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- (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 R³, 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 R³. In this paper we explore the relationship between the focal
set of a line congruence (or 2-parameter family of oriented lines in R³) 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 R³ 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 R³ and relate it to the classical Weierstrass representation.
- (with W. Klingenberg) Geodesic flow on global holomorphic sections of TS², Bull. Belg. Math. Soc. 13 (2006) 1-9. Preprint
We study the geodesic flow on the global holomorphic sections of the bundle TS²--> S² induced by the neutral Kaehler metric on the space of
oriented lines of R³, which we identify with TS². 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 R³ with the tangent bundle of S². Thus, the round metric on S² 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 R³.
The geodesics of this metric are either planes or helicoids in R³. The signature of the metric induced on a surface in T is determined
by the degree of twisting of the associated line congruence in R³, 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(H³) of oriented geodesics of
hyperbolic 3-space and investigate its properties. We prove that (L(H³),J) is biholomorphic
to P¹x P¹minus 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(H³) 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 H³, 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(H³) 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 H³ orthogonal to the geodesics of
S.
We prove that the induced metric on a Lagrangian surface in L(H³) has zero Gauss curvature iff the orthogonal surfaces
in H³ are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null
surfaces in L(H³) and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in H³.
- (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 TP¹ equipped with a natural neutral Kaehler structure, and prove that
they must form real curves. By virtue of the identification of TP¹ with the space L(R³) of oriented affine
lines in Euclidean 3-space, these Lagrangian curves give rise to ruled surfaces in R³, which we prove have zero
Gauss curvature.
Each ruled surface is shown to be the tangent lines to a curve in R³, called the edge of regression of the ruled surface.
We give an alternative characterization of these curves as the points in R³ 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 R³
induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in TS² or TH² 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 C³-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 R³ 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 TS², viewed as the space of
oriented geodesics in R³. 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 TS².
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 S² and H², respectively. In addition, we construct the beta-surfaces of the space of oriented geodesics of hyperbolic 3-space.
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