Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Analysis of Willmore Surfaces and its Generalizations in Higher Dimensions

Tian Lan, Dorian Martino, Tristan Rivière

Abstract

We review recent progress concerning the analysis of Lagrangians on immersions into $\mathbb{R}^d$ depending on the first and second fundamental forms and their covariant derivatives.

The Analysis of Willmore Surfaces and its Generalizations in Higher Dimensions

Abstract

We review recent progress concerning the analysis of Lagrangians on immersions into depending on the first and second fundamental forms and their covariant derivatives.

Paper Structure

This paper contains 37 sections, 47 theorems, 449 equations.

Table of Contents

  1. Introduction
  2. Structure of the article.
  3. Acknowledgments.
  4. Open questions
  5. In dimension 2
  6. Construction of Willmore surfaces
  7. Conformal Gauss map approach
  8. Willmore flow
  9. Critical points of the Willmore energy under various constraints
  10. In higher dimensions
  11. Graham--Reichert functional
  12. Alternatives
  13. Analysis in high dimension
  14. Preliminaries
  15. Notation
  16. ...and 22 more sections

Key Result

Theorem I.1

Let $\vec{\Phi}$ be a $W^{2,2}$ weak immersion defined on a smooth closed orientable surface $\Sigma$ then there exists a constant Gauss curvature metric $h$ on $\Sigma$ and a function $\alpha\in C^0\cap W^{1,2}(\Sigma,{\mathbb R})$ such that Moreover $\alpha$ satisfies the Liouville equation The Gauss--Bonnet theorem holds, denoting $\gamma(\Sigma)$ is the genus of $\Sigma$,

Theorems & Definitions (93)

  • Theorem I.1
  • Theorem I.2
  • Theorem I.3
  • Conjecture II.2: Willmore conjecture in codimension 1 kusner1989
  • Conjecture II.3: 16$\pi$-conjecture
  • Remark II.23
  • Theorem II.40
  • Theorem II.41
  • Lemma III.1
  • proof
  • ...and 83 more