Geometric, algebraic, and analytic descendants of Nash isometric embedding theorems

Abstract

Abstract. Is there anything interesting left in isometric embeddings after the problem had been solved by John Nash? We do not venture a definite answer, but we outline the boundary of our knowledge and indicate conjectural directions one may pursue further.Our presentation is by no means comprehensive. The terrain of isometric embeddings and the fields surrounding this terrain are vast and craggy with valleys separated by ridges of unreachable mountains; people cultivating their personal gardens in these "valleys" only vaguely aware of what happens away from their domains and the authors of general accounts on isometric embeddings have a limited acquaintance with the original papers. Even the highly cited articles by Nash have been carefully read only by a handful of mathematicians.In order not to mislead the reader, we try be open about what we do and what we do not know firsthand and to provide references to what is missing from the present paper.

On Nash’s unique contribution to analysis in just three of his papers

“…On the regularity side the best result, due to Jacobowitz [47], requires metrics of class C 2,α , α > 0. It is not known if the result can be extended to C 2 metrics; see discussion in [31] and [18]. 6 I understand the h-principle here, loosely, as high flexibility of the space of solutions to a system of partial differential equations underlying a geometric or a physical problem.…”

“…This high flexibility is particularly striking, as is the case of Nash's theorem, when it is due to limited regularity rather than indeterminacy of the system. See Gromov [31] and [32] for a precise and more inclusive understanding. 7 The result, at least in so far as it provides existence of smooth embeddings to high-dimensional spaces, was highly expected even though it required an analyst of the class of Nash to deal with the intrinsic difficulties of the problem.…”

“…8 In his survey [31] Gromov seems to restrict the domain of PDEs to boundary or initial value problems for which uniqueness prevails. Though this is the case for deterministic problems in classical physics, Gromov's definition ignores a vast array of other problems, such as those that appear routinely in quantum physics.…”

“…In his survey [31] Gromov gives an impressive list of 85 open problems in geometry directly connected to the work of Nash on isometric embedding. My personal view is that the most important discoveries made by Nash resonate way beyond the isometric embedding context to the broader field on nonlinear PDEs.…”

“…I note in particular Gromov's article [31] on the descendants of Nash's isometric embedding theorems as well as the more comprehensive C. De Lellis survey [18] on the masterpieces of J. Nash. The May 2016 commemorative article in the Notices of American Mathematical Society [19] gives also an excellent account on Nash's work.…”