arXiv subjects
Publications and source records attributed to David L. Johnson.
A correspondence is established between a class of minimal immersed surfaces of $\mathbb{S}^3(2)$ and area-minimizing unit vector fields defined on the antipodally punctured unit sphere $\mathbb{S}^2\backslash\{N,S\}$. As a consequence, we establish a stability relation for Lawson cylinders in $\mathbb{S}^3(2)$.
The existence of orthogonal local coordinates is a generalization of the manifold being conformally flat. It is always possible to construct orthogonal coordinates on 2-manifolds, using geometric normal coordinates or isothermal coordinates. In 1984, Dennis DeTurck and Dean Yang [4] showed the existence of orthogonal coordinates on any Riemannian 3-manifold. Thus there are manifolds which have orthogonal coordinates, but are not conformally flat, since the Cotton tensor presents an obstruction to conformal flatness in dimension 3. They also showed that, for dimensions at least 4, there is apparently an obstruction to the existence of orthogonal coordinates, in that curvature components of the form R_{ijkl}, with all 4 indices distinct, will vanish if the directions correspond to orthogonal coordinates. Thus, in high dimensions, the existence of orthogonal coordinates implies a certain sparseness of the Riemannian curvature tensor. Recently, Paul Gauduchon and Andrei Moroianu showed [5] that there are no orthogonal coordinates on complex or quaternionic projective space except for trivial cases. The main results of this work are that no nontrivial self-dual Kähler 4-manifold (4 real dimensions) supports orthogonal local coordinates, and also no nontrivial Ricci-flat Kähler 4-manifold supports orthogonal coordinates. The first result uses the same technique developed by Gauduchon and Moroianu in the special case of complex projective 2-space with the Fubini-Study metric, but the second result uses purely algebraic methods.
If one could assume that local coordinates in a Riemannian manifold were orthogonal, then local expressions for differential operators, and curvature computations, would be simplified. It is always possible on 2-manifolds, using geometric normal coordinates or isothermal coordinates. In 1984, Dennis DeTurck and Dean Yang constructed smooth orthogonal coordinates on any Riemannian 3-manifold. In fact, they showed that, at any point in a 3-manifold, and for any orthonormal frame at that point, there is a set of local orthogonal coordinates so that the partial derivatives at that point are that frame. Recently, Paul Gauduchon and Andrei Moroianu showed, by contrast, that there are no orthogonal coordinates on $\mathbb{CP}^{n}$ or $\mathbb{HP}^{n}$. Their proof strongly uses the simplicity of the curvature tensor for these spaces. The main theorem of the present article is to show that the only 4 real-dimensional Kähler manifolds which admit real orthogonal coordinates are, up to a cover, a Riemannian product of 2 Riemann surfaces.
We use the Discrete Element Method (DEM) to understand the underlying attenuation mechanism in granular media, with special applicability to the measurements of the so-called effective mass developed earlier. We consider that the particles interact via Hertz-Mindlin elastic contact forces and that the damping is describable as a force proportional to the velocity difference of contacting grains. We determine the behavior of the complex-valued normal mode frequencies using 1) DEM, 2) direct diagonalization of the relevant matrix, and 3) a numerical search for the zeros of the relevant determinant. All three methods are in strong agreement with each other. The real and the imaginary parts of each normal mode frequency characterize the elastic and the dissipative properties, respectively, of the granular medium. We demonstrate that, as the interparticle damping, $ξ$, increases, the normal modes exhibit nearly circular trajectories in the complex frequency plane and that for a given value of $ξ$ they all lie on or near a circle of radius $R$ centered on the point $-iR$ in the complex plane, where $R\propto 1/ξ$. We show that each normal mode becomes critically damped at a value of the damping parameter $ξ\approx 1/ω_n^0$, where $ω_n^0$ is the (real-valued) frequency when there is no damping. The strong indication is that these conclusions carry over to the properties of real granular media whose dissipation is dominated by the relative motion of contacting grains. For example, compressional or shear waves in unconsolidated dry sediments can be expected to become overdamped beyond a critical frequency, depending upon the strength of the intergranular damping constant.
