EVENT DETAILS AND ABSTRACT

Title: Isometric Embedding of 2-dim Riemannian Manifolds in R³ (I): Overview
Speaker: Professor Qing Han
Speaker Info: University of Notre Dame
Brief Description:
Special Note: Special Time
Abstract:

It is an old problem whether a 2-dimensional smooth Riemannian manifold admits a smooth isometric embedding in R³. 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 R³. It was shown by Darboux in 1905 that isometrically embedding a 2-dimensional smooth Riemannian manifold in R³ 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 S² with a positive Guass curvature admits a smooth isometric embedding in R³. 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.