The aim of this talk is to report on a recent result on the structure of the set of smooth Riemannian metrics on a connected manifold such that the corresponding Laplace-Beltrami operator has an eigenvalue of a given multiplicity (this is joint work with Josef Greilhuber). We introduce a simple geometric condition on metrics and show it is equivalent to the Strong Arnold Hypothesis, which essentially posits that multiple Laplace eigenvalues split up under perturbation like those of symmetric matrices. We prove our condition is satisfied except on a set of infinite codimension, and use this to obtain a maximal non-crossing rule for Laplace eigenvalues, thus proving that conjecture of Arnold is morally correct. In this talk, we will explain the significance and wider context of this result and try to outline many of the essential ideas which went into the proof.
On Arnold's transversality conjecture for the Laplace-Beltrami operator
10.01.2024 11:30 - 13:00
Organiser:
T. Körber, A. Molchanova, F. Rupp
Location: