Seminar: Josias Reppekus (G.15.25)
I will discuss results on the existence of analytic discs in $C^0$ boundaries of pseudoconvex domains, that is, pseudoconvex domains whose boundary is locally the graph of a continuous (real-valued) function. The central tool is the solution to the Dirichlet problem for the Levi form with continuous boundary data by Nikolay Shcherbina.
In complex dimension $2$, we conclude that a disk in the closure of such a domain $D$ will always be completely contained inside the domain or completely contained in its boundary. Moreover, a boundary point lies on a boundary disk if and only if one can delete a neighbourhood of the point from the closure of $D$ and recover the boundary point in the plurisubharmonic hull of this set.
The first result is motivated by limit maps of recurrent Fatou components. The latter result has been proven by David Catlin in the case of smooth boundaries in dimension $2$ and generalised to higher dimensions by Sahutoglu and Straube under strict pseudoconvexity conditions. We conjecture a different generalisation holds also in higher dimensions in the continuous case.
This is joint work with Nikolay Shcherbina and Tobias Harz.
