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Corrections to: "The Willmore flow of Hopf-tori in the 3-sphere"

Ruben Jakob

Abstract

This erratum addresses a logical mistake in the author's article [Jakob, R. The Willmore flow of Hopf-tori in the $3$-sphere. Journal of Evolution Equations 23, No. 72 (2023)] which resulted in two wrong assertions in parts (II) and (III) of Theorem 1 in the author's cited paper and in an inaccuracy in the formulation of the second part of Proposition 6 in that paper. We will not only point out these mistakes and their corrections, but we will additionally give a concrete counterexample to Statement (III) in Theorem 1 of the author's cited article which will automatically imply the optimality of the corrected version of that statement, as it is formulated in this erratum.

Corrections to: "The Willmore flow of Hopf-tori in the 3-sphere"

Abstract

This erratum addresses a logical mistake in the author's article [Jakob, R. The Willmore flow of Hopf-tori in the -sphere. Journal of Evolution Equations 23, No. 72 (2023)] which resulted in two wrong assertions in parts (II) and (III) of Theorem 1 in the author's cited paper and in an inaccuracy in the formulation of the second part of Proposition 6 in that paper. We will not only point out these mistakes and their corrections, but we will additionally give a concrete counterexample to Statement (III) in Theorem 1 of the author's cited article which will automatically imply the optimality of the corrected version of that statement, as it is formulated in this erratum.
Paper Structure (2 sections, 4 theorems, 16 equations)

This paper contains 2 sections, 4 theorems, 16 equations.

Table of Contents

  1. Corrections of mistakes in Ruben.Hopf.Willmore.2023
  2. Optimality of the initial energy-threshold "$4 \pi^2$" in Theorem \ref{['corrected.StatementIII.Theorem.1']}

Key Result

Theorem 1.1

Let $F_0:\Sigma \longrightarrow {\mathbb S}^3$ be a smooth immersion of a smooth compact torus $\Sigma$ into ${\mathbb S}^3$ which parametrizes a Hopf-torus with Willmore energy ${\mathcal{W} }(F_0) < 4\pi^2$. Then there is a smooth family of smooth diffeomorphisms $\Psi_t:\Sigma \longrightarrow \Si

Theorems & Definitions (5)

  • Definition 1.1
  • Theorem 1.1: Corrected Statement (III) in Theorem 1 of Ruben.Hopf.Willmore.2023
  • Theorem 1.2: Alternative revised version of Statement (III) in Theorem 1 of Ruben.Hopf.Willmore.2023
  • Theorem 2.1
  • Theorem 2.2