Connected surfaces with boundary minimizing the Willmore energy

Matteo Novaga; Marco Pozzetta; Matteo Novaga; Marco Pozzetta

doi:10.3934/mine.2020024

Mathematics in Engineering

2020, Volume 2, Issue 3: 527-556. doi: 10.3934/mine.2020024

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Connected surfaces with boundary minimizing the Willmore energy

Matteo Novaga ^,,
Marco Pozzetta

Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

Received: 02 October 2019 Accepted: 17 March 2020 Published: 23 March 2020

For a given family of smooth closed curves $\gamma^1, ..., \gamma^\alpha\subset\mathbb{R}^3$ we consider the problem of finding an elastic connected compact surface $M$ with boundary $\gamma = \gamma^1\cup...\cup\gamma^\alpha$. This is realized by minimizing the Willmore energy $\mathcal{W}$ on a suitable class of competitors. While the direct minimization of the Area functional may lead to limits that are disconnected, we prove that, if the infimum of the problem is $ < 4\pi$, there exists a connected compact minimizer of $\mathcal{W}$ in the class of integer rectifiable curvature varifolds with the assigned boundary conditions. This is done by proving that varifold convergence of bounded varifolds with boundary with uniformly bounded Willmore energy implies the convergence of their supports in Hausdorff distance. Hence, in the cases in which a small perturbation of the boundary conditions causes the non-existence of Area-minimizing connected surfaces, our minimization process models the existence of optimal elastic connected compact generalized surfaces with such boundary data. We also study the asymptotic regime in which the diameter of the optimal connected surfaces is arbitrarily large. Under suitable boundedness assumptions, we show that rescalings of such surfaces converge to round spheres. The study of both the perturbative and the asymptotic regime is motivated by the remarkable case of elastic surfaces connecting two parallel circles located at any possible distance one from the other. The main tool we use is the monotonicity formula for curvature varifolds (^[15,31]) that we extend to varifolds with boundary, together with its consequences on the structure of varifolds with bounded Willmore energy.
- Willmore energy,
- monotonicity formula,
- varifolds,
- connectedness,
- existence
Citation: Matteo Novaga, Marco Pozzetta. Connected surfaces with boundary minimizing the Willmore energy[J]. Mathematics in Engineering, 2020, 2(3): 527-556. doi: 10.3934/mine.2020024

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Abstract

For a given family of smooth closed curves $\gamma^1, ..., \gamma^\alpha\subset\mathbb{R}^3$ we consider the problem of finding an elastic connected compact surface $M$ with boundary $\gamma = \gamma^1\cup...\cup\gamma^\alpha$. This is realized by minimizing the Willmore energy $\mathcal{W}$ on a suitable class of competitors. While the direct minimization of the Area functional may lead to limits that are disconnected, we prove that, if the infimum of the problem is $ < 4\pi$, there exists a connected compact minimizer of $\mathcal{W}$ in the class of integer rectifiable curvature varifolds with the assigned boundary conditions. This is done by proving that varifold convergence of bounded varifolds with boundary with uniformly bounded Willmore energy implies the convergence of their supports in Hausdorff distance. Hence, in the cases in which a small perturbation of the boundary conditions causes the non-existence of Area-minimizing connected surfaces, our minimization process models the existence of optimal elastic connected compact generalized surfaces with such boundary data. We also study the asymptotic regime in which the diameter of the optimal connected surfaces is arbitrarily large. Under suitable boundedness assumptions, we show that rescalings of such surfaces converge to round spheres. The study of both the perturbative and the asymptotic regime is motivated by the remarkable case of elastic surfaces connecting two parallel circles located at any possible distance one from the other. The main tool we use is the monotonicity formula for curvature varifolds (^[15,31]) that we extend to varifolds with boundary, together with its consequences on the structure of varifolds with bounded Willmore energy.

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