Frederick Justin Almgren, 1933–1997

Suppose curves are moving by curvature in a plane, but one embeds the plane in R³ and looks at the plane from an angle. Then circles shrinking to a round point would appear to be ellipses shrinking to an “elliptical point,” and the surface energy would appear to be anisotropic as would the mobility. The result of this paper is that if one uses the apparent surface energy and the apparent mobility, then the motion by weighted curvature with mobility in the apparent plane is the same as motion by curvature in the original plane but then viewed from the angle. This result applies not only to the isotropic case but to arbitrary surface energy functions and mobilities in the plane, to surfaces in 3-space, and (in the case that the surface energy function is twice differentiable) to the case of motion viewed through distorted lenses (i.e., diffeomorphisms) as well. This result is to be contrasted with an earlier result [4], which states that for area-preserving affine transformations of the plane where the energy and mobility are not also transformed, motion by curvature to the power 1/3 (rather than 1) is invariant.     

An estimate of the attraction domain with a specified exponential stability index in a wheeled robot control problem

The problem of synthesizing a law for the control of the plane motion of a wheeled robot is investigated. The rear wheels are the drive wheels and the front wheels are responsible for the turning of the platform. The aim of the control is to steer a target point to a specified trajectory and to stabilize the motion along it. The trajectory is assumed to be specified by a smooth curve. The actual curvature of the trajectory of the target point, which is related to the angle of rotation of the front wheels by a simple algebraic relation, is considered as the control. The control is subjected to bilateral constraints by virtue of the fact that the angle of rotation of the front wheels is limited. The attraction domain in the distance to trajectory - orientation space, is investigated for the proposed control law. Arrival at a trajectory with a specified exponential stability index is guaranteed in the case of initial conditions belonging to the given domain. An estimate of the attraction domain in the form of an ellipse is given.     

Nonlinear Oscillations of a Spring Pendulum at the 1 : 1 : 2 Resonance: Theory,Experiment, and Physical Analogies

Nonlinear spatial oscillations of a material point on a weightless elastic suspension are considered. The frequency of vertical oscillations is assumed to be equal to the doubled swinging frequency (the 1 : 1 : 2 resonance). In this case, vertical oscillations are unstable, which leads to the transfer of the energy of vertical oscillations to the swinging energy of the pendulum. Vertical oscillations of the material point cease, and, after a certain period of time, the pendulum starts swinging in a vertical plane. This swinging is also unstable, which leads to the back transfer of energy to the vertical oscillation mode, and again vertical oscillations occur. However, after the second transfer of the energy of vertical oscillations to the pendulum swinging energy, the apparent plane of swinging is rotated through a certain angle. These phenomena are described analytically: the period of energy transfer, the time variations of the amplitudes of both modes, and the change of the angle of the apparent plane of oscillations are determined. The analytic dependence of the semiaxes of the ellipse and the angle of precession on time agrees with high degree of accuracy with numerical calculations and is confirmed experimentally. In addition, the problem of forced oscillations of a spring pendulum in the presence of friction is considered, for which an asymptotic solution is constructed by the averaging method. An analogy is established between the nonlinear problems for free and forced oscillations of a pendulum and for deformation oscillations of a gas bubble. The transfer of the energy of radial oscillations to a resonance deformation mode leads to an anomalous increase in its amplitude and, as a consequence, to the break-up of a bubble.     

Motion by curvature of planar curves with end points moving freely on a line

This paper deals with the motion by curvature of planar curves having end points moving freely along a line with fixed contact angles to this line. We first prove the existence and uniqueness of self-similar shrinking solution. Then we show that the curve shrinks to a point in a self-similar manner, if initially the curve is a graph.     

On affine Sawada–Kotera equation

It is shown that motion of plane curves in affine geometry induces naturally the Sawada–Kotera hierarchy. The affine Sawada–Kotera equation is obtained in view of the equivalence of equations for the curvature and graph of plane curves when the curvature satisfies the Sawada–Kotera equation. The affine Sawada–Kotera equation can be viewed as an affine version of the WKI equation since they have similarity properties, such as they have loop-solitons, they are solved by the AKNS-scheme and are obtained by choosing the normal velocity to be the derivative of the curvature with respect to the arc-length. Its symmetry reductions to ordinary differential equations corresponding to an one-dimensional optimal system of its Lie symmetry algebras are discussed.     