We develop the concept of frequency dependent effective mass, M(omega), of jammed granular materials which occupy a rigid cavity to a filling fraction of 48%, the remaining volume being air of normal room condition or controlled humidity. The dominant features of M(omega) provide signatures of the dissipation of acoustic modes, elasticity and aging effects in the granular medium. We perform humidity controlled experiments and interpret the data in terms of a continuum model and a "trap" model of thermally activated capillary bridges at the contact points. The results suggest that attenuation in the granular materials is influenced significantly by the kinetics of capillary condensation between the asperities at the contacts.
A zero-temperature critical point has been invoked to control the anomalous behavior of granular matter as it approaches jamming or mechanical arrest. Criticality manifests itself in an anomalous spectrum of low-frequency normal modes and scaling behavior near the jamming transition. The critical point may explain the peculiar mechanical properties of dissimilar systems such as glasses and granular materials. Here, we study the critical scenario via an experimental measurement of the normal modes frequencies of granular matter under stress from a pole decomposition analysis of the effective mass. We extract a complex-valued characteristic frequency which displays scaling $|ω^*(σ)|\simσ^{Ω'}$ with vanishing stress $σ$ for a variety of granular systems. The critical exponent is smaller than that predicted by mean-field theory opening new challenges to explain the exponent for frictional and dissipative granular matter. Our results shed light on the anomalous behavior of stress-dependent acoustics and attenuation in granular materials near the jamming transition.
An extrinsic representation of a Ricci flow on a differentiable n-manifold M is a family of submanifolds S(t), each smoothly embedded in R^{n+k}, evolving as a function of time t such that the metrics induced on the submanifolds S(t) by the ambient Euclidean metric yield the Ricci flow on M. When does such a representation exist? We formulate this question precisely and describe a new, comprehensive way of addressing it for surfaces of revolution in R^3. Of special interest is the Ricci flow on a toroidal surface of revolution, that is, a surface of revolution whose profile curve is an immersed curve which does not intersect the axis of revolution. In In this case, the extrinsic representation of the Ricci flow on a Riemannian cover of S is eternal. This flow can also be realized as a compact family of non-smooth, but isometric, embeddings of the torus into R^3.
For a compact 3-manifold $M$ which is a circle bundle over a compact Riemann surface $Σ$ with even Euler number $e(M)$, and with a Riemannian metric compatible with the bundle projection, there exists a compact minimal surface $S$ in $M$. $S$ is embedded and is a section of the restriction of the bundle to the complement of a finite number of points in $Σ$.
Let B be a fiber bundle with compact fiber F over a compact Riemannian n-manifold M. There is a natural Riemannian metric on the total space B consistent with the metric on M. With respect to that metric, the volume of a rectifiable section s:M--> B is the mass of the image s(M) as a rectifiable n-current in B. Theorem: For any homology class of sections of B, there is a mass-minimizing Cartesian current T representing that homology class which is the graph of a C^1 section on an open dense subset of M.
The nullity of a minimal submanifold $M\subset S^{n}$ is the dimension of the nullspace of the second variation of the area functional. That space contains as a subspace the effect of the group of rigid motions $SO(n+1)$ of the ambient space, modulo those motions which preserve $M$, whose dimension is the Killing nullity $kn(M)$ of $M$. In the case of 2-dimensional tori $M$ in $S^{3}$, there is an additional naturally-defined 2-dimensional subspace; the dimension of the sum of the action of the rigid motions and this space is the natural nullity $nnt(M)$. In this paper we will study minimal tori in $S^{3}$ with natural nullity less than 8. We construct minimal immersions of the plane $R^{2}$ in $S^{3}$ that contain all possible examples of tori with $nnt(M)<8$. We prove that the examples of Lawson and Hsiang with $kn(M)=5$ also have $nnt(M)=5$, and we prove that if the $nnt(M)\le6$ then the group of isometries of $M$ is not trivial.
