[Paper Review] Embedded Weingarten tori in S^3

Simon Brendle|arXiv (Cornell University)|May 13, 2013

Geometric Analysis and Curvature Flows9 references4 citations

TL;DR

This paper proves that any embedded Weingarten torus in the 3-sphere $S^3$ satisfying a specific curvature relation $\lambda_1 + \lambda_2 = \psi(\lambda_1 - \lambda_2)$, where $\psi$ meets certain analytic conditions, must be rotationally symmetric. The proof adapts the maximum principle applied to a two-point function, extending techniques from Brendle's work on minimal and constant mean curvature tori to a broader class of Weingarten surfaces.

ABSTRACT

In this paper, we show that an embedded Weingarten surface in S^3 of genus 1 must be rotationally symmetric, provided that certain structure conditions are satisfied. The argument involves an adaptation of our proof of Lawson's Conjecture for minimal tori.

Motivation & Objective

• To establish rigidity for embedded Weingarten tori in $S^3$ under curvature relations generalizing minimal and constant mean curvature surfaces.
• To extend the maximum principle technique used in proving Lawson’s conjecture to a broader class of surfaces governed by a Weingarten equation.
• To show that under specific analytic conditions on the function $\psi$, such tori must be rotationally symmetric.
• To generalize previous rigidity results for minimal and constant mean curvature tori to the Weingarten setting.
• To demonstrate that the absence of umbilical points and the structure of the two-point function lead to symmetry via Killing vector fields.

Proposed method

• Adapts the two-point function method of Huisken and Andrews, using the maximum principle on a function $Z(x,y) = \Phi(x)(1 - \langle F(x), F(y) \rangle) + \langle \nu(x), F(y) \rangle \geq 0$.
• Employs conformal coordinates on the torus $\Sigma = \mathbb{C}/\Lambda$ to express curvature equations in terms of the Hopf differential $h_{zz}$.
• Proves that $h_{zz}$ has no zeros via a Hopf index count, implying no umbilical points, using the Beltrami-type PDE derived from the Weingarten equation.
• Uses the vanishing of the third derivative of a nonnegative function $f(t)$ to deduce $D_1\lambda_1 = D_1\lambda_2 = 0$, showing constancy of principal curvatures along one curvature line.
• Constructs a vector field $V = \varphi(\lambda_1 - \lambda_2) e_1$ with $\varphi' = -\frac{1 + \psi'(s)}{2s} \varphi(s)$, showing $[V, e_i] = 0$ and $\mathcal{L}_V g = \mathcal{L}_V h = 0$.
• Extends $V$ to a Killing vector field on $S^3$, proving existence of an anti-symmetric matrix $Q \in \mathfrak{so}(4)$ of rank 2 such that $V(x) = Q F(x)$, implying rotational symmetry.

Experimental results

Research questions

• RQ1 Can the rigidity result for minimal tori in $S^3$ be extended to a broader class of Weingarten surfaces?
• RQ2 Under what analytic conditions on $\psi$ does the Weingarten equation $\lambda_1 + \lambda_2 = \psi(\lambda_1 - \lambda_2)$ imply rotational symmetry for embedded tori?
• RQ3 Does the absence of umbilical points persist in Weingarten tori satisfying the curvature relation, as in the minimal case?
• RQ4 Can the two-point function maximum principle technique be adapted to prove symmetry for non-minimal, non-constant mean curvature Weingarten surfaces?
• RQ5 Is the symmetry of such tori characterized by the existence of a Killing vector field induced from the ambient $S^3$?

Key findings

• Any embedded Weingarten torus in $S^3$ satisfying $\lambda_1 + \lambda_2 = \psi(\lambda_1 - \lambda_2)$ with $\psi$ even and satisfying $0 \leq s\psi'(s) < \min\{\psi(s), s\}$ and $0 \leq s\psi''(s) \leq 1 - \psi'(s)^2$ must be rotationally symmetric.
• The function $\psi(s) = \sqrt{a + b s^2} + c$ for $a > 0$, $b \in [0,1]$, $c \geq 0$ satisfies the required conditions, so tori satisfying $(\lambda_1 + \lambda_2 - c)^2 = a + b(\lambda_1 - \lambda_2)^2$ with $\lambda_1 + \lambda_2 \geq c$ are rotationally symmetric.
• The absence of umbilical points is established via a Hopf index argument on the Hopf differential $h_{zz}$, which has no zeros due to the torus topology and the structure of the Beltrami-type PDE.
• The principal curvatures are constant along one family of curvature lines, as $D_1\lambda_1 = D_1\lambda_2 = 0$ at every point.
• The symmetry is realized via a Killing vector field $V(x) = Q F(x)$ for some $Q \in \mathfrak{so}(4)$ of rank 2, proving rotational symmetry of the surface.
• The vector field $V$ constructed from $\varphi$ and the principal frame satisfies $\mathcal{L}_V g = \mathcal{L}_V h = 0$, confirming it generates a one-parameter group of isometries preserving the surface.