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Free boundary CMC annuli in spherical and hyperbolic balls

Alberto Cerezo, Isabel Fernandez, Pablo Mira

Abstract

We construct, for any $H\in \mathbb{R}$, infinitely many free boundary annuli in geodesic balls of $\mathbb{S}^3$ with constant mean curvature $H$ and a discrete, non-rotational, symmetry group. Some of these free boundary CMC annuli are actually embedded if $H\geq 1/\sqrt{3}$. We also construct embedded, non-rotational, free boundary CMC annuli in geodesic balls of $\mathbb{H}^3$, for all values $H>1$ of the mean curvature $H$.

Free boundary CMC annuli in spherical and hyperbolic balls

Abstract

We construct, for any , infinitely many free boundary annuli in geodesic balls of with constant mean curvature and a discrete, non-rotational, symmetry group. Some of these free boundary CMC annuli are actually embedded if . We also construct embedded, non-rotational, free boundary CMC annuli in geodesic balls of , for all values of the mean curvature .
Paper Structure (31 sections, 27 theorems, 152 equations, 5 figures)

This paper contains 31 sections, 27 theorems, 152 equations, 5 figures.

Table of Contents

  1. Introduction and statement of the results
  2. Historical introduction
  3. Main results
  4. Organization of the paper and sketch of the proof
  5. Open problems
  6. Preliminaries
  7. Special solutions of the sinh-Gordon equation
  8. Wente's overdetermined system
  9. Constructions of special solutions to the system
  10. CMC surfaces with spherical curvature lines
  11. A special class of CMC surfaces with spherical curvature lines
  12. Study of the center map
  13. Study of the symmetries of
  14. The period map
  15. Construction of CMC annuli
  16. ...and 16 more sections

Key Result

Theorem 1.1

There exists an open interval $\mathcal{J}=\mathcal{J}(H,\varepsilon)$ contained in $(0,1)$ such that, for any irreducible $q=m/n\in \mathcal{J}\cap \mathbb{Q}$, there exists a real analytic $1$-parameter family $\mathcal{F}_q:=\{\mathbb{A}_q (\eta): \eta\in [0,\epsilon_0(q))\}$ of compact annuli in

Figures (5)

  • Figure 8.1: The parameter domain $\widehat{\mathcal{W}}$ in \ref{['ogra']} for the cases $\varepsilon = 1$ (left) and $\varepsilon = -1$ (right). In the right picture, the value $r_0$ stands for $r_0 = \frac{H + \mu}{2\mu}$.
  • Figure 8.2: Example of a curve $\Upsilon(r)$ in the conditions of Theorem \ref{['th:upsilon']}.
  • Figure 9.1: Level lines $\Upsilon$ of the period map on the parameter domain $(r_1,r_3)$ for the cases $\varepsilon = 1$ (left) and $\varepsilon = -1$ (right). If $8H^2 - \varepsilon > 0$, there are infinitely many level lines for which a segment of $\Upsilon$ satisfies the hypotheses of Theorem \ref{['th:upsilon']}.
  • Figure 9.2: The level line $\Upsilon$ and the line $L$ meet at a point $\Upsilon(\overline r) \in \widehat{\mathcal{W}}$.
  • Figure A.1: Integration path $C_n$ for $f(z)$.

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Theorem 3.1: CFM
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Definition 4.1
  • Remark 4.2
  • ...and 46 more