[Paper Review] The smooth Riemannian extension problem: completeness

Stefano Pigola, Giona Veronelli|arXiv (Cornell University)|Jan 19, 2016

Geometric Analysis and Curvature Flows12 references3 citations

TL;DR

This paper establishes that any smooth, metrically complete Riemannian manifold with smooth boundary can be isometrically embedded as a closed domain in a smooth, geodesically complete Riemannian manifold without boundary, using a general gluing and conformal deformation construction. The key contribution is the existence of complete Riemannian extensions preserving metric completeness, with applications to Sobolev spaces and proper Nash embeddings.

ABSTRACT

By means of a general gluing and conformal-deformation construction, we prove that any smooth, metrically complete Riemannian manifold with smooth boundary can be realized as a closed domain into a smooth, geodesically complete Riemannan manifold without boundary. Applications to Sobolev spaces, Nash embedding and local extensions with strict curvature bounds are presented.

Motivation & Objective

• To address the fundamental problem of extending a smooth, metrically complete Riemannian manifold with boundary into a geodesically complete manifold without boundary.
• To investigate whether such extensions can preserve curvature constraints, such as bounds on sectional, Ricci, or scalar curvature.
• To explore topological obstructions in the extension process, particularly in relation to curvature bounds and fundamental group restrictions.
• To establish density results for Sobolev spaces on manifolds with boundary using the existence of complete extensions.
• To demonstrate the existence of proper isometric embeddings into Euclidean space via extension techniques.

Proposed method

• Constructs a smooth, geodesically complete Riemannian extension via a general gluing and conformal deformation procedure.
• Uses a partition of unity subordinate to a locally finite, relatively compact smooth atlas on the extended manifold.
• Applies standard mollification and approximation techniques in local coordinates to approximate functions in $W^{1,p}$-spaces.
• Employs a smooth, globally Lipschitz exhaustion function $\rho_M$ on the original manifold to control approximation on compact subsets.
• Utilizes cutoff functions $\psi_k = \psi(\rho_N/k)$ with $\psi$ smooth and compactly supported to approximate functions in $C_c^\infty(N)$.
• Reduces the problem of extending Sobolev functions to the existence of a proper isometric embedding via extension of a complete Riemannian manifold.

Experimental results

Research questions

• RQ1 Can every smooth, metrically complete Riemannian manifold with smooth boundary be isometrically embedded as a closed domain in a geodesically complete Riemannian manifold without boundary?
• RQ2 Under what conditions can a Riemannian extension preserve curvature bounds such as $\mathrm{Ric} \geq C$ or $\mathrm{Sect} \leq C$?
• RQ3 What topological constraints arise in the original manifold that obstruct the existence of a complete Einstein extension with $\mathrm{Ric} = \lambda$?
• RQ4 Can Sobolev functions on a complete manifold with boundary be approximated by smooth, compactly supported functions in the $W^{1,p}$-norm?
• RQ5 Is it possible to achieve a proper isometric embedding of a complete Riemannian manifold with boundary into some Euclidean space $\mathbb{R}^\ell$?

Key findings

• Any smooth, metrically complete Riemannian manifold with smooth boundary admits a geodesically complete Riemannian extension without boundary.
• The space $W^{1,p}(\mathrm{int}M)$ is the closure of $C_c^\infty(M)$ under the $W^{1,p}$-norm, implying that compactly supported smooth functions are dense in Sobolev spaces on complete manifolds with boundary.
• A smooth, globally Lipschitz exhaustion function $\rho_M$ exists on any complete Riemannian manifold with boundary, satisfying $\|\nabla \rho_M\|_{L^\infty} \leq L$.
• The classical Nash embedding theorem can be extended to manifolds with boundary by first extending to a complete manifold and then embedding properly into $\mathbb{R}^\ell$.
• The construction ensures that the extended metric is smooth and geodesically complete, preserving the isometric embedding of the original manifold.
• The method provides a general framework for constructing complete extensions with controlled curvature and topological constraints, applicable to problems in geometric analysis and relativity.