We propose a novel method to apply Teichmüller space theory to study the signature of a family non-intersecting closed 3D curves on a general genus zero closed surface. Our algorithm provides an efficient method to encode both global surface and local contour shape information. The signature - Teichmüller shape descriptor - is computed by surface Ricci flow method, which is equivalent to solving an elliptic partial differential equation on surfaces and is quite stable. We propose to apply the new signature to analyze abnormalities in brain cortical morphometry. Experimental results with 3D MRI data from ADNI dataset (12 healthy controls versus 12 Alzheimer's disease (AD) subjects) demonstrate the effectiveness of our method and illustrate its potential as a novel surface-based cortical morphometry measurement in AD research.