[Paper Review] Geometric Structures of Collapsing Riemannian Manifolds I

Aaron Naber, Gang Tian|ArXiv.org|Apr 14, 2008

Geometric Analysis and Curvature Flows8 references10 citations

TL;DR

This paper establishes sharp Hausdorff dimension bounds on the singular set of collapsed Riemannian manifolds under bounded sectional curvature, proving that the set of non-orbifold points has dimension at most min{n−5, dimX−3}. It shows that for n ≤ 4, the limit space is a smooth Riemannian orbifold, and applies this to Einstein four-manifolds, proving they are orbifold away from finitely many points with a −dist⁻² curvature lower bound.

ABSTRACT

Let (M^n_i,g_i,p_i) be a sequence of smooth pointed complete n-dimensional Riemannian Manifolds with uniform bounds on the sectional curvatures and let (X,d,p) be a metric space such that (M^n_i,g_i,p_i) -> (X,d,p) in the Gromov-Hausdorff sense. Let O \subseteq X be the set of points x \in X such that there exists a neighborhood of x which is isometric to an open set in a Riemannian orbifold and let B = O^c be the complement set. Then we have the sharp estimates dim_Haus(B) \leq min{n-5, dim_Haus(X)-3}, and further for arbitrary x \in X we have that x \in O iff a neighborhood of x has bounded Alexandroff curvature. In particular, if n \leq 4 then B is empty and (X,d) is a Riemannian orbifold. Our main application is to prove that a collapsed limit of Einstein four manifolds has a smooth Riemannian orbifold structure away from a finite number of points, and that near these points the curvatures has a -dist^{-2} lower bound.

Motivation & Objective

• To understand the metric orbifold structure of the Gromov-Hausdorff limit of collapsing Riemannian manifolds with bounded curvature.
• To determine when points in the limit space have neighborhoods isometric to Riemannian orbifolds.
• To classify the structure of singularities in collapsed limits, especially in low dimensions.
• To apply the results to the compactification of Einstein moduli spaces, particularly for four-dimensional Einstein manifolds.

Proposed method

• Analyzes the Gromov-Hausdorff limit of sequences of n-dimensional Riemannian manifolds with uniform bounds on sectional curvature and higher-order curvature derivatives.
• Introduces the set O of C^{k+1,α}-orbifold points and its complement B, the non-orbifold points, and studies the Hausdorff dimension of B.
• Uses metric approximation results from [1], [2], [19] to extend results under weaker curvature assumptions, such as Ricci curvature bounds and conjugacy radius bounds.
• Applies ε-regularity estimates from [11] to control curvature blow-up in collapsed Einstein four-manifolds.
• Employs group action and reflection group techniques to classify local group actions arising in collapse and prove conjugacy of reflection subgroups.
• Lifts isometries from orbifold regular loci to affine maps on R^n, proving conjugacy of holonomy groups and hence isomorphism of orbifold structures.

Experimental results

Research questions

• RQ1 Under what conditions does a point in the Gromov-Hausdorff limit of a collapsing sequence of Riemannian manifolds admit a neighborhood isometric to a Riemannian orbifold?
• RQ2 What is the maximal possible Hausdorff dimension of the set of points in the limit space that are not locally isometric to a Riemannian orbifold?
• RQ3 How does the presence of bounded curvature and higher-order derivative bounds affect the regularity of the limit space?
• RQ4 Can the singular set in the limit of Einstein four-manifolds be controlled, and what curvature behavior is expected near singularities?
• RQ5 To what extent can the orbifold structure of the limit space be recovered from the group actions and stratified geometry of the collapsing sequence?

Key findings

• The set B of non-orbifold points in the limit space X has Hausdorff dimension at most min{n−5, dimX−3}.
• If n ≤ 4, then B is empty, so the limit space (X,d) is a C^{k+1,α} Riemannian orbifold.
• For collapsed Einstein four-manifolds, the limit space is a smooth Riemannian orbifold away from a finite number of points.
• Near these isolated singular points in four-dimensional Einstein collapse, the curvature satisfies a lower bound of order −dist⁻².
• The existence of a C^{k+1,α} orbifold structure at a point is equivalent to the existence of a neighborhood with bounded stratified curvature.
• The limit space inherits a well-defined Hausdorff dimension, and the orbifold structure is determined by the conjugacy class of the holonomy group and reflection subgroups.