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

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

We study the geodesic flow on the normal line congruence of a minimal surface in R induced by the neutral Kähler 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.

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

Area-stationary Surfaces in Neutral Kähler 4-Manifolds

We study surfaces in TN that are area-stationary with respect to a neutral Kähler metric constructed on TN from a Riemannian metric g on N. We show that holomorphic curves in TN are areastationary. However, in general, area-stationary surfaces are not holomorphic. We prove this by constructing counter-examples. In the case where g is rotationally symmetric, we find all area-stationary surfaces ...

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...

On Area-stationary Surfaces in Certain Neutral Kähler 4-manifolds Brendan Guilfoyle and Wilhelm Klingenberg

We study surfaces in TN that are area-stationary with respect to a neutral Kähler metric constructed on TN from a riemannian metric g on N. We show that holomorphic curves in TN are area-stationary, while lagrangian surfaces that are area-stationary are also holomorphic and hence totally null. However, in general, area stationary surfaces are not holomorphic. We prove this by constructing count...