[Paper Review] Bounded geometry and leaves

Jesús A. Álvarez López, Ramón Barral Lijó|arXiv (Cornell University)|Jan 9, 2017

Geometric Analysis and Curvature Flows28 references9 citations

TL;DR

This paper establishes that any complete, connected Riemannian manifold of bounded geometry can be isometrically embedded as a leaf with trivial holonomy in a compact Riemannian foliated space. The authors construct a universal space $\overline{\mathcal{M}}^\infty_{*,\text{imm}}(n)$ using $C^\infty$-convergent sequences of pointed immersions into a separable Hilbert space, prove its Polish topology and compactness of closures, and show that the image of the canonical map $\hat{\imath}_{M,f}$ is a leaf with trivial holonomy in a compact foliated subspace, thus proving the main theorem on realization of bounded geometry manifolds as such leaves.

ABSTRACT

The main theorem states that any complete connected Riemannian manifold of bounded geometry can be isometrically realized as a leaf with trivial holonomy in a compact Riemannian foliated space.

Motivation & Objective

• To prove that every complete, connected Riemannian manifold of bounded geometry arises as a leaf with trivial holonomy in a compact Riemannian foliated space.
• To construct a universal parameter space $\overline{\mathcal{M}}^\infty_{*,\text{imm}}(n)$ using $C^\infty$-convergent sequences of pointed immersions into a Hilbert space.
• To establish that the closure of the image of the canonical map $\hat{\imath}_{M,f}$ is compact and forms a Riemannian foliated subspace.
• To show that the holonomy of the leaf $\widehat{\imath}_{M,f}(M)$ is trivial under suitable choices of the immersion $f$.
• To resolve an open problem in geometric analysis by providing a complete and rigorous proof of the long-conjectured realization theorem for bounded geometry manifolds.

Proposed method

• Define the space $b\mathcal{M}^*_*(n)$ as the set of isometry classes of pointed, complete, connected Riemannian $n$-manifolds equipped with $C^\infty$-maps into a separable Hilbert space $E$, modulo pointed isometry and pullback equivalence.
• Introduce a $C^m$-topology on $b\mathcal{M}^*_*(n)$ via convergence of pullbacks of metrics and maps on compact domains under local embeddings.
• Prove that the $C^\infty$-convergence topology on $b\mathcal{M}^*_*(n)$ is Polish, ensuring completeness and separability.
• Construct the canonical map $\hat{\imath}_{M,f}: M \to b\mathcal{M}^\infty_{*,\text{imm}}(n)$ by sending each point to its $C^\infty$-equivalence class.
• Show that the closure $\overline{\text{Cl}}^\infty(\widehat{\imath}_{M,f}(M))$ is compact in $b\mathcal{M}^\infty_{*,\text{imm}}(n)$ when $M$ has bounded geometry.
• Prove that the image $\widehat{\imath}_{M,f}(M)$ is a leaf with trivial holonomy in the compact foliated subspace $\overline{\text{Cl}}^\infty(\widehat{\imath}_{M,f}(M))$ by ensuring the holonomy group of the leaf is trivial.

Experimental results

Research questions

• RQ1 Can every complete, connected Riemannian manifold of bounded geometry be realized as a leaf with trivial holonomy in a compact Riemannian foliated space?
• RQ2 Is the closure of the image of the canonical map $\widehat{\imath}_{M,f}$ compact in the $C^\infty$-topology on the universal space $b\mathcal{M}^\infty_{*,\text{imm}}(n)$?
• RQ3 Can the immersion $f$ be chosen so that the holonomy group of the leaf $\widehat{\imath}_{M,f}(M)$ is trivial?
• RQ4 Does there exist a finite-valued decoration of the Cayley graph of a bounded geometry graph such that all images in the closure have trivial automorphism group?
• RQ5 Can the universal space $b\mathcal{M}^\infty_{*,\text{imm}}(n)$ be endowed with a $C^\infty$-foliated structure making the image of $\widehat{\imath}_{M,f}$ a leaf?

Key findings

• The $C^\infty$-convergence topology on $b\mathcal{M}^*_*(n)$ is Polish, ensuring completeness and separability, which is a non-trivial result in the theory of convergence of Riemannian manifolds.
• The closure $\overline{\text{Cl}}^\infty(\widehat{\imath}_{M,f}(M))$ is compact in $b\mathcal{M}^\infty_{*,\text{imm}}(n)$ if and only if the manifold $M$ has bounded geometry.
• The image $\widehat{\imath}_{M,f}(M)$ is an isometric embedding of $M$ into a compact Riemannian foliated subspace of $b\mathcal{M}^\infty_{*,\text{imm}}(n)$, with the induced foliation structure.
• The leaf $\widehat{\imath}_{M,f}(M)$ has trivial holonomy if the immersion $f$ is chosen so that $\text{Iso}(N,h) = \{\text{id}\}$ for all $[N,h,y]$ in the closure of $\widehat{\imath}_{M,f}(M)$.
• The main theorem is proven: every complete, connected Riemannian manifold of bounded geometry is isometrically realized as a leaf with trivial holonomy in a compact Riemannian foliated space.
• The construction is universal and relies on embedding $M$ into a Hilbert space via a $C^\infty$-embedding $f$ with uniformly bounded derivatives and non-vanishing volume form, ensuring the image is a well-defined leaf in the universal space.