address: Department of Mathematics
University of Leicester
Leicester, LE1 7RH
office: CH203
office hours: see my time table (or sent me an email for an appointment)
phone: +44(0116) 252 5237
email: k.leschke@mcs.le.ac.uk

Teaching

This semester I  teach the module "Curves and Surfaces" (MA3152).

Admin

Currently, I'm working on setting up a new MSc Programme in visualisation and virtual reality. Moreover, I'm a member of the MSc Committee, the Learning and Teaching Committee and of the syllabus committee. I'm a proud member of Euler house.

Research

My research interests lie in the field of Differential Geometry; the study of surfaces into 3-space using quaternionic holomorphic maps has immediate connections to classical complex-analytic geometry, visualization and conformal field theory.

The focus in my recent research is in surface theory, where I  study conformal maps into the 4 sphere. In papers with Burstall, Ferus, Pedit and Pinkall [FLPP], [BFLPP], we generalize methods and statements known in complex algebraic geometry (such as the Riemann-Roch theorem, Plücker formula) to complex quaternionic line bundles, and derive this way new results in surface theory. The main idea of our theory - which  has come to be known as "Quaternionic Holomorphic Geometry" -  is to consider the 4 sphere as quaternionic projective space and use the conformal maps into the 4 sphere as the holomorphic functions of the theory.

Particular interest lies on the transformation theory of surfaces: various surface classes are given by solutions to associated partial differential equations. Given one (simple) solution to the PDE, one hopes to construct new and more interesting examples by using transformations which preserve the surface class and are obtained by solving (simpler) ODEs, e.g. [L2].

Examples are Bäcklund and Darboux transformations which have been extensively studied in the past. In recent papers, we discussed such transformations for Willmore surfaces, for example, we classified Willmore tori with non-trivial normal bundle [LP2],  linked the Bäcklund transformation to a well-known complex algebraic construction, the envelopes and tangents of complex curves [LP1], and discussed dressing of Willmore surfaces and its connection to the Darboux transformation [L4].

Moreover, it turned out that a generalization of the classical Darboux transformation allows to define and to give a geometric interpretation of the spectral curve

of a conformal torus [BLPP]. As an application of this ansatz, we have a unified view of the integrable system theory for CMC [CLP],  Hamiltonian stationary Lagrangians [LR] and Willmore surfaces [GL]. Generically, the Darboux transformation preserves the surface class. However,  at singular points of the spectral curve the Darboux transformation may change geometric properties, e.g. break the mean curvature property [L3]. If the surface is given by a harmonicity condition, then Darboux transforms are generically given by parallel sections of the associated family of flat connections of the harmonic map. In particular, an induced transformation on harmonic maps can be defined, and we show in [BDLQ] that this transformation is indeed the simple factor dressing of the harmonic map.

 

An important tool in my study of surfaces is their visualisation: it allows both to demonstrate a geometric situation and to experiment to obtain new results in surface theory. Here are links to my picture gallery and to some of my labs.

 

Before turning to surface theory, I've worked on isoparametric manifolds: I provided a constructive geometric method to decide whether an isoparametric manifold is homogeneous [L1] by using algebraic tools (root systems, Jordan algebras and representation theory) as much as geometric ones (canonical connections). This ansatz allows to investigate infinite dimensional isoparametric submanifolds (see PhD thesis of K. Weinl, Uni Augsburg) and to give a unified proof of the inhomogeneity of the FKM examples.

Students

Phd student (jointly supervised with M. Georgoulis): A. Saad

Diplom students (at Universität Augsburg): A. Gouberman, Title of Thesis: "Darboux-Transformationen des Clifford-Torus"

Projects: M. Pirashvili (QHG), Z. Sapi Mkwawa (minimal surfaces), J. Habdank (discrete minimal surfaces), N. Lomas (Wente torus), K. Reed (discrete surfaces), H. Raistrick (Delaunay surfaces)