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In this seminar, we focus on the minimal immersions of the 2-dimensional sphere in a space of constant sectional curvature. We will present some results and their proofs following the below resources. 

 

Topics to be covered are: 

 

1. Higher osculating spaces and higher fundamental forms of a submanifold.

2. Higher fundamental forms of a submanifold.

3. Minimal Surfaces.

4. Elliptic differential systems. 

5. Minimal immersions of the 2-sphere.

6. The case of N-sphere as the ambient space.

 

References:

1. Chern, Shiing Shen - On the minimal immersions of the two-sphere in a space of constant curvature. Problems in analysis (Lectures at the Sympos. in honor of Salomon Bochner, Princeton Univ., Princeton, N.J., 1969), pp. 27–40. Princeton Univ. Press, Princeton, N.J., 1970.    
2. Calabi, Eugenio - Minimal immersions of surfaces in Euclidean spheres. J. Differential Geometry 1 (1967), 111–125.