Optimum velocity for a person moving under rain

The total rain received by a moving body has previously been modelled by defining a wetness function. Several cases such as one-dimensional motion of an inclined plane, two-dimensional motion of an inclined plane, motion with time-varying velocity, inclined rain for an inclined plane, rain on a cylindrical surface and three-dimensional motion of a convex body were treated in detail. One of the major conclusions was that for a fixed distance, assuming vertical rain, the body should travel as fast as possible since the total wetness decreases with increasing velocity. The wetness function was shown to decrease asymptotically to a constant value as the velocity increases and in the high-speed range, increasing the velocity does not decrease the wetness substantially. One might think that the excess amount of energy required for a higher speed does not compensate for the small fraction of decrease in wetness. In this work, criteria are developed for a critical speed (optimum speed), for which the wetness is small enough for a reasonable energy consumption. Three cases are investigated: (1) vertical rain; (2) rain inclined towards the body; (3) rain inclined away from the body. In the first two cases, there is no absolute minimum for the wetness function and the optimum velocity is determined by special criteria. The third case is somewhat different, however, and if the inclination angle is higher than a critical value, an absolute minimum for wetness is obtained and the optimum velocity for this case is defined to be the velocity corresponding to this absolute minimum. Therefore the definition of optimum velocity is qualitatively different from the first two cases.     

Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature

Summary. Motion by (weighted) mean curvature is a geometric evolution law for surfaces, representing steepest descent with respect to (an)isotropic surface energy. It has been proposed that this motion could be computed by solving the analogous evolution law using a ``crystalline' approximation to the surface energy. We present the first convergence analysis for a numerical scheme of this type. Our treatment is restricted to one dimensional surfaces (curves in the plane) which are graphs. In this context, the scheme amounts to a new algorithm for solving quasilinear parabolic equations in one space dimension. Received January 28, 1993     

The conjugate heat equation and Ancient solutions of the Ricci flow

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of Type I κ-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman?s previous result on backward limits of κ-solutions in dimension 3, in which case the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23], where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.     

The capillary contact angle,II: The inclined plane

In an earlier paper an hypothesis on the nature of the forces resisting motion of a liquid drop on a support surface II was introduced and tested in the (rotationally symmetric) case for which ll is a horizontal plane. In the present work II is chosen as an inclined plane, so that rotational symmetry fails. The hypothesis leads to a formal series development; in a restricted case the result is verified by a perturbation analysis, which yields the identical series. The series predictions are compared with results of computer calculations. A consequence of the analysis is an estimate for a Bohd number B₀, such that a further increase in B would result in the fluid penetrating the support surface; thus an absolute upper bound B₀, for stability of the envisaged configuration is obtained.     

Boundary value problems for rotationally symmetric mean curvature flows

The motion of surfaces by their mean curvature has been studied by several authors from different points of view. K. A. Brake studied this problem from the geometric measure theory point of view, the parametric problem was studied by G. Huisken [5]. Nonparametric mean curavture flow with boundary conditions was studied in [6] and [7]. Rotationally symmetric mean curvature flows have been treated by G. Dziuk, B. Kawohl [3], but also by S. Altschuler, S. B. Angenent and Y. Giga [2]. In this paper we consider the case in which the initial surface has rotational symmetry and we shall generalize the results in [3] in the sense that we shall give more general boundary conditions which enforce the formation of a singularity in finite time. The proofs rely entirely on parabolic maximum principles. Received: 6 September 2006     

Positivity of curvature and convexity of faces

Two-dimensional polyhedra homeomorphic to closed two-dimensional surfaces are considered in the three-dimensional Euclidean space. While studying the structure of an arbitrary face of a polyhedron, an interesting particular case is revealed when the magnitude of only one plane angle determines the sign of the curvature of the polyhedron at the vertex of this angle. Due to this observation, the following main theorem of the paper is obtained: If a two-dimensional polyhedron in the three-dimensional Euclidean space is isometric to the surface of a closed convex three-dimensional polyhedron, then all faces of the polyhedron are convex polygons.     

Invariants and Bonnet-type theorem for surfaces in ℝ4

In the tangent plane at any point of a surface in the four-dimensional Euclidean space we consider an invariant linear map ofWeingarten-type and find a geometrically determined moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of surfaces in the four-dimensional Euclidean space, determined by conditions on their invariants, can be interpreted in terms of the properties of two geometric figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We construct a family of surfaces with flat normal connection.     

