Abstract
We are interested in surfaces with boundary satisfying a sixth order non-linear elliptic partial differential equation associated with extremal surfaces of the L2-norm of the gradient of the mean curvature. We show that such surfaces satis-fying so-called ‘flat boundary conditions’ and small L2-norm of the second fundamental form are necessarily planar.
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References
Y. Bernard, G. Wheeler, V.-M. Wheeler, Spherocytosis and the Helfrich model, Interfaces And Free Boundaries, 19 (2017) no. 4, 495–523.
E. Kuwert, R. Schätzle, The Willmore flow with small initial energy, J. Differential Geom. 57 (2001), 409–441.
J. McCoy, S. Parkins, G. Wheeler, The geometric triharmonic heat flow of immersed surfaces near spheres, Nonlinear Anal. 161 (2017), 44–86.
J. H. Michael, L. Simon, Sobolev and mean-value inequalities on generalized submanifolds of \( {\mathbb{R}}^{n} \), Comm. Pure Appl. Math. 26 (1973), 361–379.
J. McCoy, G. Wheeler, A classification theorem for Helfrich surfaces, Math. Ann. 357 (2013) no. 4, 1485–1508.
S. Parkins, A selection of higher-order parabolic curvature flows, PhD thesis, Uni. Wollongong, 2017.
G. Wheeler, Surface diffusion flow near spheres, Calc. Var. 44 (2012), no. 1–2, 131–151.
G. Wheeler, Chen’s conjecture and \( \varepsilon \)-superbiharmonic submanifolds of Riemannian manifolds, Int. J. Math. 24 (2013) no. 4, 1350028.
G. Wheeler, Gap phenomena for a class of fourth-order geometric differential operators on surfaces with boundary, Proc. Amer. Math. Soc. 143 (2014) no. 4.
Acknowledgements
Research supported by the Australian Research Council DP150100375 and DP180100431. The authors also acknowledge Benjamin Maldon (University of Newcastle) for assistance with typesetting through the University of Newcastle Priority Research Centre for Computer-Assisted Research Mathematics and its Applications (CARMA).
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McCoy, J., Wheeler, G. (2020). A Rigidity Theorem for Ideal Surfaces with Flat Boundary. In: de Gier, J., Praeger, C., Tao, T. (eds) 2018 MATRIX Annals. MATRIX Book Series, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-030-38230-8_19
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DOI: https://doi.org/10.1007/978-3-030-38230-8_19
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