  In this paper, we consider the transmission eigenvalue problem associated
with a general conductive transmission condition and study the geometric
structures of the transmission eigenfunctions. We prove that under a mild
regularity condition in terms of the Herglotz approximations of one of the pair
of the transmission eigenfunctions, the eigenfunctions must be vanishing around
a corner on the boundary. The Herglotz approximation can be regarded as the
Fourier transform of the transmission eigenfunction in terms of the plane
waves, and the growth rate of the transformed function can be used to
characterize the regularity of the underlying wave function. The geometric
structures derived in this paper include the related results in [5,19] as
special cases and verify that the vanishing around corners is a generic local
geometric property of the transmission eigenfunctions.
