Some structure theorems for Weingarten surfaces

TL;DR

This work extends symmetry and containment results known for constant mean curvature surfaces to the broader class of elliptic Weingarten surfaces with boundary on a planar curve. By leveraging the Alexandrov reflection principle and the monotonicity of the Alexandrov function, the authors establish conditions under which the boundary symmetries of $C$ propagate to the entire surface, including the rotational end scenario when $C$ is a circle. In the linear Weingarten case, they develop a flux-balancing framework and define end mass via convergence to W-Delaunay surfaces, yielding containment in a halfspace or strong structural restrictions on ends, and in balanced mass cases, compactness or rotational end behavior. These results generalize Rosenberg–Sa Earp’s cmc findings to Weingarten surfaces and provide a unified approach to symmetry, end behavior, and end-counting via flux formulas in the elliptic setting.

Abstract

Let $M\subset\mathbb{R}^3$ be a properly embedded, connected, complete surface with boundary a convex planar curve $C$, satisfying an elliptic equation $H=f(H^2-K)$, where $H$ and $K$ are the mean and the Gauss curvature respectively - which we will refer to as Weingarten equation. When $M$ is contained in one of the two halfspaces determined by $C$, we give sufficient conditions for $M$ to inherit the symmetries of $C$. In particular, when $M$ is vertically cylindrically bounded, we get that $M$ is rotational if $C$ is a circle. In the case in which the Weingarten equation is linear, we give a sufficient condition for such a surface to be contained in a halfspace. Both results are generalizations of results of Rosenberg and Sa Earp, for constant mean curvature surfaces, to the Weingarten setting. In particular, our results also recover and generalize the constant mean curvature case.

Figures (7)

• Figure 1: The compact surface with boundary $\tilde{M}$.
• Figure 2: The loops $\gamma_{i}$ and $\gamma_{j}$, with $\vec{H}$ along $\gamma_i$ indicated.
• Figure 3: Constructing a closed surface starting from $S$.
• Figure 4: The construction of $D_0$.
• Figure 5: The Alexandrov procedure.

Theorems & Definitions (17)

• Lemma 1: Korevaar1989TheSO
• Theorem 3.1
• proof
• Theorem 2: saearpweingarten, espinar
• Definition 4.1
• Theorem 3: saearpweingarten,Korevaar1989TheSO
• Lemma 4.2
• proof
• Lemma 4.3
• proof