On Weyl's Problem of Isometric Embedding

In this talk I shall discuss some classical results on isometric embedding of positively/nonegatively curved surfaces into $\mathbb{R}^3$. 

    The isometric embedding problem has played a crucial role in the development of geometric analysis and nonlinear PDE techniques--Nash invented his Nash-Moser techniques to prove the embeddability of general manifolds; later Gromov recast the problem into his ``h-Principle", which recently led to a major breakthrough by C. De Lellis et al. in the analysis of Euler/Navier-Stokes. Moreover, Nirenberg settled (positively) the Weyl Problem: given a smooth metric with strictly positive Gaussian curvature on a closed surface, does there exist a global isometric embedding into the Euclidean space $\mathbb{R}^3$? This work is proved by the continuity method and based on the regularity theory of the Monge-Ampere Equation, which led to Cheng-Yau's renowned works on the Minkowski Problem and the Calabi Conjecture. 

    Today we shall summarise Nirenberg's original proof for the Weyl problem. Also, we shall describe Hamilton's simplified proof using Nash-Moser Inverse Function Theorem, and Guan-Li's generalisation to the case of nonnegative Gaussian curvature. We shall also mention the status-quo of the related problems.