Research

  • (with W. Klingenberg) On area-stationary surfaces in certain neutral Kaehler 4-manifolds, Beitraege Algebra Geom. 49 (2008) 481-490. Preprint
    We study surfaces in TN that are area-stationary with respect to a neutral Kaehler metric constructed on TN from a Riemannian metric g on N. When (N,g) is the round 2-sphere, TN can be identified with the space of oriented affine lines in R, and we exhibit a two parameter family of area-stationary tori that are neither holomorphic nor Lagrangian.

  • (with W. Klingenberg) A neutral Kaehler surface with applications in geometric optics, in Recent developments in pseudo-Riemannian Geometry, European Mathematical Society Publishing House, Zurich (2008) 149-178. Preprint
    The space L of oriented lines, or rays, in Euclidean 3-space is a 4-dimensional space with an abundance of natural geometric structure. In particular, it boasts a neutral Kaehler metric which is closely related to the Euclidean metric on R. In this paper we explore the relationship between the focal set of a line congruence (or 2-parameter family of oriented lines in R) and the geometry induced on the associated surface in L. The physical context of such sets is geometric optics in a homogeneous isotropic medium, and so, to illustrate the method, we compute the focal set of the k-th reflection of a point source off the inside of a cylinder. The focal sets, which we explicitly parameterize, exhibit unexpected symmetries, and are found to fit well with observable phenomena.

  • (with W. Klingenberg) Geodesic flow on the normal congruence of a minimal surface, Progr. Math. 265 (2007) 427-436. Preprint
    We study the geodesic flow on the normal line congruence of a minimal surface in R induced by the neutral Kaehler metric on the space of oriented lines. The metric is lorentz with isolated degenerate points and the flow is shown to be completely integrable. In addition, we give a new holomorphic description of minimal surfaces in R and relate it to the classical Weierstrass representation.

  • (with W. Klingenberg) Geodesic flow on global holomorphic sections of TS, Bull. Belg. Math. Soc. 13 (2006) 1-9. Preprint
    We study the geodesic flow on the global holomorphic sections of the bundle TS--> S induced by the neutral Kaehler metric on the space of oriented lines of R, which we identify with TS. This flow is shown to be completely integrable when the sections are symplectic, and the behaviour of the geodesics is described.

  • (with W. Klingenberg) An indefinite Kaehler metric on the space of oriented lines, J. London Math. Soc. 72 (2005) 497-509. Preprint
    The total space of the tangent bundle of a Kaehler manifold admits a canonical Kaehler structure. Parallel translation identifies the space T of oriented affine lines in R with the tangent bundle of S. Thus, the round metric on S induces a Kaehler structure on T which turns out to have a metric of neutral signature. It is shown that the identity component of the isometry group of this metric is isomorphic to the identity component of the isometry group of the Euclidean metric on R. The geodesics of this metric are either planes or helicoids in R. The signature of the metric induced on a surface in T is determined by the degree of twisting of the associated line congruence in R, and we show that, for a Lagrangian surface, the metric is either Lorentz or totally null. For such surfaces it is proven that the Keller-Maslov index counts the number of isolated complex points of J inside a closed curve on the surface.

  • (with N. Georgiou) On the space of oriented geodesics of hyperbolic 3-space, Rocky Mountain J. Math. Preprint
    We construct a Kaehler structure (J,W,G) on the space L(H) of oriented geodesics of hyperbolic 3-space and investigate its properties. We prove that (L(H),J) is biholomorphic to Px Pminus the reflected diagonal, and that the Kaehler metric G is of neutral signature, conformally flat and scalar flat. We establish that the identity component of the isometry group of the metric G on L(H) is isomorphic to the identity component of the hyperbolic isometry group. Finally, we show that the geodesics of G correspond to ruled minimal surfaces in H, which are totally geodesic iff the geodesics are null.

  • (with W. Klingenberg) On Weingarten surfaces in Euclidean and Lorentzian 3-space, Differential Geom. Appl. Preprint
    We study the neutral Kaehler metric on the space of time-like lines in Lorentzian, which we identify with the total space of the tangent bundle to the hyperbolic plane. We find all of the infinitesimal isometries of this metric, as well as the geodesics, and interpret them in terms of the Lorentzian metric. In addition, we give a new characterisation of Weingarten surfaces in Euclidean 3-space and Lorentzian 3-space as the vanishing of the scalar curvature of the associated normal congruence in the space of oriented lines. Finally, we relate our construction to the classical Weierstrass representation of minimal and maximal surfaces.

