Geodesic Flow on the Normal Congruence of a Minimal Surface Brendan Guilfoyle and Wilhelm Klingenberg

Geodesic Flow on Global Holomorphic Sections of Ts Brendan Guilfoyle and Wilhelm Klingenberg

We study the geodesic flow on the global holomorphic sections of the bundle π : TS → S induced by the neutral Kähler 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.

Proof of the Carathéodory Conjecture by Mean Curvature Flow in the Space of Oriented Affine Lines Brendan Guilfoyle and Wilhelm Klingenberg

ar X iv : m at h / 04 06 21 2 v 1 [ m at h . D G ] 1 0 Ju n 20 04 REFLECTION OF A WAVE OFF A SURFACE

Recent advances in twistor theory are applied to geometric optics in R 3. The general formulae for reflection of a wavefront in a surface are derived and in three special cases explicit descriptions are provided: when the reflecting surface is a plane, when the incoming wave is a plane and when the incoming wave is spherical. In each case particular examples are computed exactly and the results...

Level Sets of Functions and Symmetry Sets of Surface Sections

We prove that the level sets of a real C function of two variables near a non-degenerate critical point are of class C [s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at an elliptic or hyperbolic point, and in particular at an umbilic point. We go on to use the results to study symmetry sets of the planar sections. We also anal...

Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

Given an oriented Riemannian surface (Σ, g), its tangent bundle TΣ enjoys a natural pseudo-Kähler structure, that is the combination of a complex structure J, a pseudo-metric G with neutral signature and a symplectic structure Ω. We give a local classification of those surfaces of TΣ which are both Lagrangian with respect to Ω and minimal with respect to G. We first show that if g is non-flat, ...