Stagnation Points of Harmonic Vector Fields and the Domain Topology - Some Applications of Morse Theory - Koppenhöfer,Theo (Master’s Theses in Mathematical Sciences, 2024)
Given a harmonic vector field u on a bounded domain we call the set on which u enters the domain entrant boundary and the set on which u exits the domain emergent boundary. In the first part of this thesis we ask the question of whether it is possible to have interior stagnation points and connected entrant and emergent boundaries. The answer in two dimensions for a domain homeomorphic to a disc is ‘no’. If one allows for holes in the domain the answer becomes a ‘yes’ for which we give explicit examples. In four dimensions a simple example shows that this is possible with the ball as the domain. In three dimensions we use Morse theory to argue that the number of stagnation points is even. With the help of this result and numerical methods we found a harmonic polynomial on the ball which has interior stagnation points and simply connected entrant and emergent sets. The second part revolves around harmonic vector fields with interior stagnation points and without boundary stagnation points. This question yields in two dimensions an elegant relation between the domain topology and the number of interior stagnation points. As a special case we obtain statements about harmonic vector fields which are tangential and nonvanishing at the boundary. We also give examples of vector fields illustrating this point. In three dimensions we do not find an explicit example but the Morse index theorem implies that for such an example the number of stagnation points has to be even and the Euler characteristic of the domain has to vanish. As part of the thesis we formulate a version of Morse theory by [Braess, 1974] and [Agrachëv, 1991] for manifolds with corners. This relates the domain topology with the number of stagnation points. Additionally we give a proof for the density of Morse functions for this type of Morse theory.
Keywords: Morse theory; Partial differential equations; harmonic function; stagnation points; critical points