Plane Domains Which Are Spectrally Determined

This paper studies the trace of the heat kernelZ(t) _j=1 exp (_jt), where{_j} are the eigenvalues of atwo-dimensional Dirichlet or Neumann Laplace operator. FromZ(t), a sequence of invariants (geometrical invariants)such as area, boundary measure, Euler characteristics, etc., can bedetermined. Using these invariants, the existence of the nondisk domainswhich are determined from the information of Dirichlet and Neumannspectrum, can be shown. In addition, we prove that the number of suchdomains is infinite (uncountable) and these domains are not similar eachother.