Diffraction of reaction-diffusion waves: the conformal-mapped eikonal equation

The interaction of reaction-diffusion (RD) waves with obstacles is considered. The conformal map transformation of the eikonal equation is used for investigating the behavior of waves in the presence of movable boundaries. It is shown that using the conformal map, the complicated boundary conditions become simple Neumann boundary conditions and can be easily dealt with numerically. The process is applied to diffraction of RD waves by two disks in two dimensions. The approach is extended to three-dimensions and the obstacles considered are 2 two-tori. A stable stationary solution, in the form of an unduloid, trapped between 2 two-tori, is obtained. It is shown that, if the obstacles are located a distance apart, the wave moves away from its stationary position giving rise to regular and irregular motions, depending on the choice of initial solutions.     

Classification of conformal minimal immersions of constant curvature from S^2 to { HP}^2

In this paper, we study geometry of conformal minimal two-spheres immersed in quaternionic projective spaces. We firstly use Bahy-El-Dien and Wood’s results to obtain some characterizations of the harmonic sequences generated by conformal minimal immersions from \(S^2\) to the quaternionic projective space \({ HP}^2\) . Then we give a classification theorem of linearly full totally unramified conformal minimal immersions of constant curvature from \(S^2\) to the quaternionic projective space \({ HP}^2\) .     

Flat metrics,cubic differentials and limits of projective holonomies

F. Labourie and the author independently have shown that a convex real projective structure on an oriented closed surface S of genus at least two is equivalent to a pair of a conformal structure plus a holomorphic cubic differential. Along certain paths, we find the limiting holonomy of convex real projective structures on a surface S corresponding corresponding to a given fixed conformal structure S and a holomorphic cubic differential λ U ₀ as . We explicitly give part of the data needed to identify the boundary point in Inkang Kim’s compactification of the deformation space of convex real projective structures. The proof follows similar analysis to that studied by Mike Wolf is his application of harmonic map theory to reproduce Thurston’s boundary of Teichmüller space.      

Harmonic conjugation in harmonic matroids

We study a generalization of the concept of harmonic conjugation from projective geometry and full algebraic matroids to a larger class of matroids called harmonic matroids. We use harmonic conjugation to construct a projective plane of prime order in harmonic matroids without using the axioms of projective geometry. As a particular case we have a combinatorial construction of a projective plane of prime order in full algebraic matroids.     

λ-Automorphisms of a Riemannian foliation

We study geometric properties of -automorphisms of a Riemannian foliationF which is not harmonic. This notion was first introduced in [KTT] for the case whereF is harmonic. Transversal Killing, affine, conformal, projective fields are all examples of -automorphisms. We derive several general identities for a -automorphism. In particular, we extend the results on the transversal conformal and Killing fields obtained in [PrY], [NY1,2]. Furthermore, we analyse the geometric meaning of the condition appearing in our results.The present studies were supported (in part) by the Basic Science Research Institute Program, Ministry of Education, 1994, Project No. BSRI-94-1404     

Harmonic mappings with hereditary starlikeness

We study a hereditary starlikeness property for planar harmonic mappings on a disk and on an annulus. While such a property is a common trait of conformal mappings, it may be absent in harmonic mappings. It turns out that a sufficient condition for a harmonic mapping f to possess this hereditary property is to have a harmonic argument — a striking feature of conformal mappings that does not extend to all harmonic mappings.     

Harmonic maps in unfashionable geometries

 We describe some general constructions on a real smooth projective 4-quadric which provide analogues of the Willmore functional and conformal Gauss map in Lie sphere and projective differential geometry. Extrema of these functionals are characterized by harmonicity of this Gauss map. Received: 3 August 2001     

Integrable Equations Arising from Motions of Plane Curves. II

Integrable equations satisfied by the curvature of plane curves or curves on the real line under inextensible motions in some Klein geometries are identified. These include the Euclidean, similarity, and projective geometries on the real line, and restricted conformal, conformal, and projective geometries in the plane. Together with Chou and Qu [Physica D 162 (2002), 9–33], we determine inextensible motions and their associated integrable equations in all Klein geometries in the plane. The relations between several pairs of these geometries provide a natural geometric explanation of the existence of transformations of Miura and Cole-Hopf type.     

Projective complex Finsler metrics

In this paper we obtain the conditions under which two complex Finsler metrics are projective, i.e. have the same geodesics as point sets. Two important classes of such metrics are considered: conformal projective and weakly projective complex Finsler spaces. For each of them we study the transformations of the canonical connection. We pay attention to local projectivity in a pure Hermitian or Kähler space.     

Minimal 2-spheres in a complex projective space

In this paper conformal minimal 2-spheres immersed in a complex projective space are studied by applying Lie theory and moving frames. We give differential equations of Kähler angle and square length of the second fundamental form. By applying these differential equations we give characteristics of conformal minimal 2-spheres of constant Kähler angle and obtain pinching theorems for curvature. We also discuss conformal minimal 2-spheres of constant normal curvature and prove that there does not exist any linearly full minimal 2-sphere immersed in a complex projective space CPn (n>2) with non-positive constant normal curvature. We also prove that a linearly full minimal 2-sphere immersed in a complex projective space CPn (n>2) with constant normal curvature and constant Kähler angle is of constant curvature.     

