Quantitative 𝑊^{2,𝑝}-stability for almost Einstein hypersurfaces

Gioffrè, Stefano

doi:10.1090/tran/7504

Quantitative $W^{2, p}$-stability for almost Einstein hypersurfaces
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by Stefano Gioffrè PDF

Trans. Amer. Math. Soc. 371 (2019), 3505-3528 Request permission

Abstract:

Let $n \ge 3$, $p \in (1, +\infty )$ be given. Let $\Sigma$ be an $n$-dimensional, closed hypersurface in $\mathbb {R}^{n+1}$. It is a well known fact that if $\Sigma$ is an Einstein hypersurface with positive scalar curvature, then it is a round sphere. Here we prove that if a hypersurface is almost Einstein in an $L^p$-sense, then it is $W^{2, p}$-close to a sphere and we give a quantitative version of this fact.

References

Luigi Ambrosio, Lecture notes on partial differential equations, http://cvgmt.sns.it/paper/1280/, 2010.
Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990), no. 2, 429–445. MR 1074481, DOI 10.1007/BF01233434
Jeff Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61–74. MR 263092, DOI 10.2307/2373498
Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
Aaron Fialkow, Hypersurfaces of a space of constant curvature, Ann. of Math. (2) 39 (1938), no. 4, 762–785. MR 1503435, DOI 10.2307/1968462
Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine, Riemannian geometry, 3rd ed., Universitext, Springer-Verlag, Berlin, 2004. MR 2088027, DOI 10.1007/978-3-642-18855-8
Stefano Gioffrè, A $W^{2, \, p}$-estimate for nearly umbilical hypersurfaces, ArXiv e-prints, December 2016.
Jürgen Jost, Riemannian geometry and geometric analysis, 4th ed., Universitext, Springer-Verlag, Berlin, 2005. MR 2165400
Camillo De Lellis and Peter M. Topping, Almost-Schur lemma, Calc. Var. Partial Differential Equations 43 (2012), no. 3-4, 347–354. MR 2875643, DOI 10.1007/s00526-011-0413-z
Daniel Perez, On nearly umbilical surfaces, http://user.math.uzh.ch/delellis/uploads/media/Daniel.pdf, 2011.
Peter Petersen, Convergence theorems in Riemannian geometry, Comparison geometry (Berkeley, CA, 1993–94) Math. Sci. Res. Inst. Publ., vol. 30, Cambridge Univ. Press, Cambridge, 1997, pp. 167–202. MR 1452874, DOI 10.2977/prims/1195166127
Peter Petersen, Riemannian geometry, 2nd ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. MR 2243772
Julien Roth, Pinching of the first eigenvalue of the Laplacian and almost-Einstein hypersurfaces of the Euclidean space, Ann. Global Anal. Geom. 33 (2008), no. 3, 293–306. MR 2390836, DOI 10.1007/s10455-007-9086-4
Michael Spivak, A comprehensive introduction to differential geometry. Vol. IV, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532833
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
T. Y. Thomas, On Closed Spaces of Constant Mean Curvature, Amer. J. Math. 58 (1936), no. 4, 702–704. MR 1507192, DOI 10.2307/2371240