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Equivariant Smoothing Processes on Currents and Spaces with Bounded Curvature

Andrés Ahumada Gómez

TL;DR

The paper addresses symmetry-preserving smoothing in geometric analysis by introducing equivariant versions of De Rham's current smoothing and Nikolaev's metric smoothing for spaces with bounded curvature. It constructs the equivariant operators $\mathcal{Z}_{\mathcal{G}}$ and $\mathcal{H}^{\mathcal{G}}$ through averaging over a compact Lie group $\mathcal{G}$, ensuring $\mathcal{G}$-invariance while preserving convergence properties: currents converge weakly to the original and metrics converge in the Lipschitz sense with preserved curvature bounds. The main contributions are the Equivariant De Rham's Approximation Theorem, the Equivariant Nikolaev's Approximation Theorem, and an Equivariant Sphere Theorem, which together enable symmetry-preserving regularization and topological conclusions on spaces with curvature bounds. This framework facilitates transferring geometric structure to smooth models while respecting group actions, with potential implications for orbit space analysis and symmetry in geometric topology.

Abstract

We introduce actions of a compact Lie group in two regularization processes: in De Rham's approximation process of currents on a smooth manifold by smooth currents, and in a smoothing operator of Riemannian metrics of metric spaces with bounded curvature.

Equivariant Smoothing Processes on Currents and Spaces with Bounded Curvature

TL;DR

The paper addresses symmetry-preserving smoothing in geometric analysis by introducing equivariant versions of De Rham's current smoothing and Nikolaev's metric smoothing for spaces with bounded curvature. It constructs the equivariant operators and through averaging over a compact Lie group , ensuring -invariance while preserving convergence properties: currents converge weakly to the original and metrics converge in the Lipschitz sense with preserved curvature bounds. The main contributions are the Equivariant De Rham's Approximation Theorem, the Equivariant Nikolaev's Approximation Theorem, and an Equivariant Sphere Theorem, which together enable symmetry-preserving regularization and topological conclusions on spaces with curvature bounds. This framework facilitates transferring geometric structure to smooth models while respecting group actions, with potential implications for orbit space analysis and symmetry in geometric topology.

Abstract

We introduce actions of a compact Lie group in two regularization processes: in De Rham's approximation process of currents on a smooth manifold by smooth currents, and in a smoothing operator of Riemannian metrics of metric spaces with bounded curvature.
Paper Structure (4 sections, 11 theorems, 56 equations)

This paper contains 4 sections, 11 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Currents
  3. Spaces with bounded curvature
  4. Equivariant Sphere Theorem

Key Result

Theorem 1

Let $(\mathcal{M},d(\textbf{g}_{0}))$ be a space with bounded curvature with $d$ its metric and $\texttt{g}_{0}$ the induced Riemannian metric by this. Then, on the differentiable manifold $\mathcal{M}$, one can define a sequence of infinitely differentiable Riemannian metrics $\{\textbf{g}_{m}\}_{m

Theorems & Definitions (17)

  • Theorem 1: Nikolaev, 1991
  • Theorem 2: De Rham's Approximation Theorem
  • Theorem 3: Equivariant De Rham's Approximation Theorem
  • Theorem 4: Equivariant Nikolaev's Approximation Theorem
  • Theorem 5: Equivariant Sphere Theorem
  • Proposition 6: Proposition 1, Section 14, Chapter III, of zbMATH03848254
  • Proposition 7: Proposition 2, Section 15, Chapter III, of zbMATH03848254
  • Proposition 8
  • proof
  • Proposition 9
  • ...and 7 more