Prof. Dr. Rainer Schimming: Personal statements, Research, Publications



Professional activities:

Supervised dissertation works:

Selected invited talks:


  1. - Deutsche Mathematiker-Vereinigung since 1990
  2. - Deutsche Physikalische Gesellschaft since 1990
  3. - American Mathematical Society since 1993
  4. - London Mathematical Society since 2001



Let (M,g) be a plane-wave spacetime of even dimension n. If tex2html_wrap_inline124 , tex2html_wrap_inline126 , then the De Rham wave equation for p-forms satisfies Huygens' principle. The like, if tex2html_wrap_inline130 , tex2html_wrap_inline132 , then the p-form Maxwell equations satisfy Huygens principle [1].
The recursive structure of the Hadamard coefficients to the p-form De Rham-Laplace equations tex2html_wrap_inline138 leads to the ``fantastic cancellation'' of McKean and Singer and to a new simple proof of the Gauss-Bonnet-Chern integral theorem [3].
Necessary conditions (in tensorial form) as well as sufficient conditions (=classes of examples) for Huygens' principle (HP) to a hyperbolic operator tex2html_wrap_inline140 , acting on sections of a vector bundle, are derived. The existence of a logarithm-free elementary solution for any type of L, elliptic, hyperbolic, ultrahyperbolic, or holomorphic, is studied as a natural generalization of HP [4,12,16].
Cauchy's problem to the Bach field equations of general relativity is studied: existence and uniqueness of a solution, extension theorem, the constraints as nonlinear elliptic equations [6].
An existence theorem for the Bach-Einstein gravitational field equations is proved: these admit a one-parameter family of non-conformally-flat centrally symmetric static solutions [7].
An algebraic algorithm, suited for computer implementation, for the Taylor expansion of Hadamard's coefficients is presented. The algorithm also works, with modifications, for other geodesic transport equations [4,12,15].
An explicit formula for the Korteweg-de Vries hierarchy is found, together with other results on this hierarchy [10,18].
The volume of truncated light cones in a Lorentzian manifold contains the information whether or not the manifold is flat or is Ricci-flat [11].
A natural generalization of harmonic manifolds is proposed: harmonic Laplace-like operators. This new concept leads to interesting necessary conditions and sufficient conditions [13].
The Helmholtz exceptional numbers of an even-dimensional harmonic manifold are introduced as the numbers tex2html_wrap_inline144 for wich the Helmholtz equation tex2html_wrap_inline146 admits a logarithm-free elementary solution. These exceptional numbers are found for the symmetric spaces of rank one [16].

Last modified: Tue Apr 1 11:13:07 MEST 2003