Georgia Institute of Technology College of Sciences

Nash's isometric embedding theorem is a fundamental result in differential geometry, establishing that every smooth Riemannian manifold can be isometrically embedded into a sufficiently high-dimensional Euclidean space. Nash's original proof relies on a sophisticated iterative scheme to overcome the loss of derivatives arising from the nonlinear isometric embedding equations. In the late 1980s, Matthias Günther introduced an alternative approach that avoids the Nash–Moser iteration scheme and the associated tame estimates, considerably simplifying the argument. In this talk, we will introduce the isometric embedding problem, discuss the main difficulties in solving the underlying nonlinear system of partial differential equations, and outline the key ideas behind Günther's approach. The talk will be expository, emphasizing the geometric intuition and analytical techniques underlying the proof.