PDE and applications: Schiffer operators on Riemann surfaces and the jump formula for quasicircles

  • Date:
  • Location: Akademiska sjukhuset 64119
  • Lecturer: Eric Schippers
  • Contact person: Kaj Nyström
  • Seminarium

Abstract: Consider a Riemann surface $R$ separated into two connected components by a smooth closed curve. The jump formula states that, given a sufficiently regular function $u$ on this curve satisfying certain algebraic conditions, there are holomorphic functions on the components whose difference on the curve is $u$. We extend this classical result to quasicircles, which are not rectifiable, and functions $u$ in a certain conformally invariant function space related to the Dirichlet space. This is accomplished by relating the problem to the Schiffer operators, which are singular integral operators on $L^2$ holomorphic or anti-holomorphic one-forms on the components. Joint work with Wolfgang Staubach.