Let $\Omega$ be a $C^{2+\gamma}$ domain in $\mathbb{R}^N$ ,
$N\geq 2$ , $0 < \gamma < 1$ . Let $T > 0$ and let $L$ be a uniformly
parabolic operator $Lu ={\partial u}/{\partial t}-\sum_{i,j}({\partial}/{\partial x_{i}})(a_{ij}({\partial u}/{\partial x_{j}})) +\sum_{j}b_{j}({\partial u}/{\partial x_{i}})+ a_{0}u$ , $a_{0}\geq 0$ , whose coefficients, depending on
$(x,t)\in\Omega \times\mathbb{R}$ , are $T$ periodic in $t$ and
satisfy some regularity assumptions. Let $A$ be the $N \times N$
matrix whose $i,j$ entry is $a_{ij}$ and let $\nu$ be the unit
exterior normal to $\partial\Omega$ . Let $m$ be a $T$ -periodic
function (that may change sign) defined on $\partial\Omega$ whose
restriction to $\partial\Omega \times\mathbb{R}$ belongs to
$W_{q}^{2-{1}/{q},1 -{1}/{2q}}(\partial\Omega \times(0,T))$ for some large enough $q$ .
In this paper, we give necessary and sufficient conditions on $m$
for the existence of principal eigenvalues for the periodic
parabolic Steklov problem $Lu = 0$ on $\Omega \times\mathbb{R}$ ,
$\langle A\nabla u,\nu\rangle =\lambda mu$ on
$\partial\Omega \times\mathbb{R}$ , u( x,t) =u(x,t+T) , u>0 on
$\Omega \times\mathbb{R}$ . Uniqueness and simplicity of the
positive principal eigenvalue is proved and a related maximum
principle is given.