A Decomposition Lemma in Convex Integration via Classical Algebraic Geometry

Abstract

In this paper, we prove a decomposition lemma for symmetric matrix fields on bounded domains: $D+\mathrm{Sym}\nablaΦ=\sum_i a_i^2ξ_i\otimesξ_i$ with uniform control on $Φ$ and $a_i^2$, using fewer than the usual $n(n+1)/2$ rank-one symmetric terms. Except possibly in dimensions $n=8,16$, the decomposition is shown to be optimal through algebraic arguments. This reduces the number of steps in convex integration for a nonlinear PDE system, improving Hölder regularity of flexible solutions in dimension $n\ge3$. This PDE is a partial linearization of the codimension-one local isometric embedding equation in the Nash--Kuiper theorem, and also yields improved regularity for very weak solutions of related 2D Monge--Ampére and $2$-Hessian systems. The improved Hölder exponent is any $α<(n^2+1)^{-1}$ for $n=2,4,8,16$ and any $α<(n^2+n-2ρ(n/2)-1)^{-1}$ otherwise, where $ρ$ is the Radon--Hurwitz number, related to Bott periodicity. The proof involves novel applications of algebraic geometry and topology that yield the optimality of decomposition, including Adams&#39; theorem on vector fields on spheres, intersections of projective varieties, and projective duality, combined with an elliptic method that avoids loss of differentiability.