Your minimization problem is equivalent to
\begin{equation*}
\min_{R^TR=I}\quad\prod_{i=1}^p r_i^T\Sigma r_i,
\end{equation*}
and it can be shown (using *Hadamard's determinant inequality* and some more argumentation) that this minimum overall $p$ orthonormal tuples is achieved by choosing the $r_i$ corresponding to the smallest $p$ eigenvectors.