PDE Seminar

Title: Isometric Embedding of 2-dim Riemannian Manifolds in $R^3$((II): Local Theory
Speaker: Professor Qing Han
Speaker Info: University of Notre Dame
Brief Description:
Special Note: Special Time

It is an old problem whether a 2-dimensional smooth Riemannian manifold admits a smooth isometric embedding in $R^3$. The local version has a long history. In 1887, Schlaefli conjectured that any 2-dimensional smooth Riemannian manifold always admits a local smooth isometric embedding in $R^3$. It was shown by Darboux in 1905 that isometrically embedding a 2-dimensional smooth Riemannian manifold in $R^3$ is equivalent to solving a fully nonlinear differential equation of Monge-Ampere type. A key step is to analyze its linearized equations. It is open whether these linearized equations always admit a solution when they are degenerate hyperbolic or of mixed type, corresponding to Gauss curvatures which are nonpositive or of mixed sign. The global version of the isometric embedding of 2-dimensional Riemannian manifolds also has a long history. In 1905, Weyl asked whether any smooth metric on $S^2$ with a positive Guass curvature admits a smooth isometric embedding in $R^3$. This question was solved affirmatively by Nirenberg in 1953. This remains to be the only global result so far. In this series of talks, I will present some results on both local and global isometric embeddings.
Date: Wednesday, May 02, 2007
Time: 3:00pm
Where: Lunt 107
Contact Person: Prof. Gui-Qiang Chen
Contact email: gqchen@math.northwestern.edu
Contact Phone: 847-491-5553
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