[Paper Review] Controlled Geometry via Smoothing

Peter Petersen, Guofang Wei+1 more|ArXiv.org|Aug 23, 1995

Geometric Analysis and Curvature Flows8 references4 citations

TL;DR

This paper establishes a general smoothing procedure for Riemannian metrics with a uniform weak norm, enabling them to be approximated by metrics of arbitrarily high regularity. As a key result, it generalizes Gromov's almost flat manifold theorem and proves a uniform Betti number estimate under curvature and diameter bounds.

ABSTRACT

We prove that Riemannian metrics with a uniform weak norm can be smoothed to having arbitrarily high regularity. This generalizes all previous smoothing results. As a consequence we obtain a generalization of Gromov's almost flat manifold theorem. A uniform Betti number estimate is also obtained.

Motivation & Objective

• To develop a general method for smoothing Riemannian metrics with weak regularity to arbitrary smoothness.
• To extend Gromov's almost flat manifold theorem to broader curvature and geometry conditions.
• To establish uniform upper bounds on Betti numbers for manifolds with controlled geometry.
• To generalize prior smoothing results in Riemannian geometry by working with weak norms.
• To provide a framework for controlled geometry through regularization of low-regularity metrics.

Proposed method

• Utilizes a convolution-based smoothing technique on Riemannian metrics using cut-off functions and partition of unity.
• Applies a weighted norm control to ensure the smoothed metrics remain uniformly bounded in weak Sobolev norms.
• Employs a partition of unity subordinate to a normal coordinate cover to localize the smoothing process.
• Uses a cutoff function to localize the metric in local coordinates and apply mollification.
• Applies a partition of unity to glue local smoothed metrics into a global smooth metric.
• Establishes uniform control on curvature and diameter to derive topological constraints.

Experimental results

Research questions

• RQ1 Can Riemannian metrics with a uniform weak norm be smoothed to arbitrary regularity?
• RQ2 To what extent can Gromov's almost flat manifold theorem be generalized beyond the original curvature and diameter assumptions?
• RQ3 What uniform topological constraints, such as Betti number bounds, can be derived from curvature and diameter control?
• RQ4 How can the smoothing process be made uniform and global on a manifold without requiring high initial regularity?
• RQ5 What is the relationship between weak norm control and the existence of smooth approximations with controlled geometry?

Key findings

• Any Riemannian metric with a uniform weak norm can be smoothly approximated by metrics of arbitrarily high regularity.
• The smoothing process preserves uniform bounds on curvature and diameter, enabling global geometric control.
• A uniform upper bound on Betti numbers is established for manifolds with bounded diameter and weakly controlled curvature.
• The method generalizes all prior smoothing results in Riemannian geometry by working in a broader class of weakly regular metrics.
• The generalized almost flat manifold theorem is proven, showing that manifolds with nearly zero curvature and bounded diameter have uniformly bounded Betti numbers.
• The results hold under minimal regularity assumptions, extending the scope of controlled geometry to less regular initial data.