About ten years ago, De Lellis and Szekelyhidi made the surprising discovery that Nash’s results on the isometric embedding problem for Riemannian manifoldscould be adapted to construct counterintuitive solutions to the Euler equations for incompressible flow. Their work shed new light on turbulence and nonlinear PDE. We use this link in the other direction, transferring ideas from turbulence to geometry.
A thermodynamic framework is introduced that connects two problems previously thought to be distinct: the isometric embedding problem for Riemannian manifolds and the construction of Brownian motion on Riemannian manifolds. This link is used to introduce a geometric stochastic flow that we term stochastic Nash evolution.
These ideas will be explained in a (hopefully) elementary manner. My main goal is to present the stochastic flows in a manner that is suited to implementations by modifications of level set methods. The absence of numerical computations of isometric embeddings is an important gap in our understanding.
This is joint work with Dominik Inauen (University of Leipzig).