0.995. Vapor pressure equations for which d(2)p(r)/dT(r)(2) diverges at the critical point present a minimum in their curvature. Therefore, the point of minimum curvature can be used as a marker for the critical region. By using the well-known Ambrose-Walton (AW) vapor pressure equation we obtain the reduced temperatures of the maximum and minimum curvature in terms of the Pitzer acentric factor. The AW predictions are checked against those obtained from NIST data. (C) 2013 Elsevier Ltd. All rights reserved.

Infinitely divisible metrics and curvature inequalities for operators in the Cowen-Douglas class

The curvature (T)(w) of a contraction T in the Cowen-Douglas class B-1() is bounded above by the curvature (S*)(w) of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this paper, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E-T corresponding to the operator T in the Cowen-Douglas class B-1() which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the class B-1() for a bounded domain in C-m.

Curvature based mobility analysis and form closure of smooth planar curves with multiple contacts

This paper presents a simple second-order, curvature based mobility analysis of planar curves in contact. The underlying theory deals with penetration and separation of curves with multiple contacts, based on relative configuration of osculating circles at points of contact for a second-order rotation about each point of the plane. Geometric and analytical treatment of mobility analysis is presented for generic as well as special contact geometries. For objects with a single contact, partitioning of the plane into four types of mobility regions has been shown. Using point based composition operations based on dual-number matrices, analysis has been extended to computationally handle multiple contacts scenario. A novel color coded directed line has been proposed to capture the contact scenario. Multiple contacts mobility is obtained through intersection of the mobility half-spaces. It is derived that mobility region comprises a pair of unbounded or a single bounded convex polygon. The theory has been used for analysis and synthesis of form closure configurations, revolute and prismatic kinematic pairs. (C) 2013 Elsevier Ltd. All rights reserved.

Enhanced Gas Adsorption on Graphitic Substrates via Defects and Local Curvature: A Density Functional Theory Study

Using van-der-Waals-corrected density functional theory calculations, we explore the possibility of engineering the local structure and morphology of high-surface-area graphene-derived materials to improve the uptake of methane and carbon dioxide for gas storage and sensing. We test the sensitivity of the gas adsorption energy to the introduction of native point defects, curvature, and the application of strain. The binding energy at topological point defect sites is inversely correlated with the number of missing carbon atoms, causing Stone-Wales defects to show the largest enhancement with respect to pristine graphene (similar to 20%). Improvements of similar magnitude are observed at concavely curved surfaces in buckled graphene sheets under compressive strain, whereas tensile strain tends to weaken gas binding. Trends for CO2 and CH4 are, similar, although CO2 binding is generally stronger by similar to 4 to 5 kJ mol(-1). However, the differential between the adsorption of CO2 and CH4 is much higher on folded graphene sheets and at concave curvatures; this could possibly be leveraged for CH4/CO2 flow separation and gasselective sensors.

On the stability of the L-p-norm of the Riemannian curvature tensor

We consider the Riemannian functional defined on the space of Riemannian metrics with unit volume on a closed smooth manifold M where R(g) and dv (g) denote the corresponding Riemannian curvature tensor and volume form and p a (0, a). First we prove that the Riemannian metrics with non-zero constant sectional curvature are strictly stable for for certain values of p. Then we conclude that they are strict local minimizers for for those values of p. Finally generalizing this result we prove that product of space forms of same type and dimension are strict local minimizer for for certain values of p.

SOME UNSTABLE CRITICAL METRICS FOR THE L-n/2-NORM OF THE CURVATURE TENSOR

We consider the Riemannian functional defined on the space of Riemannian metrics with unit volume on a closed smooth manifold M given by R-n/2(g) := integral(M) vertical bar R(g)vertical bar(n//2) dv(g) where R(g), dv(g) denote the Riemannian curvature and volume form corresponding to g. We show that there are locally symmetric spaces which are unstable critical points for this functional.

Curvature-constrained trajectory generation for waypoint following for miniature air vehicle

This paper addresses trajectory generation problem of a fixed-wing miniature air vehicle, constrained by bounded turn rate, to follow a given sequence of waypoints. An extremal path, named as g-trajectory, that transitions between two consecutive waypoint segments (obtained by joining two waypoints in sequence) in a time-optimal fashion is obtained. This algorithm is also used to track the maximum portion of waypoint segments with the desired shortest distance between the trajectory and the associated waypoint. Subsequently, the proposed trajectory is compared with the existing transition trajectory in the literature to show better performance in several aspects. Another optimal path, named as loop trajectory, is developed for the purpose of tracking the waypoints as well as the entire waypoint segments. This paper also proposes algorithms to generate trajectories in the presence of steady wind to meet the same objective as that of no-wind case. Due to low computational burden and simplicity in the design procedure, these trajectory generation approaches are implementable in real time for miniature air vehicles.

