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A finiteness principle for distance functions on Riemannian surfaces with Hölder continuous curvature

Rotem Assouline

Abstract

We study distance functions from geodesics to points on Riemannian surfaces with Hölder continuous Gauss curvature, and prove a finiteness principle in the spirit of Whitney extension theory for such functions. Our result can be viewed as a finiteness principle for isometric embedding of a certain type of metric spaces into Riemannian surfaces, with control over the Hölder seminorm of the Gauss curvature.

A finiteness principle for distance functions on Riemannian surfaces with Hölder continuous curvature

Abstract

We study distance functions from geodesics to points on Riemannian surfaces with Hölder continuous Gauss curvature, and prove a finiteness principle in the spirit of Whitney extension theory for such functions. Our result can be viewed as a finiteness principle for isometric embedding of a certain type of metric spaces into Riemannian surfaces, with control over the Hölder seminorm of the Gauss curvature.
Paper Structure (8 sections, 35 theorems, 175 equations, 2 figures)

This paper contains 8 sections, 35 theorems, 175 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $C^{2,\alpha}$ Riemannian surfaces
  4. Polar coordinates, the Riccati equation and curvature comparison
  5. Whitney's extension theorem
  6. Coefficients of $C^{2,\alpha}$ metrics in polar coordinates
  7. Properties of distance functions on $C^{2,\alpha}$ surfaces
  8. Proof of Theorem \ref{['mainthm']}

Key Result

Theorem 1

There exist universal constants $C_1, C_2$ such that the following holds: Let $\rho : I \to (0,C_1]$, where $I \subseteq \mathbb R$ is an open bounded interval. Assume that for every subset $S \subseteq I$ consisting of at most 12 points, there exists a $C^{2,\alpha}$ Riemannian surface $(M_S, g_S)$

Figures (2)

  • Figure 1: Proof of Lemma \ref{['phi0bound']}
  • Figure 2: The set $A_k'$.

Theorems & Definitions (61)

  • Theorem 1
  • Remark 1.1
  • Theorem 2
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: Kazdan and Deturck KD
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Corollary 2.7
  • ...and 51 more