A key challenge in the accurate reconstruction of cortical surfaces is the automated correction of geometric and topological outliers in tissue boundaries. Conventionally these two types of errors are handled separately. In this work, we propose a unified analysis framework for the joint correction of geometric and topological outliers in cortical reconstruction. Using the Reeb graph of intrinsically defined Laplace-Beltrami eigenfunctions, our method automatically locates spurious branches, handles and holes on tissue boundaries and corrects them with image information and geometric regularity derived from paired boundary evolutions. In our experiments, we demonstrate on 200 MR images from two datasets that our method is much faster and achieves better performance than FreeSurfer in population studies.