Structure,classification, and conformal symmetry of elementary particles over non-archimedean space-time

It is well known that at distances shorter than Planck length, no length measurements are possible. The Volovich hypothesis asserts that at sub-Planckian distances and times, spacetime itself has a non-Archimedean geometry. We discuss the structure of elementary particles, their classification, and their conformal symmetry under this hypothesis. Specifically, we investigate the projective representations of the p-adic Poincaré and Galilean groups, using a new variant of the Mackey machine for projective unitary representations of semidirect products of locally compact and second countable (lcsc) groups. We construct the conformal spacetime over p-adic fields and discuss the imbedding of the p-adic Poincaré group into the p-adic conformal group. Finally, we show that the massive and the so called eventually masssive particles of the Poincaré group do not have conformal symmetry. The whole picture bears a close resemblance to what happens over the field of real numbers, but with some significant variations.     

The double bubble problem on the flat two-torus

We characterize the perimeter-minimizing double bubbles on all flat two-tori and, as corollaries, on the flat infinite cylinder and the flat infinite strip with free boundary. Specifically, we show that there are five distinct types of minimizers on flat two-tori, depending on the areas to be enclosed.

     

Note on equivariant harmonic maps in complex projective spaces

We give a simple criterion for equivariant harmonic maps into complex projective spaces CP ⁿ . As an application of the criterion, we give examples of equivariant harmonic cylinders. We also give examples of non-equivariant harmonic cylinders as perturbations of equivariant harmonic cylinders.     

Jordan Homomorphisms and Harmonic Mappings

 We show that each Jordan homomorphism R → R′ of rings gives rise to a harmonic mapping of one connected component of the projective line over R into the projective line over R′. If there is more than one connected component then this mapping can be extended in various ways to a harmonic mapping which is defined on the entire projective line over R. Received December 7, 2001; in revised form April 28, 2002 Published online January 7, 2003     

Second-order differential invariants of the rotation group O(n) and of its extensions: E(n), P(1, n), G(1, n)

Functional bases of second-order differential invariants of the Euclid, Poincaré, Galilei, conformal, and projective algebras are constructed. The results obtained allow us to describe new classes of nonlinear many-dimensional invariant equations.     

Envelopes and osculates of Willmore surfaces

We view conformal surfaces in the 4-sphere as quaternionic holomorphiccurves in quaternionic projective space. By constructing envelopingand osculating curves, we obtain new holomorphic curves in quaternionicprojective space and thus new conformal surfaces. Applying theseconstructions to Willmore surfaces, we show that the osculatingand enveloping curves of Willmore spheres remain Willmore.     

GJMS operators and Q-curvature for conformal Codazzi structures

The conformal Codazzi structure is an intrinsic geometric structure on strictly convex hypersurfaces in a locally flat projective manifold. We construct the GJMS operators and the Q-curvature for conformal Codazzi structures by using the ambient metric. We relate the total Q-curvature to the logarithmic coefficient in the volume expansion of the Blaschke metric, and derive the first and second variation formulas for a deformation of strictly convex domains.     

Projective vector fields on Finsler manifolds

In this paper, we give the equation that characterizes projective vector fields on a Finsler manifold by the local coordinate. Moreover, we obtain a feature of the projective fields on the compact Finsler manifold with non-positive flag curvature and the non-existence of projective vector fields on the compact Finsler manifold with negative flag curvature. Furthermore, we deduce some expectable, but non-trivial relationships between geometric vector fields such as projective, affine, conformal, homothetic and Killing vector fields on a Finsler manifold.     

Conformal Riemannian Maps between Riemannian Manifolds,Their Harmonicity and Decomposition Theorems

Riemannian maps were introduced by Fischer (Contemp. Math. 132:331–366, 1992) as a generalization isometric immersions and Riemannian submersions. He showed that such maps could be used to solve the generalized eikonal equation and to build a quantum model. On the other hand, horizontally conformal maps were defined by Fuglede (Ann. Inst. Fourier (Grenoble) 28:107–144, 1978) and Ishihara (J. Math. Kyoto Univ. 19:215–229, 1979) and these maps are useful for characterization of harmonic morphisms. Horizontally conformal maps (conformal maps) have their applications in medical imaging (brain imaging)and computer graphics. In this paper, as a generalization of Riemannian maps and horizontally conformal submersions, we introduce conformal Riemannian maps, present examples and characterizations. We show that an application of conformal Riemannian maps can be made in weakening the horizontal conformal version of Hermann’s theorem obtained by Okrut (Math. Notes 66(1):94–104, 1999). We also give a geometric characterization of harmonic conformal Riemannian maps and obtain decomposition theorems by using the existence of conformal Riemannian maps.