  • (with N. Georgiou) A characterization of Weingarten surfaces in hyperbolic 3-space, (2007) Preprint
    We study 2-dimensional submanifolds of the space L(H) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kaehler structure. Such a surface is Lagrangian iff there exists a surface in H orthogonal to the geodesics of S. We prove that the induced metric on a Lagrangian surface in L(H) has zero Gauss curvature iff the orthogonal surfaces in H are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in L(H) and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in H.

  • (with M. Khalid and J. J. Ramon Mari) Lagrangian curves on spectral curves of monopoles, (2007) Preprint
    We study Lagrangian points on smooth holomorphic curves in TP equipped with a natural neutral Kaehler structure, and prove that they must form real curves. By virtue of the identification of TP with the space L(R) of oriented affine lines in Euclidean 3-space, these Lagrangian curves give rise to ruled surfaces in R, which we prove have zero Gauss curvature. Each ruled surface is shown to be the tangent lines to a curve in R, called the edge of regression of the ruled surface. We give an alternative characterization of these curves as the points in R where the number of oriented lines in the complex curve C that pass through the point is less than the degree of C. We then apply these results to the spectral curves of certain monopoles and construct the ruled surfaces and edges of regression generated by the Lagrangian curves.

  • (with H. Anciaux and P. Romon) Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface, (2008) Preprint
    Given an oriented Riemannian surface (S,g), its tangent bundle TS enjoys a natural pseudo-Kaehler structure, that is the combination of a complex structure J, a pseudo-metric G with neutral signature and a symplectic structure W. We give a local classification of those surfaces of TS which are both Lagrangian with respect to W and minimal with respect to G. We first show that if g is non-flat, the only such surfaces are affine normal bundles over geodesics. In the flat case there is, in contrast, a large set of Lagrangian minimal surfaces, which is described explicitly. As an application, we show that motions of surfaces in R induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in TS or TH respectively. We relate the area of the congruence to a second-order functional on the original surface.

  • (with W. Klingenberg) Proof of the Caratheodory conjecture by mean curvature flow in the space of oriented affine lines, (2008) Preprint
    We prove that the index of an isolated umbilic point on a C-smooth surface in Euclidean 3-space is less than or equal to one. As a corollary, we establish the Caratheodory conjecture, that the number of umbilic points on a closed convex surface in R must be greater than one. We do this by first reformulating the problem in terms of the index of an isolated complex point on a Lagrangian surface in TS, viewed as the space of oriented geodesics in R. The main step in the proof is to establish the existence of stable holomorphic discs with boundary contained on the Lagrangian surface enclosing the complex point. We first show that the existence of such discs implies that the Keller-Maslov index must be greater than or equal to one, which for topological reasons, places a bound on the index of the isolated complex point on the Lagrangian surface. To construct the holomorphic disc we utilize mean curvature flow with respect to the canonical neutral Kaehler metric on TS. We prove long-time existence of this flow by a priori estimates and show that, for small enough initial disc, the flowing disc is asymptotically holomorphic. Convergence to a bubbled holomorphic disc is then proven by a version of compactness for J-holomorphic discs with boundary contained in a totally real surface. Continuity up to the boundary assures that the Keller-Maslov index is retained in the limit and this establishes our main result.

  • (with N. Georgiou and W. Klingenberg) Totally null surfaces in neutral Kaehler 4-manifolds, (2008) Preprint
    We study the totally null surfaces of the neutral Kaehler metric on certain 4-manifolds. The tangent spaces of totally null surfaces are either self-dual (alpha-planes) or anti-self-dual (beta-planes) and so we consider alpha-surfaces and beta-surfaces. The metric of the examples we study, which include the spaces of oriented geodesics of 3-manifolds of constant curvature, are anti-self-dual, and so it is well-known that the alpha-planes are integrable and alpha-surfaces exist. These are holomorphic Lagrangian surfaces, which for the geodesic spaces correspond to totally umbilic foliations of the underlying 3-manifold. The beta-surfaces are less known and our interest is mainly in their description. In particular, we classify the beta-surfaces of the neutral Kaehler metric on TN, the tangent bundle to a Riemannian 2-manifold N. These include the spaces of oriented geodesics in Euclidean and Lorentz 3-space, for which we show that the beta-surfaces are affine tangent bundles to curves of constant geodesic curvature on S and H, respectively. In addition, we construct the beta-surfaces of the space of oriented geodesics of hyperbolic 3-space.