Nondegeneracy of standard double bubbles
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Abstract:
In this paper we prove that all Jacobi fields of a standard double bubble in $\mathbb {R}^{m+1}$ are generated by infinitesimal translations and rotations. This implies that the degeneracy of the area functional on double bubbles is only generated by global isometries, with dimension $2m+1$. The proof relies on a Fourier expansion of the normal components of the Jacobi fields and on an accurate analysis of some ordinary differential equations.References
- Thierry Aubin, Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1636569, DOI 10.1007/978-3-662-13006-3
- F. J. Almgren Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), no. 165, viii+199. MR 420406, DOI 10.1090/memo/0165
- Antonio Ambrosetti and Andrea Malchiodi, Perturbation methods and semilinear elliptic problems on $\textbf {R}^n$, Progress in Mathematics, vol. 240, Birkhäuser Verlag, Basel, 2006. MR 2186962, DOI 10.1007/3-7643-7396-2
- Antonio Ambrosetti and Andrea Malchiodi, Nonlinear analysis and semilinear elliptic problems, Cambridge Studies in Advanced Mathematics, vol. 104, Cambridge University Press, Cambridge, 2007. MR 2292344, DOI 10.1017/CBO9780511618260
- L. Ambrosio, Lecture Notes on Elliptic Partial Differential Equations, 2013.
- Bruno Bianchini, Luciano Mari, and Marco Rigoli, On some aspects of oscillation theory and geometry, Mem. Amer. Math. Soc. 225 (2013), no. 1056, vi+195. MR 3112813, DOI 10.1090/s0065-9266-2012-00681-2
- Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
- Marco Cicalese, Gian Paolo Leonardi, and Francesco Maggi, Sharp stability inequalities for planar double bubbles, Interfaces Free Bound. 19 (2017), no. 3, 305–350. MR 3713891, DOI 10.4171/IFB/384
- Marco Cicalese, Gian Paolo Leonardi, and Francesco Maggi, Improved convergence theorems for bubble clusters I. The planar case, Indiana Univ. Math. J. 65 (2016), no. 6, 1979–2050. MR 3595487, DOI 10.1512/iumj.2016.65.5932
- G. P. Leonardi and F. Maggi, Improved convergence theorems for bubble clusters. II. The three-dimensional case, Indiana Univ. Math. J. 66 (2017), no. 2, 559–608. MR 3641486, DOI 10.1512/iumj.2017.66.6016
- G. Di Matteo and A. Malchiodi, in progress.
- Olivier Druet, Sharp local isoperimetric inequalities involving the scalar curvature, Proc. Amer. Math. Soc. 130 (2002), no. 8, 2351–2361. MR 1897460, DOI 10.1090/S0002-9939-02-06355-4
- Nina Virchenko and Iryna Fedotova, Generalized associated Legendre functions and their applications, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. With a foreword by Semyon Yakubovich. MR 1836207, DOI 10.1142/9789812811783
- J. Foisy, Soap bubble clusters in $R^2$ and $R^3$, undergraduate thesis, Williams College, 1991.
