[Paper Review] On the manifold of closed hypersurfaces in R^n

Jan Pruess, Gieri Simonett|arXiv (Cornell University)|Dec 28, 2012

Geometric Analysis and Curvature Flows9 references21 citations

TL;DR

This paper develops a geometric toolbox for analyzing moving hypersurfaces in R^n using the direct mapping method, focusing on hypersurfaces defined via normal height functions over a reference surface. It establishes key differential geometric quantities—mean curvature, Weingarten tensor, surface gradient, and divergence—in terms of the height function and proves that the space of compact embedded hypersurfaces satisfying a uniform ball condition can be identified with a submanifold of C²(Ω̄), enabling compactness and embedding results crucial for studying evolution equations with moving interfaces.

ABSTRACT

Several results from differential geometry of hypersurfaces in R^n are derived to form a tool box for the direct mapping method. The latter technique has been widely employed to solve problems with moving interfaces, and to study the asymptotics of the induced semiflows.

Motivation & Objective

• To provide a comprehensive toolkit of differential geometric results for hypersurfaces in R^n, especially for problems involving moving interfaces.
• To formalize the manifold structure of compact embedded hypersurfaces in R^n, particularly under uniform ball conditions.
• To enable the application of maximal regularity theory to moving interface problems by identifying the space of hypersurfaces as a C²-manifold.
• To derive explicit representations of geometric quantities (mean curvature, Weingarten tensor, Laplace-Beltrami operator) in terms of height functions over a reference surface.
• To prove that the class of C² hypersurfaces satisfying a uniform ball condition can be embedded into C²(Ω̄), ensuring compactness and embedding properties for analysis.

Proposed method

• Parameterize perturbed hypersurfaces Γρ as normal graphs over a fixed reference surface Σ using a height function ρ ∈ C²(Σ, ℝ).
• Derive explicit formulas for principal curvatures, mean curvature, surface gradient, surface divergence, and Laplace-Beltrami operator on Γρ in terms of ρ and the geometry of Σ.
• Use the tubular neighborhood construction and a uniformly defined level function φΓ = g(dΓ(x)) with a fixed cut-off function χ to ensure uniform regularity across the family of hypersurfaces.
• Establish an injective, isomorphic embedding Φ: M²(Ω,r) → C²(Ω̄) by mapping each hypersurface to its level function, enabling topological and analytic control.
• Prove that convergence in the C²(Ω̄)-norm of level functions implies Hausdorff convergence of hypersurfaces and their normal bundles, linking function space convergence to geometric convergence.
• Apply Rellich’s theorem to show that Wˢᵖ(Ω,r) embeds compactly into W^σ_q(Ω,r) when s−n/p > σ−n/q and s > σ, ensuring compactness in Sobolev spaces.

Experimental results

Research questions

• RQ1 How can geometric quantities like mean curvature and surface divergence be expressed explicitly in terms of a height function ρ over a reference hypersurface Σ?
• RQ2 What is the manifold structure of the space of compact embedded C²-hypersurfaces in R^n under a uniform ball condition?
• RQ3 How can the direct mapping method be rigorously justified for moving interface problems by identifying the space of hypersurfaces with a Banach or C²-manifold?
• RQ4 What conditions ensure that convergence of level functions in C²(Ω̄) implies geometric convergence of the corresponding hypersurfaces and their normal bundles?
• RQ5 Under what conditions does the embedding of the hypersurface space into C²(Ω̄) preserve compactness and allow for embedding theorems in Sobolev spaces?

Key findings

• The mean curvature of a hypersurface Γρ defined as a normal graph over Σ satisfies the formula κ′(0) = tr(LΣ²) + ΔΣ, where LΣ is the Weingarten tensor and ΔΣ is the Laplace-Beltrami operator on Σ.
• The space M²(Ω,r) of compact embedded C²-hypersurfaces in a bounded domain Ω satisfying a uniform ball condition with radius r > 0 can be identified with a subset of C²(Ω̄), forming a C²-manifold.
• The embedding Φ: M²(Ω,r) → C²(Ω̄) defined by Γ ↦ φΓ is an isomorphism, and convergence in the C²(Ω̄)-norm of φΓ₁ and φΓ₂ implies Hausdorff convergence of the hypersurfaces and their normal bundles.
• For s−(n−1)/p > 2, the space Wˢᵖ(Ω,r) of hypersurfaces with level functions in Wˢᵖ(Ω) inherits a manifold structure, and compactness holds when s−n/p > σ−n/q and s > σ.
• The construction of a uniformly defined level function φΓ using a fixed cut-off function χ ensures that the tubular neighborhood width exceeds r/2, enabling uniform estimates across the family M²(Ω,r).
• The direct mapping method is applicable to moving interface problems even for C²-hypersurfaces by approximating them with real analytic hypersurfaces in the second normal bundle, thus preserving regularity.