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A Table Theorem for Surfaces with Odd Euler Characteristic

Ali Naseri Sadr

Abstract

We use the square peg problem for smooth curves to prove a generalized table Theorem for real valued functions on Riemannian surfaces with odd Euler characteristic. We then use this result to prove the table conjecture for even functions on the two sphere.

A Table Theorem for Surfaces with Odd Euler Characteristic

Abstract

We use the square peg problem for smooth curves to prove a generalized table Theorem for real valued functions on Riemannian surfaces with odd Euler characteristic. We then use this result to prove the table conjecture for even functions on the two sphere.

Paper Structure

This paper contains 6 sections, 23 theorems, 34 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Tables on Riemannian Surfaces
  3. A Submersion
  4. Star-Shaped Squares
  5. Proof of the Main Theorem
  6. Square Peg for Star-Shaped Curves

Key Result

Theorem 1.1

Let $(\Sigma,g)$ be a Riemannian surface with $\chi(\Sigma)$ odd and $f$ a continuous real valued function on it. Then for every $d>0$, $f$ admits a table with diameter $d$.

Figures (3)

  • Figure 1: The four regions that cannot contain the origin
  • Figure 2: Line $l$ and origin $o$
  • Figure 3: The blue curve is our star-shaped curve.

Theorems & Definitions (62)

  • Theorem 1.1: Table Theorem for Surfaces with Odd Euler Characteristic
  • Corollary 1.2
  • Definition
  • Definition
  • Remark
  • Conjecture 2.1: The Table Problem for $S^2$
  • Remark
  • Definition
  • Remark
  • Lemma 3.1
  • ...and 52 more