In this paper we propose a novel system for the accurate reconstruction of cortical surfaces from magnetic resonance images. At the core of our system is a novel framework for outlier detection and pruning by integrating intrinsic Reeb analysis of Laplace-Beltrami eigen-functions with topology-preserving evolution for localized filtering of outliers, which avoids unnecessary smoothing and shrinkage of cortical regions with high curvature. In our experiments, we compare our method with FreeSurfer and illustrate that our results can better capture cortical geometry in deep sulcal regions. To demonstrate the robustness of our method, we apply it to over 1300 scans from the Alzheimer's Disease Neuroimaging Initiative (ADNI). We show that cross-sectional group differences and longitudinal changes can be detected successfully with our method.