Paul Günther’s Work on Hadamard’s Conjecture

While Professor Günthers mathematical work embraces many aspects of differential geometry and partial differential equations his work on 2nd order hyperbolic equations in relation to Huygens principle stands out as particularly significant. The wave equation in flat space R' has the remarkable property (Huygens' principle) that the value u(x, t) of the solution at time t only depends on the initial data on the surface of a sphere with center x and radius t. This is verified by everyday experience: a flash of light is seen and the sound of a shot is heard at a distance t after time t (the velocity of light and sound taken as 1) but then no more. This three dimensional phenomenon contrasts markedly with wave propagation in two dimensions as illustrated by the sequence of spherical wave from a pebble falling on a flat water surface.