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Rigidity of volume-minimising hypersurfaces in Riemannian 5-manifolds

  • ABRAÃO MENDES (a1)

Abstract

In this paper we generalise the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimising closed hypersurface Σ of a Riemannian 5-manifold M with scalar curvature bounded from below by a positive constant in terms of the total traceless Ricci curvature of Σ. Furthermore, if Σ saturates the respective upper bound and M has nonnegative Ricci curvature, then Σ is isometric to 𝕊4 up to scaling and M splits in a neighbourhood of Σ. Also, we obtain a rigidity result for the Riemannian cover of M when Σ minimises the volume in its homotopy class and saturates the upper bound.

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[1] Andersson, L., Cai, M. and Galloway, G. J. Rigidity and positivity of mass for asymptotically hyperbolic manifolds. Ann. Henri Poincaré 9 (2008), no. 1, 133.
[2] Aubin, T. Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pure Appl. (9) 55 (1976), no. 3, 269296.
[3] Aubin, T. Some nonlinear problems in Riemannian geometry. Springer Monogr. Math. (Springer-Verlag, Berlin, 1998).
[4] Barros, A., Cruz, C., Batista, R. and Sousa, P.. Rigidity in dimension four of area-minimising Einstein manifolds. Math. Proc. Camb. Phils. Soc. 158 (2015), no. 2, 355363.
[5] Bray, H., Brendle, S. and Neves, A. Rigidity of area-minimising two-spheres in three-manifolds. Comm. Anal. Geom. 18 (2010), no. 4, 821830.
[6] Cai, M. Volume minimising hypersurfaces in manifolds of nonnegative scalar curvature. In Minimal surfaces, geometric analysis and symplectic geometry (Baltimore, MD, 1999), Adv. Stud. Pure Math. vol. 34 (Math. Soc. Japan, Tokyo, 2002), pp. 17.
[7] Gromov, M. and Lawson, H. B. Jr., The classification of simply connected manifolds of positive scalar curvature. Ann. of Math. (2) 111 (1980), no. 3, 423434.
[8] Gursky, M. J. Locally conformally flat four- and six-manifolds of positive scalar curvature and positive Euler characteristic. Indiana Univ. Math. J. 43 (1994), no. 3, 747774.
[9] Hang, F. and Wang, X. Rigidity theorems for compact manifolds with boundary and positive Ricci curvature. J. Geom. Anal. 19 (2009), no. 3, 628642.
[10] Marques, F. C. and Neves, A. Rigidity of min-max minimal spheres in three-manifolds. Duke Math. J. 161 (2012), no. 14, 27252752.
[11] Micallef, M. and Moraru, V. Splitting of 3-manifolds and rigidity of area-minimising surfaces. Proc. Amer. Math. Soc. 143 (2015), no. 7, 28652872.
[12] Nunes, I. Rigidity of area-minimising hyperbolic surfaces in three-manifolds. J. Geom. Anal. 23 (2013), no. 3, 12901302.
[13] Obata, M. The conjectures on conformal transformations of Riemannian manifolds. J. Differential Geometry 6 (1971/72), 247258.
[14] Schoen, R. Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom. 20 (1984), no. 2, 479495.
[15] Toponogov, V. A. Evaluation of the length of a closed geodesic on a convex surface. Dokl. Akad. Nauk SSSR 124 (1959), 282284.
[16] Trudinger, N. S. Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 265274.
[17] Yamabe, H. On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12 (1960), 2137.

Rigidity of volume-minimising hypersurfaces in Riemannian 5-manifolds

  • ABRAÃO MENDES (a1)

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