Trace Formulae for Curvature of Jet Bundles over Planar Domains

For a domain Omega in C and an operator T in B-n(Omega), Cowen and Douglas construct a Hermitian holomorphic vector bundle E-T over Omega corresponding to T. The Hermitian holomorphic vector bundle E-T is obtained as a pull-back of the tautological bundle S(n, H) defined over by Gr(n, H) a nondegenerate holomorphic map z bar right arrow ker(T - z), z is an element of Omega. To find the answer to the converse, Cowen and Douglas studied the jet bundle in their foundational paper. The computations in this paper for the curvature of the jet bundle are rather intricate. They have given a set of invariants to determine if two rank n Hermitian holomorphic vector bundle are equivalent. These invariants are complicated and not easy to compute. It is natural to expect that the equivalence of Hermitian holomorphic jet bundles should be easier to characterize. In fact, in the case of the Hermitian holomorphic jet bundle J(k)(L-f), we have shown that the curvature of the line bundle L-f completely determines the class of J(k)(L-f). In case of rank Hermitian holomorphic vector bundle E-f, We have calculated the curvature of jet bundle J(k)(E-f) and also obtained a trace formula for jet bundle J(k)(E-f).

Influence of curvature on the device physics of thin film transistors on flexible substrates

Thin film transistors (TFTs) on elastomers promise flexible electronics with stretching and bending. Recently, there have been several experimental studies reporting the behavior of TFTs under bending and buckling. In the presence of stress, the insulator capacitance is influenced due to two reasons. The first is the variation in insulator thickness depending on the Poisson ratio and strain. The second is the geometric influence of the curvature of the insulator-semiconductor interface during bending or buckling. This paper models the role of curvature on TFT performance and brings to light an elegant result wherein the TFT characteristics is dependent on the area under the capacitance-distance curve. The paper compares models with simulations and explains several experimental findings reported in literature. (C) 2014 AIP Publishing LLC.

Role of interface curvature on stress distribution under indentation for ZrN/Zr multilayer coating

Contact damage in curved interface nano-layeredmetal/nitride (150 (ZrN)/10 (Zr) nm) multilayer is investigated in order to understand the role of interface morphology on contact damage under indentation. A finite element method (FEM) model was formulated with different wavelengths of 1000 nm, 500 nm, 250 nm and common height of 50 nm, which gives insight on the effect of different curvature on stress field generated under indentation. Elastic-plastic properties were assigned to the metal layer and substrate while the nitride layer was assigned perfectly elastic properties. Curved interface multilayers show delamination along the metal/nitride interface and vertical cracks emanating from the ends of the delamination. FEM revealed the presence of tensile stress normal to the interface even under the contact, along with tensile radial stresses, both present at the valley part of the curve, which leads to vertical cracks associated with interfacial delamination. Stress enhancement was seen to be relatively insensitive to curvature. (C) 2014 Elsevier B.V. All rights reserved.

On the curvature ODE associated to the Ricci flow

In the vector space of algebraic curvature operators we study the reaction ODE which is associated to the evolution equation of the Riemann curvature operator along the Ricci flow. More precisely, we give a partial classification of the zeros of this ODE up to suitable normalization and analyze the stability of a special class of zeros of the same. In particular, we show that the ODE is unstable near the curvature operators of the Riemannian product spaces where is an Einstein (locally) symmetric space of compact type and not a spherical space form when .

The tarani mutation alters surface curvature in Arabidopsis leaves by perturbing the patterns of surface expansion and cell division

The leaf surface usually stays flat, maintained by coordinated growth. Growth perturbation can introduce overall surface curvature, which can be negative, giving a saddle-shaped leaf, or positive, giving a cup-like leaf. Little is known about the molecular mechanisms that underlie leaf flatness, primarily because only a few mutants with altered surface curvature have been isolated and studied. Characterization of mutants of the CINCINNATA-like TCP genes in Antirrhinum and Arabidopsis have revealed that their products help maintain flatness by balancing the pattern of cell proliferation and surface expansion between the margin and the central zone during leaf morphogenesis. On the other hand, deletion of two homologous PEAPOD genes causes cup-shaped leaves in Arabidopsis due to excess division of dispersed meristemoid cells. Here, we report the isolation and characterization of an Arabidopsis mutant, tarani (tni), with enlarged, cup-shaped leaves. Morphometric analyses showed that the positive curvature of the tni leaf is linked to excess growth at the centre compared to the margin. By monitoring the dynamic pattern of CYCLIN D3;2 expression, we show that the shape of the primary arrest front is strongly convex in growing tni leaves, leading to excess mitotic expansion synchronized with excess cell proliferation at the centre. Reduction of cell proliferation and of endogenous gibberellic acid levels rescued the tni phenotype. Genetic interactions demonstrated that TNI maintains leaf flatness independent of TCPs and PEAPODs.