- Joel Foisy, Manuel Alfaro, Jeffrey Brock, Nickelous Hodges, and Jason Zimba, The standard double soap bubble in $\textbf {R}^2$ uniquely minimizes perimeter, Pacific J. Math. 159 (1993), no. 1, 47–59. MR 1211384
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443
- Joel Hass and Roger Schlafly, Double bubbles minimize, Ann. of Math. (2) 151 (2000), no. 2, 459–515. MR 1765704, DOI 10.2307/121042
- Michael Hutchings, The structure of area-minimizing double bubbles, J. Geom. Anal. 7 (1997), no. 2, 285–304. MR 1646776, DOI 10.1007/BF02921724
- Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros, Proof of the double bubble conjecture, Ann. of Math. (2) 155 (2002), no. 2, 459–489. MR 1906593, DOI 10.2307/3062123
- Nicolaos Kapouleas, Compact constant mean curvature surfaces in Euclidean three-space, J. Differential Geom. 33 (1991), no. 3, 683–715. MR 1100207
- Stefano Nardulli, The isoperimetric profile of a smooth Riemannian manifold for small volumes, Ann. Global Anal. Geom. 36 (2009), no. 2, 111–131. MR 2529468, DOI 10.1007/s10455-008-9152-6
- Francesco Maggi, Sets of finite perimeter and geometric variational problems, Cambridge Studies in Advanced Mathematics, vol. 135, Cambridge University Press, Cambridge, 2012. An introduction to geometric measure theory. MR 2976521, DOI 10.1017/CBO9781139108133
- Frank Morgan, Geometric measure theory, 5th ed., Elsevier/Academic Press, Amsterdam, 2016. A beginner’s guide; Illustrated by James F. Bredt. MR 3497381
- F. Pacard and X. Xu, Constant mean curvature spheres in Riemannian manifolds, Manuscripta Math. 128 (2009), no. 3, 275–295. MR 2481045, DOI 10.1007/s00229-008-0230-7
- Mauro Picone, Sui valori eccezionali di un parametro da cui dipende un’equazione differenziale lineare ordinaria del second’ordine, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 11 (1910), 144 (Italian). MR 1556637
- Ben W. Reichardt, Cory Heilmann, Yuan Y. Lai, and Anita Spielman, Proof of the double bubble conjecture in $\textbf {R}^4$ and certain higher dimensional cases, Pacific J. Math. 208 (2003), no. 2, 347–366. MR 1971669, DOI 10.2140/pjm.2003.208.347
- Ben W. Reichardt, Proof of the double bubble conjecture in $\mathbf R^n$, J. Geom. Anal. 18 (2008), no. 1, 172–191. MR 2365672, DOI 10.1007/s12220-007-9002-y
- Xiaofeng Ren and Juncheng Wei, A double bubble in a ternary system with inhibitory long range interaction, Arch. Ration. Mech. Anal. 208 (2013), no. 1, 201–253. MR 3021547, DOI 10.1007/s00205-012-0593-5
- Xiaofeng Ren and Juncheng Wei, Asymmetric and symmetric double bubbles in a ternary inhibitory system, SIAM J. Math. Anal. 46 (2014), no. 4, 2798–2852. MR 3238496, DOI 10.1137/140955720
- Xiaofeng Ren and Juncheng Wei, A double bubble assembly as a new phase of a ternary inhibitory system, Arch. Ration. Mech. Anal. 215 (2015), no. 3, 967–1034. MR 3302114, DOI 10.1007/s00205-014-0798-x
- Ros A., The isoperimetric Problem, http://www.ugr.es/$\sim$aros/
- R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu; Translated from the Chinese by Ding and S. Y. Cheng; With a preface translated from the Chinese by Kaising Tso. MR 1333601
- L.A. Slobozhanin and D. J. A. Iwan, On the stability of double bubbles and double drops, J. Colloid Interface Sci. 262 (2003), no. 1, 212–220.
- Wacharin Wichiramala, Proof of the planar triple bubble conjecture, J. Reine Angew. Math. 567 (2004), 1–49. MR 2038304, DOI 10.1515/crll.2004.011
- Rugang Ye, Foliation by constant mean curvature spheres, Pacific J. Math. 147 (1991), no. 2, 381–396. MR 1084717
Additional Information
- Gianmichele Di Matteo
- Affiliation: Department of Mathematics, Queen Mary University of London, London E1 4NS, United Kingdom
- Email: g.dimatteo@qmul.ac.uk
- Received by editor(s): December 6, 2018
- Received by editor(s) in revised form: January 11, 2019, and January 21, 2019
- Published electronically: June 14, 2019
- Communicated by: Jia-Ping Wang
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 4379-4395
- MSC (2010): Primary 34C10, 53A10, 53C24, 53A24, 35J57
- DOI: https://doi.org/10.1090/proc/14551
- MathSciNet review: 4002550