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A decomposition lemma in convex integration via classical algebraic geometry

Zhitong Su, Weijun Zhang

TL;DR

The paper develops a decomposition lemma that reduces the number of rank-one symmetric matrices needed to correct deficits in convex integration, enabling higher Hölder regularity for $C^{1,\alpha}$ solutions to the nonlinear PDE $\frac{1}{2}\nabla v\otimes\nabla v+\mathrm{Sym}\nabla w=A$ in dimension $n\ge2$. The main innovation combines elliptic constructions with Algebraic Geometry/Topology, notably projective duality and the Radon–Hurwitz theory, to identify an optimal subspace $L$ of symmetric matrices so that the deficit lies in a dual space $L^\vee$ and can be represented by at most $\Xi_n$ primitive terms; this yields $\alpha$-regimes better than prior results (e.g., $\alpha<(n^2+n+1)^{-1}$ for $n\ge3$). The approach connects to the Nash–Kuiper local embedding framework and to very weak solutions of Monge–Ampère and $2$-Hessian systems, providing a pathway to higher-regularity flexible solutions via an elliptic-corrugation strategy. It also clarifies exceptional dimensions ($n=2,4,8,16$) where the maximal $L$-dimension is reduced, raising interesting conjectures about quadrics base loci and Bott-type periodicity, with potential impact on geometric PDEs and isometric embedding problems.

Abstract

In this paper, we introduce a decomposition lemma that allows error terms to be expressed using fewer rank-one symmetric matrices than $\frac{n(n+1)}{2}$ within the convex integration scheme of constructing flexible $C^{1,α}$ solutions to a system of nonlinear PDEs in dimension $n\geq 2$, which can be viewed as a kind of truncation of the codimension one local isometric embedding equation in Nash-Kuiper Theorem. This leads to flexible solutions with higher Hölder regularity, and consequently, improved very weak solutions to certain induced equations for any $n$, including Monge-Ampère systems and $2$-Hessian systems. The Hölder exponent of the solutions can be taken as any $α<(n^2+1)^{-1}$ for $n=2,4,8,16$, and any $α<(n^2+n-2ρ(\frac{n}{2})-1)^{-1}$ for other $n$, thereby improving the previously known bound $α<(n^2+n+1)^{-1}$ for $n\geq 3$. Here, $ρ(n)$ is the Radon-Hurwitz number, which exhibits an $8$-fold periodicity on $n$ that is related to Bott periodicity. Our arguments involve novel applications of several results from algebraic geometry and topology, including Adams' theorem on maximum linearly independent vector fields on spheres, the intersection of projective varieties, and projective duality. We also use an elliptic method ingeniously that avoids loss of differentiability.

A decomposition lemma in convex integration via classical algebraic geometry

TL;DR

The paper develops a decomposition lemma that reduces the number of rank-one symmetric matrices needed to correct deficits in convex integration, enabling higher Hölder regularity for solutions to the nonlinear PDE in dimension . The main innovation combines elliptic constructions with Algebraic Geometry/Topology, notably projective duality and the Radon–Hurwitz theory, to identify an optimal subspace of symmetric matrices so that the deficit lies in a dual space and can be represented by at most primitive terms; this yields -regimes better than prior results (e.g., for ). The approach connects to the Nash–Kuiper local embedding framework and to very weak solutions of Monge–Ampère and -Hessian systems, providing a pathway to higher-regularity flexible solutions via an elliptic-corrugation strategy. It also clarifies exceptional dimensions () where the maximal -dimension is reduced, raising interesting conjectures about quadrics base loci and Bott-type periodicity, with potential impact on geometric PDEs and isometric embedding problems.

Abstract

In this paper, we introduce a decomposition lemma that allows error terms to be expressed using fewer rank-one symmetric matrices than within the convex integration scheme of constructing flexible solutions to a system of nonlinear PDEs in dimension , which can be viewed as a kind of truncation of the codimension one local isometric embedding equation in Nash-Kuiper Theorem. This leads to flexible solutions with higher Hölder regularity, and consequently, improved very weak solutions to certain induced equations for any , including Monge-Ampère systems and -Hessian systems. The Hölder exponent of the solutions can be taken as any for , and any for other , thereby improving the previously known bound for . Here, is the Radon-Hurwitz number, which exhibits an -fold periodicity on that is related to Bott periodicity. Our arguments involve novel applications of several results from algebraic geometry and topology, including Adams' theorem on maximum linearly independent vector fields on spheres, the intersection of projective varieties, and projective duality. We also use an elliptic method ingeniously that avoids loss of differentiability.
Paper Structure (20 sections, 23 theorems, 107 equations, 2 figures, 1 table)

This paper contains 20 sections, 23 theorems, 107 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Convex integration: A review
  3. Main ideas of the decomposition lemma
  4. Preliminaries
  5. Preparations on iterations of convex integration
  6. Corrugation functions
  7. Mollification
  8. The elliptic theory to linear systems
  9. Radon-Hurwitz number: an introduction
  10. Decomposition lemma
  11. Elimination lemma and nonnegative coefficient lemma
  12. Algebraic description
  13. The intersections in real projective spaces
  14. Projective duality
  15. Radon-Hurwitz number: symmetric matrices case
  16. ...and 5 more sections

Key Result

Theorem 1.3

Let $n\geq 2$, $\Omega\subset\mathbb{R}^n$ be a bounded domain with $C^2$ boundary. Given a function $v^\flat\in C^0(\overline{\Omega})$, a vector field $w^\flat\in C^0(\overline{\Omega},\mathbb{R}^n)$ and a matrix field $A\in C^{2,\beta}(\overline{\Omega},\mathbb{R}_{sym}^{n\times n})$, then for an there exist $v\in C^{1,\alpha}(\overline{\Omega})$, $w\in C^{1,\alpha}(\overline{\Omega},\mathbb{R}

Figures (2)

  • Figure 1: Projective duality of $\boldsymbol{\mathrm{Y}}$ and $\boldsymbol{\mathrm{Z}}$ when $n=2$.
  • Figure 2: The choice of $\boldsymbol{\mathrm{L}}_{cs}$ in Cao-Székelyhidi cao_very_2019

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1.3
  • Theorem 1.4: Adams Adams_1962_vecotrfields
  • Theorem 1.5
  • Conjecture 1.6: Quadrics base locus conjecture \ref{['conjecture']}
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1: Mollification Lemma
  • Definition 2.2
  • Remark 2.3
  • ...and 38 more