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The goal of this course is to cover topics such as the theory of minimal surfaces from the view point of Elie Cartan. We will also cover basic topics in Riemannian geometry such as the use of Jacobi field, Laplace comparison theorem and the volume comparison theorem on Riemannian manifolds. Of course, we will prove the famous Gauss Bonnet theorem in the higher even dimension due to S.S. Chern. We will also prove the famous gradient estimate for harmonic functions due to S.T.Yau on a Riemannian manifold. All and all, we will try to demonstrate the interaction between differential geometry and elliptic partial differential equations.

A student taking this advanced graduate level course should have a solid background in elementary differential geometry which includes the theory of surface in the Euclidean 3 space. A student should know basic vector calculus and the use of differential forms.

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attendance (25precent), notes taking (30 percent ), and a Final Project (25 percent), and may be a take home exam (20 percent).

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There is no specific textbook required. However, we have the following important reference. Global Differential Geometry, Volume 2, Moving frame by Wu-Hsiung Huang Global Differential Geometry, Volume 3, Calculus of Variations in Geometry by Wu-Hsiung Huang

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