  Trimerization-polarization domains in ferroelectric hexagonal YMnO$_3$ were
resolved in all three spatial dimensions by piezoresponse force microscopy.
Their topology is dominated by electrostatic effects with a range of 100 unit
cells and reflects the unusual electrostatic origin of the spontaneous
polarization. The response of the domains to locally applied electric fields
explains difficulties in transferring YMnO$_3$ into a single-domain state. Our
results demonstrate that the wealth of non-displacive mechanisms driving
ferroelectricity that emerged from the research on multiferroics are a rich
source of alternative types of domains and domain-switching phenomena.
