Export file:

Format

• RIS(for EndNote,Reference Manager,ProCite)
• BibTex
• Text

Content

• Citation Only
• Citation and Abstract

Connected surfaces with boundary minimizing the Willmore energy

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

This contribution is part of the Special Issue: Variational Models in Elasticity
Guest Editors: Lucia De Luca; Marcello Ponsiglione

Special Issues: Variational Models in Elasticity

## Abstract    Full Text(HTML)    Figure/Table    Related pages

For a given family of smooth closed curves $γ^1,...,γ^\alpha⊂\mathbb{R}^3$ we consider the problem of finding an elastic connected compact surface $M$ with boundary $γ=γ^1\cup...\cupγ^\alpha$. 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.
Figure/Table
Supplementary
Article Metrics

Citation: Matteo Novaga, Marco Pozzetta. Connected surfaces with boundary minimizing the Willmore energy. Mathematics in Engineering, 2020, 2(3): 527-556. doi: 10.3934/mine.2020024

References

• 1. Alessandroni R, Kuwert E (2016) Local solutions to a free boundary problem for the Willmore functional. Calc Var Partial Dif 55: 1-29.
• 2. Bauer M, Kuwert E (2003) Existence of minimizing Willmore surfaces of prescribed genus. Int Math Res Notices 10: 553-576.
• 3. Bergner M, Dall'Acqua A, Fröhlich S (2010) Symmetric Willmore surfaces of revolution satisfying natural boundary conditions. Calc Var Partial Dif 39: 361-378.
• 4. Bergner M, Dall'Acqua A, Fröhlich S (2013) Willmore surfaces of revolution with two prescribed boundary circles. J Geom Anal 23: 283-302.
• 5. Bergner M, Jakob R (2014) Sufficient conditions for Willmore immersions in $\mathbb{R}^3$ to be minimal surfaces. Ann Glob Anal Geom 45: 129-146.
• 6. Dall'Acqua A, Deckelnick K, Grunau H (2008) Classical solutions to the Dirichlet problem for Willmore surfaces of revolution. Adv Calc Var 1: 379-397.
• 7. Dall'Acqua A, Deckelnick K, Wheeler G (2013) Unstable Willmore surfaces of revolution subject to natural boundary conditions. Calc Var Partial Dif 48: 293-313.
• 8. Dall'Acqua A, Fröhlich S, Grunau H, et al. (2011) Symmetric Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data. Adv Calc Var 4: 1-81.
• 9. Deckelnick K, Grunau H (2009) A Navier boundary value problem for Willmore surfaces of revolution. Analysis 29: 229-258.
• 10. Eichmann S (2016) Nonuniqueness for Willmore surfaces of revolution satisfying Dirichlet boundary data. J Geom Anal 26: 2563-2590.
• 11. Eichmann S (2019) The Helfrich boundary value problem. Calc Var Partial Dif 58: 1-26.
• 12. Elliott CM, Fritz H, Hobbs G (2017) Small deformations of Helfrich energy minimising surfaces with applications to biomembranes. Math Mod Meth Appl Sci 27: 1547-1586.
• 13. Gazzola F, Grunau H, Sweers G (2010) Polyharmonic boundary value problems. Lect Notes Math 1991: xviii+423.
• 14. Hutchinson J (1986) Second fundamental form for varifolds and the existence of surfaces minimizing curvature. Indiana U Math J 35: 45-71.
• 15. Kuwert E, Schätzle R (2004) Removability of point singularities of Willmore surfaces. Ann Math 160: 315-357.
• 16. Li P, Yau ST (1982) A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue on compact surfaces. Invent Math 69: 269-291.
• 17. Mandel R (2018) Explicit formulas, symmetry and symmetry breaking for Willmore surfaces of revolution. Ann Glob Anal Geom 54: 187-236.
• 18. Mantegazza C (1996) Curvature varifolds with boundary. J Differ Geom 43: 807-843.
• 19. Marques FC, Neves A (2014) Min-Max theory and the Willmore Conjecture. Ann Math 179: 683-782.
• 20. Morgan F (2008) Geometric Measure Theory: A Beginners's Guide, 4 Eds., Academic Press.
• 21. Pozzetta M (2017) Confined Willmore energy and the Area functional. arXiv:1710.07133.
• 22. Pozzetta M (2018) On the Plateau-Douglas problem for the Willmore energy of surfaces with planar boundary curves. arXiv:1810.07662.
• 23. Rivière T (2008) Analysis aspects of Willmore surfaces. Invent Math 174: 1-45.
• 24. Rivière T (2013) Lipschitz conformal immersions from degenerating Riemann surfaces with L2-bounded second fundamental forms. Adv Calc Var 6: 1-31.
• 25. Rivière T (2014) Variational principles for immersed surfaces with L2-bounded second fundamental form. J Reine Angew Math 695: 41-98.
• 26. Schätzle R (2010) The Willmore boundary problem. Calc Var 37: 275-302.
• 27. Schoen R (1983) Uniqueness, symmetry, and embeddedness of minimal surfaces. J. Differ Geom 18: 791-809.
• 28. Schygulla J (2012) Willmore minimizers with prescribed isoperimetric ratio. Arch Ration Mech Anal 203: 901-941.
• 29. Seguin B, Fried E (2014) Microphysical derivation of the Canham-Helfrich free-energy density. J Math Biol 68: 647-665.
• 30. Simon L (1984) Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis of Australian National University.
• 31. Simon L (1993) Existence of surfaces minimizing the Willmore functional. Commun Anal Geom 1: 281-326.
• 32. Willmore TJ (1965) Note on embedded surfaces. Ann Al Cuza Univ Sect I 11B: 493-496.
• 33. Willmore TJ (1993) Riemannian Geometry, Oxford Science Publications.