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Deformable registration of cortical surfaces facilitates longitudinal and intergroup comparisons of cortical structure and function in the study of many neurodegenerative diseases. Non-rigid cortical matching is a challenging task due to the large variability between individuals and the complexity of the cortex. We present a new framework for computing cortical correspondences on brain surfaces based on 3D Shape Context and mean curvatures of partially flattened surfaces (PFS). Our approach is scale invariant and provides an accurate and anatomically meaningful alignment across the population. Registering PFS, instead of original cortical surfaces, simplifies the determination of shape correspondences, overcoming the problem of intersubject variability, while still guaranteeing the alignment of the main brain lobes and folding patterns. We validated the approach using 30 segmented brains from the OASIS database registered to a common space and compared the results with Freesurfer. In average, mean absolute distance of 0.36 and Hausdorff distance of 5.06 between moving and target surfaces are obtained. Further localization of labelled areas on each hemisphere demonstrated the accuracy of the technique.