Let $E$ be a principle bundle over a compact manifold $M$ with compact structural group $G$. For any $G$-invariant polynomial $P$, The transgressive forms $TP(ω)$ defined by Chern and Simons are shown to extend to forms $ΦP(ω)$ on associated bundles $B$ with fiber a quotient $F=G/H$ of the group. These forms satisfy a heterotic formula $$dΦP(ω)=P(Ω)-P(Ψ),$$ relating the characteristic form $P(Ω)$ to a fiber-curvature characteristic form. For certain natural bundles $B$, $P(Ψ)=0$, giving a true transgressive form on the associated bundle, which leads to the standard obstruction properties of characteristic classes as well as natural expressions for boundary terms.
Elastic and dissipative properties of granular assemblies under uniaxial compression are studied both experimentally and by numerical simulations. Following a novel compaction procedure at varying oscillatory pressures, the stress response to a step-strain reveals an exponential relaxation followed by a slow logarithmic decay. Simulations indicate that the latter arises from the coupling between damping and collective grain motion predominantly through sliding. We characterize an analogous "glass transition" for packed grains, below which the system shows aging in time-dependent sliding correlation functions.
In this article, we show that, for any compact 3-manifold, there is a $C^{1}$ volume-minimizing one-dimensional foliation. More generally, we show the existence of mass-minimizing rectifiable sections of sphere bundles without isolated "pole points" in the base manifold. This same analysis is used to show that the examples, due to Sharon Pedersen, of potentially volume-minimizing rectifiable sections (rectifiable foliations) of the unit tangent bundle to $S^{2n+1}$ are not, in fact, volume minimizing.
Experiments on isotropic compression of a granular assembly of spheres show that the shear and bulk moduli vary with the confining pressure faster than the 1/3 power law predicted by Hertz-Mindlin effective medium theories (EMT) of contact elasticity. Moreover, the ratio between the moduli is found to be larger than the prediction of the elastic theory by a constant value. The understanding of these discrepancies has been a longstanding question in the field of granular matter. Here we perform a test of the applicability of elasticity theory to granular materials. We perform sound propagation experiments, numerical simulations and theoretical studies to understand the elastic response of a deforming granular assembly of soft spheres under isotropic loading. Our results for the behavior of the elastic moduli of the system agree very well with experiments. We show that the elasticity partially describes the experimental and numerical results for a system under compressional loads. However, it drastically fails for systems under shear perturbations, particularly for packings without tangential forces and friction. Our work indicates that a correct treatment should include not only the purely elastic response but also collective relaxation mechanisms related to structural disorder and nonaffine motion of grains.
The volume of a k-dimensional foliation $\mathcal{F}$ in a Riemannian manifold $M^{n}$ is defined as the mass of image of the Gauss map, which is a map from M to the Grassmann bundle of k-planes in the tangent bundle. Generalizing a construction by Gluck and Ziller, "singular" foliations by 3-spheres are constructed on round spheres $S^{4n+3}$, as well as a singular foliation by 7-spheres on $S^{15}$, which minimize volume within their respective relative homology classes. These singular examples provide lower bounds for volumes of regular 3-dimensional foliations of $S^{4n+3}$ and regular 7-dimensional foliations of $S^{15}$ .
3D Computer simulations and experiments are employed to study random packings of compressible spherical grains under external confining stress. Of particular interest is the rigid ball limit, which we describe as a continuous transition in which the applied stress vanishes as (ϕ-ϕ_c)^β, where ϕis the (solid phase) volume density. This transition coincides with the onset of shear rigidity. The value of ϕ_c depends, for example, on whether the grains interact via only normal forces (giving rise to random close packings) or by a combination of normal and friction generated transverse forces (producing random loose packings). In both cases, near the transition, the system's response is controlled by localized force chains. As the stress increases, we characterize the system's evolution in terms of (1) the participation number, (2) the average force distribution, and (3) visualization techniques.
Experimentally it is known that the bulk modulus, K, and shear modulus, μ, of a granular assembly of elastic spheres increase with pressure, p, faster than the p^1/3 law predicted by effective medium theory (EMT) based on Hertz-Mindlin contact forces. To understand the origin of these discrepancies, we perform numerical simulations of granular aggregates under compression. We show that EMT can describe the moduli pressure dependence if one includes the increasing number of grain-grain contacts with p. Most important, the affine assumption (which underlies EMT), is found to be valid for K(p) but breakdown seriously for μ(p). This explains why the experimental and numerical values of μ(p) are much smaller than the EMT predictions.