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Energy quantization for constrained Willmore surfaces

Christian Scharrer, Alexander West

TL;DR

This work proves energy quantization and strong compactness for constrained Willmore surfaces under fixed area, volume, and total mean curvature, with the conformal structures remaining bounded. The authors develop an $\,\varepsilon$-regularity theory, derive a conservative system for the constrained equation, and establish Lorentz-type and $L^{p}$-type neck estimates to rule out energy loss in neck regions. They implement a bubble-neck decomposition to show that the total Willmore energy splits into a limiting surface plus a finite collection of bubbles, with ends leading to unconstrained Willmore surfaces under inversion. As a corollary, minimizers of the isoperimetric and total mean curvature problems are compact in $C^l$ for all $l$, and the Lagrange multipliers remain bounded, ensuring stability and robustness of constrained Willmore minimizers in geometric variational problems.

Abstract

We establish an energy quantization for constrained Willmore surfaces, where the constraints are given by area, volume, and total mean curvature, assuming that the underlying conformal structures remain bounded. Furthermore, we show strong compactness of constrained Willmore surfaces under some energy threshold, proving in particular the strong compactness of minimizers of two previously studied problems.

Energy quantization for constrained Willmore surfaces

TL;DR

This work proves energy quantization and strong compactness for constrained Willmore surfaces under fixed area, volume, and total mean curvature, with the conformal structures remaining bounded. The authors develop an -regularity theory, derive a conservative system for the constrained equation, and establish Lorentz-type and -type neck estimates to rule out energy loss in neck regions. They implement a bubble-neck decomposition to show that the total Willmore energy splits into a limiting surface plus a finite collection of bubbles, with ends leading to unconstrained Willmore surfaces under inversion. As a corollary, minimizers of the isoperimetric and total mean curvature problems are compact in for all , and the Lagrange multipliers remain bounded, ensuring stability and robustness of constrained Willmore minimizers in geometric variational problems.

Abstract

We establish an energy quantization for constrained Willmore surfaces, where the constraints are given by area, volume, and total mean curvature, assuming that the underlying conformal structures remain bounded. Furthermore, we show strong compactness of constrained Willmore surfaces under some energy threshold, proving in particular the strong compactness of minimizers of two previously studied problems.

Paper Structure

This paper contains 15 sections, 16 theorems, 334 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. Constrained Willmore surfaces
  4. Lorentz spaces
  5. epsilon-regularity for constrained Willmore surfaces
  6. Energy quantization for constrained Willmore surfaces
  7. Conservative system for constrained equation
  8. Lp-estimate for the conformal parameters
  9. Lorentz-type estimates on annuli
  10. Improved estimates on the mean curvature in neck regions
  11. L2-weak quantization of the Gauss map in neck regions
  12. Proof of main results
  13. Bounding the Lagrange multipliers
  14. The bubble-neck decomposition
  15. Derivation of conservative system

Key Result

Theorem 1.1

There exists $\varepsilon_0>0$ such that for any conformal, constrained Willmore immersion $\vec{\Phi}:B_1\to \mathbb{R}^3$ with coefficients $\alpha$, $\beta$, $\gamma$ in the sense of intro:EL equation such that it holds for any $r\in (0,1)$ for all integers $l\geq 1$. Here, $\lambda$ is the conformal factor of $\vec{\Phi}$ and $\overline{\lambda} = \frac{1}{|B_{1}|} \int _{B_{1}} \lambda \,\ma

Figures (3)

  • Figure 1: An example of the images of neck and bubble regions. The left sphere $\Phi_k(B_{\alpha^{-1} \rho^{i,1}_k}(x_{k}^{i,1}))$ and the catenoidal bridge $\Phi_k(B_{\alpha ^{-1} \rho^{i,2}_k}(x_{k}^{i,1})\setminus B_{\alpha \rho^{i,2}_k}(x_{k}^{i,1}))$ are the two resulting bubbles, connected by two neck regions (red) to the main immersion, which immerses the sphere on the right.
  • Figure 2: The three different cases belong to three different types of neck regions. The case $d>-1+\delta$ corresponds to the case when the inner end of the annulus carries no area compared to the outer end. The case $d<-1-\delta$ is the reversed situation, i.e., the outer end of the annulus carries no area compared to the inner end. The case $d\approx -1$ occurs when the area is equidistributed throughout the annulus.
  • Figure 3: The construction of the bubbles.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2: Energy quantization for constrained Willmore surfaces
  • Theorem 1.3
  • Theorem 1.4: Compactness of constrained Willmore surfaces
  • Corollary 1.5: See \ref{['thm:Compactness of constrained Willmore surfaces']}, \ref{['lem:boundedness of Lagrange multiplier for minimizers']}, and \ref{['rem:iso lagrange multipliers remain bounded']}
  • Remark 3.1
  • proof : Proof of \ref{['thm:eps regularity']}
  • Remark 3.2
  • Remark 4.1
  • Lemma 4.2: See MichelatPointwiseExpansion
  • ...and 25 more