Important anatomical features on the cortical surface are usually represented by landmark curves, called sulcal/gyral curves. Manual labeling of these landmark curves is time-consuming, especially when a large dataset is analyzed. In this paper, we propose a method to trace the landmark curves on the cortical surfaces automatically based on the principal directions of the local Weingarten matrix. Based on a global conformal parametrization of the cortical surface, our method adjusts the landmark curves iteratively on the spherical or rectangular parameter domain of the cortical surface along the principal direction field, using umbilic points of the surface as anchors. The landmark curves can then be mapped back onto the cortical surface. To speed up the iterative scheme, we obtain a good initialization by extracting the high curvature regions on the cortex using the Chan-Vese segmentation method, which solves a PDE on the manifold using our global conformal parametrization technique. Experimental results show that the landmark curves detected by our algorithm closely resemble the same curves labeled manually. We applied these automatically labeled landmark curves to build average cortical surfaces with an optimized brain conformal mapping method. Experimental results show that our method can help in automatically matching cortical surfaces of the brain across subjects.