Abstract
In this paper, we study the minimizers of curvature functionals (Willmore functional and extrinsic energy functional) subject to an area constraint in asymptotically flat manifolds. Under some certain conditions, we prove that such minimizers exist. Besides the surface theory related to the Willmore functional, the proofs also rely on the inverse mean curvature flow developed by Huisken and Ilmanen, on the positive mass theorem due to Schoen–Yau, and on the positive mass theorem for asymptotically flat manifolds with corners due to Miao and Shi–Tam. Our results may be of some interest in the General Relativity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnowitt, R., Deser, S., Misner, C.: Coordinate invariance and energy expressionsin general relativity. Phys. Rev. 122, 997–1006 (1961)
Bangert, V., Kuwert, E.: An area bound for surfaces in Riemannian manifolds.arXiv:1907.03457
Bauer, M., Kuwert, E.: Existence of minimizing Willmore surfaces of prescribed genus. Int. Math. Res. Not. 2003, 553–576 (2003)
Brendle, S.: Constant mean curvature surfaces in warped product manifolds. Publ. Math. Inst. Hautes Études Sci. 117, 247–269 (2013)
Chen, J., Li, Y.: Bubble tree of branched conformal immersions and applications to the Willmore functional. Am. J. Math. 136, 1107–1154 (2014)
Choi, H.I., Schoen, R.: The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature. Invent. Math. 81(3), 387–394 (1985)
Fan, X., Shi, Y., Tam, L.: Large-sphere and small-sphere limits of the Brown–York mass. Commun. Anal. Geom. 17, 37–72 (2009)
Hélein, F.: Harmonic Maps, Conservation Laws and Moving Frames, Cambridge Tracts in Mathematics, 150 Second Edition, Translated from the 1996 French original. Cambridge University Press, Cambridge, With a foreword by James Eells (2002)
Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59, 353–437 (2001)
Huisken, G., Yau, S.: Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Invent. Math. 124, 281–311 (1996)
Koerber, T.: The area preserving willmore flow and local maximizers of the hawking mass in asymptotically Schwarzschild manifolds. J. Geom. Anal. (2020). https://doi.org/10.1007/s12220-020-00401-6
Kuwert, E., Li, Y.: \(W^{2,2}\)-conformal immersions of a closed Riemann surface into \({\mathbb{R}}^n\). Commun. Anal. Geom. 20, 313–340 (2012)
Kuwert, E., Mondino, A., Schygulla, J.: Existence of immersed spheres minimizing curvature functionals in compact 3-manifold. Math. Ann. 359, 379–425 (2014)
Lamm, T., Metzger, J.: Small surfaces of Willmore type in Riemannian manifolds. Int. Math. Res. Not. 2010, 3786–3813 (2010)
Lamm, T., Metzger, J.: Minimizers of the Willmore functional with a small area constraint. Ann. Inst. H. Poincaré Anal. Non Linéaire 30, 497–518 (2013)
Lamm, T., Metzger, J., Schulze, F.: Foliations of asymptotically flat manifolds by surfaces of Willmore type. Math. Ann. 350, 1–78 (2011)
Li, P., Yau, S.-T.: A new conformal invariant and its applications to the Willmoreconjecture and the first eigenvalue of compact surfaces. Invent. Math. 69, 269–291 (1982)
Miao, P.: Positive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys. 6, 1163–1182 (2002)
Mondino, A., Rivière, T.: Willmore spheres in compact Riemannian manifolds. Adv. Math. 232, 608–676 (2013)
Mondino, A., Schygulla, J.: Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 707–724 (2014)
Müller, S., Šverák, V.: On surfaces of finite total curvature. J. Differ. Geom. 42, 229–258 (1995)
Rivière, T.: Analysis aspects of Willmore surfaces. Invent. Math. 174, 1–45 (2008)
Rivière, T.: Lipschitz conformal immersions from degenerating Riemann surfaces with \(L^{2}\)-bounded second fundamental forms Adv. Calc. Var. 6, 1–31 (2013)
Schoen, R., Yau, S.-T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65, 45–76 (1979)
Shi, Y., Tam, L.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Differ. Geom. 62, 79–125 (2002)
Simon, L.: Existence of surfaces minimizing the Willmore functional Comm. Anal. Geom. 1, 281–326 (1993)
Wei, G.: An Hélein’s type convergence theorem for conformal immersions from \({\mathbb{S}}^2\) to manifold. Proc. Am. Math. Soc. 147, 4969–4977 (2019)
Wei, G.: Weak convergence of branched conformal immersions with uniformly bounded areas and Willmore energies. Sci. China Math. (2019). https://doi.org/10.1007/s11425-018-9475-1
Willmore, T.J.: Riemannian Geometry. Oxford University Press, Oxford (1993)
Acknowledgements
The author would like to thank Professor Yuguang Shi for insightful discussions and many encouragements. He would also like to thank Professor Yuxiang Li and Professor Youde Wang for their constant support and encouragement. The author would like to express his sincere gratitude to the anonymous referees for their careful reading of the manuscript and their helpful comments and suggestions. This work was supported by National Natural Science Foundation of China (Grants No.11671015 and 11731001) and China Postdoctoral Science Foundation Grant (No. 2019M660274).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Wei, G. On the Minimizers of Curvature Functionals in Asymptotically Flat Manifolds. J Geom Anal 31, 5837–5853 (2021). https://doi.org/10.1007/s12220-020-00506-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-020-00506-y