[Paper Review] Compactness of Alexandrov-Nirenberg Surfaces

Qing‐Long Han, Jiaxing Hong+1 more|arXiv (Cornell University)|Feb 11, 2014

Geometric Analysis and Curvature Flows8 references3 citations

TL;DR

This paper establishes a compactness result for a class of compact surfaces in ℝ³ known as Alexandrov-Nirenberg surfaces, which are characterized by positive Gauss curvature in the interior, vanishing curvature with non-vanishing gradient on the boundary, and total curvature exactly 4π. Under uniform bounds on the induced metric, Gauss curvature gradient, and geodesic curvature of the boundary, the family of such surfaces of class $C^{k+3,\alpha}$ is precompact in the $C^k$-topology, providing a crucial step toward proving existence of isometric embeddings via the continuity method.

ABSTRACT

We study a class of compact surfaces in $\mathbb R^3$ introduced by Alexandrov and generalized by Nirenberg and prove a compactness result under suitable assumptions on induced metrics and Gauss curvatures.

Motivation & Objective

• To establish a compactness result for Alexandrov-Nirenberg surfaces, a class of compact surfaces in ℝ³ with specific curvature and boundary conditions.
• To provide a foundational step toward proving existence of isometric embeddings of metrics on compact surfaces via the continuity method.
• To control geometric quantities such as the induced metric, Gauss curvature, and geodesic curvature on the boundary to ensure sequential compactness in Hölder topology.
• To extend the rigidity results of Alexandrov and Nirenberg to a broader class of surfaces by proving that bounded sequences of such surfaces converge in $C^k$-topology.

Proposed method

• Define the class $\mathcal{S}_{J,k,\alpha,C}$ of Alexandrov-Nirenberg surfaces with $J$ boundary components, $C^{k+3,\alpha}$ regularity, and uniform bounds on the $C^{k+2,\alpha}$-norm of the metric, and reciprocal bounds on $|\nabla K|$ and $|k_g|$ on the boundary.
• Use a priori estimates for the position vector of the surface to control higher-order Hölder norms of the embedding.
• Apply elliptic regularity theory and weighted Sobolev estimates in a model problem on the half-plane $\mathbb{R}^2_+$ to derive $C^{\alpha}$-regularity of solutions to the relevant PDE system.
• Employ a bootstrapping argument using weighted Hölder norms and the maximum principle to upgrade regularity from $L^6$-integrability to $C^{\infty}$-smoothness in the model setting.
• Use cutoff functions and local coordinate charts to localize the problem and transfer regularity estimates from the model to the global surface.
• Leverage the structure of the Gauss curvature and boundary conditions to derive uniform control on geometric quantities, ensuring precompactness in $C^k$-topology.

Experimental results

Research questions

• RQ1 Under what geometric and analytic conditions is the family of Alexandrov-Nirenberg surfaces precompact in the $C^k$-topology?
• RQ2 Can uniform bounds on the induced metric, Gauss curvature gradient, and geodesic curvature on the boundary ensure sequential compactness of such surfaces?
• RQ3 How do the curvature and boundary conditions in Theorem A of Alexandrov and Nirenberg influence the regularity and convergence of sequences of surfaces?
• RQ4 To what extent can the compactness result be used to prove existence of isometric embeddings via the continuity method?
• RQ5 What role do weighted Hölder norms and elliptic estimates play in controlling the regularity of surfaces satisfying the Alexandrov-Nirenberg conditions?

Key findings

• The family $\mathcal{S}_{J,k,\alpha,C}$ of Alexandrov-Nirenberg surfaces of class $C^{k+3,\alpha}$ is precompact in the $C^k$-topology under the stated uniform bounds.
• Uniform control on $|g|_{C^{k+2,\alpha}}$, $1/|\nabla K|$, and $1/|k_g|$ on the boundary ensures that any sequence in $\mathcal{S}_{J,k,\alpha,C}$ has a convergent subsequence in $C^k$-norm.
• The proof relies on deriving a priori estimates for the position vector of the surface, which are then used to control the $C^{k,\alpha}$-norms of the embedding.
• The regularity of solutions to the model PDE on $\mathbb{R}^2_+$ is upgraded from $L^6$-integrability to $C^{\infty}$-smoothness via a bootstrapping argument using weighted Hölder norms.
• The maximum principle and Sobolev embedding are used to establish $C^{\alpha}$-regularity of the solution in the model problem, which is essential for the global regularity transfer.
• The result provides a key compactness tool for the continuity method in the global isometric embedding problem for surfaces with mixed-sign Gauss curvature.