Lavi Karp: Global Geometric Aspects of the Cauchy Problem for the Laplace Operator

This lecture will be dedicated to the discussion of problems  raised by H.S. Shapiro in a paper from 1991, and therefore entitled by the same title. We consider the Cauchy problem for the Laplacian with Cauchy data zero on a surface, and in a particular  case where the Laplacian is bounded.
 I will construct a bounded surface and a solution to the Cauchy problem with arbitrary growth at infinity. Such behavior is impossible for certain unbounded surfaces. We also solve Shapiro's conjecture claiming that the complement of null quadrature domains is a convex set. The talk is based on a joint work with A. Margulis.