On the second variation of the Graham-Witten energy
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- by Yuya Takeuchi PDF
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Abstract:
The area renormalization procedure gives an invariant of even-dimensional closed submanifolds in a conformal manifold, which we call the Graham-Witten energy, and it is a generalization of the classical Willmore energy. In this paper, we obtain an explicit formula for the second variation of this energy at minimal submanifolds in an Einstein manifold. As an application, we prove that the even-dimensional totally geodesic spheres in the unit sphere are critical points of the Graham-Witten energy with non-negative second variation.References
- Charles Fefferman and C. Robin Graham, $Q$-curvature and Poincaré metrics, Math. Res. Lett. 9 (2002), no. 2-3, 139–151. MR 1909634, DOI 10.4310/MRL.2002.v9.n2.a2
- Charles Fefferman and C. Robin Graham, The ambient metric, Annals of Mathematics Studies, vol. 178, Princeton University Press, Princeton, NJ, 2012. MR 2858236
- A. R. Gover, Laplacian operators and $Q$-curvature on conformally Einstein manifolds, Math. Ann. 336 (2006), no. 2, 311–334. MR 2244375, DOI 10.1007/s00208-006-0004-z
- C. Robin Graham, Volume and area renormalizations for conformally compact Einstein metrics, The Proceedings of the 19th Winter School “Geometry and Physics” (SrnĂ, 1999), 2000, pp. 31–42. MR 1758076
- C. Robin Graham and Nicholas Reichert, Higher-dimensional Willmore energies via minimal submanifold asymptotics, arXiv:1704.03852.
- C. Robin Graham and Edward Witten, Conformal anomaly of submanifold observables in AdS/CFT correspondence, Nuclear Phys. B 546 (1999), no. 1-2, 52–64. MR 1682674, DOI 10.1016/S0550-3213(99)00055-3
- C. Robin Graham and Maciej Zworski, Scattering matrix in conformal geometry, Invent. Math. 152 (2003), no. 1, 89–118. MR 1965361, DOI 10.1007/s00222-002-0268-1
- Colin Guillarmou, Sergiu Moroianu, and Jean-Marc Schlenker, The renormalized volume and uniformization of conformal structures, J. Inst. Math. Jussieu 17 (2018), no. 4, 853–912. MR 3835525, DOI 10.1017/S1474748016000244
- M. Henningson and K. Skenderis, The holographic Weyl anomaly, J. High Energy Phys. 7 (1998), Paper 23, 12. MR 1644988, DOI 10.1088/1126-6708/1998/07/023
- Yoshihiko Matsumoto, A GJMS construction for 2-tensors and the second variation of the total $Q$-curvature, Pacific J. Math. 262 (2013), no. 2, 437–455. MR 3069069, DOI 10.2140/pjm.2013.262.437
- Shinsei Ryu and Tadashi Takayanagi, Holographic derivation of entanglement entropy from the anti-de Sitter space/conformal field theory correspondence, Phys. Rev. Lett. 96 (2006), no. 18, 181602, 4. MR 2221050, DOI 10.1103/PhysRevLett.96.181602
- James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556
- Joel L. Weiner, On a problem of Chen, Willmore, et al, Indiana Univ. Math. J. 27 (1978), no. 1, 19–35. MR 467610, DOI 10.1512/iumj.1978.27.27003
- Yongbing Zhang, Graham-Witten’s conformal invariant for closed four dimensional submanifolds, arXiv:1703.08611.
Additional Information
- Yuya Takeuchi
- Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan
- Address at time of publication: Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan
- MR Author ID: 1253953
- Email: yu-takeuchi@cr.math.sci.osaka-u.ac.jp
- Received by editor(s): August 7, 2018
- Received by editor(s) in revised form: May 12, 2019
- Published electronically: August 7, 2019
- Additional Notes: This work was supported by JSPS Research Fellowship for Young Scientists, JSPS KAKENHI Grant Number JP16J04653, and the Program for Leading Graduate Schools, MEXT, Japan
- Communicated by: Jia-Ping Wang
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 393-402
- MSC (2010): Primary 53A30; Secondary 53C42, 58E30
- DOI: https://doi.org/10.1090/proc/14702
- MathSciNet review: 4042860