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Special Weingarten surfaces with planar convex boundary

Barbara Nelli, Giuseppe Pipoli, Marcos Paulo Tassi

Abstract

We prove a Ros-Rosenberg theorem in the setting of Special Weingarten surfaces. We show that a compact, connected, embedded, Special Weingarten surface in $\mathhb{R}^3$ with planar convex boundary is a topological disk under mild suitable assumptions.

Special Weingarten surfaces with planar convex boundary

Abstract

We prove a Ros-Rosenberg theorem in the setting of Special Weingarten surfaces. We show that a compact, connected, embedded, Special Weingarten surface in with planar convex boundary is a topological disk under mild suitable assumptions.

Paper Structure

This paper contains 8 sections, 7 theorems, 44 equations, 7 figures.

Table of Contents

  1. Preliminaries on Special Weingarten Surfaces
  2. The curvature diagram and the definition of SW-surfaces
  3. An equivalence relation among Special Weingarten surfaces
  4. The tangency principle for Special Weingarten surfaces
  5. SW-catenoids
  6. The limaçon of Pascal
  7. The graph Lemma
  8. Two Ros-Rosenberg type theorems for SW-surfaces

Key Result

Theorem 1

RoRo Let $\Gamma \subset P$ be a strictly convex curve. There is an $H(\Gamma) > 0$, depending only on the extreme values of the curvature of $\Gamma$, such that whenever $M \subset \mathbb{R}^3_{+}$ is asurface with constant mean curvature $H$ bounded by $\Gamma$, with $0 < H < H(\Gamma)$, then $M$

Figures (7)

  • Figure 1: on the left, the curvature diagram of a set of SW-surfaces; on the right, curvature diagrams of classical sets of SW-surfaces.
  • Figure 2: on the left, the connected components of $\kappa_1 + \kappa_2 - \kappa_1\kappa_2 = 0$; on the right, curvature diagrams of SW-surfaces of minimal-type, CMC-type and CGC-type.
  • Figure 3: on the left, the truncated catenoid $G^\ast$ of height $h^\ast$ and neck-size $r_0$; on the right, the curvature diagram of $G$.
  • Figure 4: on the left, the construction of the limaçon of Pascal for $a=5$ and $c=2$; on the right, the limaçon of Pascal and the disks described in Lemma \ref{['lemma2_lima']}.
  • Figure 5: on the left, the disk $D((c_\Gamma,h^\ast),R_1)$; on the right, the open region $\Omega(t)$.
  • ...and 2 more figures

Theorems & Definitions (25)

  • Theorem
  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Lemma 4.1
  • ...and 15 more