[Paper Review] Degenerations of Riemannian manifolds

Igor Belegradek|ArXiv.org|Jan 25, 2007

Geometric Analysis and Curvature Flows18 references3 citations

TL;DR

This paper provides a comprehensive survey of degenerations in Riemannian geometry, focusing on collapsing and non-collapsing phenomena under curvature bounds, particularly sectional curvature. It establishes that under $|\text{sec}| \leq c$, non-collapsing sequences converge to $C^{1,\alpha}$-smooth limit spaces, while collapsing sequences fiber over limit spaces with infranilmanifold fibers, and characterizes collapse to a point via Gromov-Ruh's theorem on almost flat manifolds.

ABSTRACT

This is an expositiry article on collapsing theory in Riemannian geometry written for the Modern Encyclopedia of Mathematical Physics (MEMPhys). We focus on describing the geometric and topological structure of collapsed/non-collapsed regions in Riemannian manifold under various curvature assumptions. Numerous applications of collapsing theory to Riemannian geometry are NOT discussed in this survey.

Motivation & Objective

• To analyze the geometric and topological structure of collapsed and non-collapsed regions in Riemannian manifolds under various curvature assumptions.
• To clarify the role of Gromov-Hausdorff convergence as a framework for studying metric degenerations.
• To survey the theory of collapsing under two-sided sectional curvature bounds, including the characterization of manifolds that collapse to a point.
• To explain the fibration structure of collapsing sequences via infranilmanifolds and the implications for limit spaces.
• To provide foundational tools, including Gromov-Hausdorff distance and ultralimits, for understanding degenerations in Riemannian geometry.

Proposed method

• Uses Gromov-Hausdorff convergence as the central framework to study degenerations of Riemannian manifolds.
• Applies Gromov's compactness theorem to ensure precompactness under Ricci curvature lower bounds.
• Employs Bishop-Gromov volume comparison to separate collapsed and non-collapsed regions.
• Utilizes Gromov-Ruh's theorem to characterize manifolds that collapse to a point as infranilmanifolds.
• Applies Fukaya’s fibration theorem to show that collapsing sequences fiber over limit spaces with infranilmanifold fibers.
• Introduces ultralimits via non-principal ultrafilters to define asymptotic cones, generalizing Gromov-Hausdorff limits.

Experimental results

Research questions

• RQ1 What is the structure of the limit space when a sequence of Riemannian manifolds collapses under bounded sectional curvature?
• RQ2 Which manifolds can collapse to a point under bounded sectional curvature, and what characterizes them?
• RQ3 How do non-collapsing sequences behave under Ricci curvature lower bounds, and what regularity do their limit spaces possess?
• RQ4 What is the fibration structure of collapsing sequences, and how do infranilmanifolds arise in this context?
• RQ5 How do ultralimits and asymptotic cones generalize Gromov-Hausdorff limits in the study of degenerations?

Key findings

• Under $|\text{sec}(M_k)| \leq c$, any non-collapsing sequence Gromov-Hausdorff converges to a $C^{1,\alpha}$-smooth Riemannian manifold with $C^{1,\alpha}$-convergent pullback metrics.
• For sequences with uniformly bounded diameter, $|\text{sec}| \leq c$, and positive volume, only finitely many diffeomorphism types exist, a result known as Cheeger-Gromov compactness.
• A manifold collapses to a point under rescaling if and only if it is an infranilmanifold, as per Gromov-Ruh's theorem.
• Collapsing sequences fiber over the limit space with infranilmanifold fibers, as established by Fukaya’s fibration theorem.
• The limit space of a Gromov-Hausdorff convergent sequence is unique up to basepoint-preserving isometry and is a complete, locally compact, path metric space.
• Ultralimits via non-principal ultrafilters generalize Gromov-Hausdorff limits and allow the definition of asymptotic cones, which for the hyperbolic plane yield uncountably branching trees.