Lotka–Volterra system and KCC theory: Differential geometric structure of competitions and predations

We consider the differential geometric structure of competitions and predations in the sense of the Lotka–Volterra system based on KCC theory. For this, we visualise the relationship between the Jacobi stability and the linear stability as a single diagram. We find the following. (I) Ecological interactions such as competition and predation can be described by the deviation curvature. In this case, the sign of the deviation curvature depends on the type of interaction, which reflects the equilibrium point type. (II) The geometric quantities in KCC theory can be expressed in terms of the mean and Gaussian curvatures of the potential surface. In this particular case, the deviation curvature can be interpreted as the Willmore energy density of the potential surface. (III) When the equations of the system have nonsymmetric structure for the species (e.g. a predation system), each species also has nonsymmetric geometric structure in the nonequilibrium region, but symmetric structure around the equilibrium point. These findings suggest that KCC theory is useful to establish the geometrisation of ecological interactions.     

The motion in a perfect fluid of a body containing a moving point mass

The motion of a system (a rigid body, symmetrical about three mutually perpendicular planes, plus a point mass situated inside the body) in an unbounded volume of a perfect fluid, which executes vortex-free motion and is at rest at infinity, is considered. The motion of the body occurs due to displacement of the point mass with respect to the body. Two cases are investigated: (a) there are no external forces, and (b) the system moves in a uniform gravity field. An analytical investigation of the dynamic equations under conditions when the point performs a specified plane periodic motion inside the body showed that in case (a) the system can be displaced as far as desired from the initial position. In case (b) it is proved that, due to the permanent addition of energy of the corresponding relative motion of the point, the body may float upwards. On the other hand, if the velocity of relative motion of the point is limited, the body will sink. The results of numerical calculations, when the point mass performs random walks along the sides of a plane square grid rigidly connected with the body, are presented.     

Existence of non-trivial harmonic functions on Cartan–Hadamard manifolds of unbounded curvature

The Liouville property of a complete Riemannian manifold M (i.e., the question whether there exist non-trivial bounded harmonic functions on M) attracted a lot of attention. For Cartan–Hadamard manifolds the role of lower curvature bounds is still an open problem. We discuss examples of Cartan–Hadamard manifolds of unbounded curvature where the limiting angle of Brownian motion degenerates to a single point on the sphere at infinity, but where nevertheless the space of bounded harmonic functions is as rich as in the non-degenerate case. To see the full boundary the point at infinity has to be blown up in a non-trivial way. Such examples indicate that the situation concerning the famous conjecture of Greene and Wu about existence of non-trivial bounded harmonic functions on Cartan–Hadamard manifolds is much more complicated than one might have expected.      

Capillary surfaces of constant mean curvature in a right solid cylinder

In this paper we investigate constant mean curvature surfaces with nonempty boundary in Euclidean space that meet a right cylinder at a constant angle along the boundary. If the surface lies inside of the solid cylinder, we obtain some results of symmetry by using the Alexandrov reflection method. When the mean curvature is zero, we give sufficient conditions to conclude that the surface is part of a plane or a catenoid.     

Gaussian Curvature and Gyroscopes

We relate Gaussian curvature to the gyroscopic force, thus giving a mechanical interpretation of the former and a geometrical interpretation of the latter. We do so by considering the motion of a spinning disk constrained to be tangent to a curved surface. It is shown that the spin gives rise to a force on the disk that is equal to the magnetic force on a point charge moving in a magnetic field normal to the surface, of magnitude equal to the Gaussian curvature, and of charge equal to the disk's axial spin. In a special case, this demonstrates that the precession of Lagrange's top is due to the curvature of a sphere determined by the parameters of the top. © 2017 Wiley Periodicals, Inc.     

Evolution of a regular precession of a solid body carrying a visco-elastic disk

The motion of inertia is studied of a system consisting of an axisymmetric solid body with fixed point and a homogeneous visco-elastic disk lying in the equatorial plane of the ellipsoid of inertia of the solid body (the center of disk coincides with the fixed point). In the case of a solid disk immobilized relative to the solid body the system accomplishes a regular precession (the case of Euler motion of a symmetric solid body with a fixed point /1/). The deformation of the disk is taking place in the plane of the disk, and is accompanied by energy dissipation is the cause of the regular precession finishing by steady rotation about the vector of the moment of momentum of the system /2/.     

有关法曲率的若干性质

以[2]中经典微分几何问题为切入点,运用复数与三角工具广泛深入地探讨了“过曲面上一点有n条切线,若相邻两条切线的交角为2nπ,曲面法线与切线所定平面截得曲线的曲率半径为ρ1,ρ2,ρ3,…,ρn时,∑ni=11ρim,∑ni=1ρi,∏ni=1ρi的结果”,得到了法曲率与相关的三个有趣定理.