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On the minimization of the Willmore energy under a constraint on total mean curvature and area

Christian Scharrer, Alexander West

Abstract

Motivated by a model for lipid bilayer cell membranes, we study the minimization of the Willmore functional in the class of oriented closed surfaces with prescribed total mean curvature, prescribed area, and prescribed genus. Adapting methods previously developed by Keller-Mondino-Rivière, Bauer-Kuwert, and Ndiaye-Schätzle, we prove existence of smooth minimizers for a large class of constraints. Moreover, we analyze the asymptotic behaviour of the energy profile close to the unit sphere and consider the total mean curvature of axisymmetric surfaces.

On the minimization of the Willmore energy under a constraint on total mean curvature and area

Abstract

Motivated by a model for lipid bilayer cell membranes, we study the minimization of the Willmore functional in the class of oriented closed surfaces with prescribed total mean curvature, prescribed area, and prescribed genus. Adapting methods previously developed by Keller-Mondino-Rivière, Bauer-Kuwert, and Ndiaye-Schätzle, we prove existence of smooth minimizers for a large class of constraints. Moreover, we analyze the asymptotic behaviour of the energy profile close to the unit sphere and consider the total mean curvature of axisymmetric surfaces.
Paper Structure (27 sections, 35 theorems, 323 equations, 4 figures)

This paper contains 27 sections, 35 theorems, 323 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The unconstrained problem
  3. The isoperimetrically constrained problem
  4. The total mean curvature constrained problem
  5. Challenges and open questions
  6. Preliminaries
  7. Definitions from differential geometry
  8. The constrained minimization problem
  9. Monotonicity of and variations of
  10. Monotonicity of
  11. Variations of
  12. Existence of minimizers of the Willmore energy under the total mean curvature constraint
  13. Weak immersions
  14. Existence of minimizers for spherical immersions
  15. The general case
  16. ...and 12 more sections

Key Result

Theorem 1.1

Suppose that $\Sigma$ is a closed, orientable and connected surface of genus $g$ and let $\mathcal{E}_\Sigma$ be the class of Lipschitz immersions defined in Section subsec:WeakImmersions. Consider the set Then, for any $R\in I$ there exists a smooth embedding $f^g_R$ of $\Sigma$ into $\mathbb{R}^3$, with $\mathcal{T}(f^g_R) = R$ and i.e., $f_R^g$ minimizes the Willmore energy among all Lipschit

Figures (4)

  • Figure 1: Intuitively, the map $\xi^1_k$ cuts away the portion of the surface with the topology and glues in a portion of a sphere, while $\xi^2_k$ cuts away the bubble and glues in an almost complete sphere while keeping the topology.
  • Figure 2: The graphic (not to scale) illustrates the 1-dimensional profile curve of the connected sum construction from this section. The dotted curves denote $q_{1/\beta}$ and $p_{\alpha}^\circ$. In the region $\gamma-\sqrt{\alpha}<r<\gamma$ respectively $1<r<1+\sqrt{\alpha}$, the interpolation between $u_\alpha^\circ$ and $p_\alpha^\circ$ respectively $q_{1/\beta}$ and $v_{1/\beta}$ is depicted in red. Finally, in the region $\gamma<r<1$, the biharmonic solution $w$ with the corresponding boundary condition is shown in blue.
  • Figure 3: A curve with $\mathop{\mathrm{Im}}\nolimits \theta = [0,2\pi]$ defining a surface with total mean curvature ratio converging to $-\infty$.
  • Figure 4: A curve with $\mathop{\mathrm{Im}}\nolimits \theta = \left [-\frac{\pi}{2}-\varepsilon, \frac{3\pi}{2}+\varepsilon \right ]$ defining a surface with negative total mean curvature.

Theorems & Definitions (65)

  • Theorem 1.1: See Theorem \ref{['thm:KMR']}, Corollary \ref{['cor:Monotonicity']}, Corollary \ref{['cor:Continuity at sphere']}
  • Theorem 1.2: See Theorem \ref{['thm:connectedsum']}
  • Proposition 1.3: See Proposition \ref{['prop:8piconstruction']}
  • Theorem 1.4: See Theorem \ref{['thm:asymptotic estimate at T(S2)']}
  • Lemma 2.1
  • proof
  • Definition 3.1
  • Proposition 3.2: Chen1974
  • Lemma 3.3
  • proof
  • ...and 55 more