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Rigidity in dimension four of area-minimising Einstein manifolds

Published online by Cambridge University Press:  23 January 2015

A. BARROS ,
C. CRUZ ,
R. BATISTA  and
P. SOUSA
A. BARROS
Affiliation:
Department of Mathematics, Universidade Federal do Ceará, 60455-760 - Fortaleza-CE, Brazil. e-mail: abbarros@mat.ufc.br, tiarlos@mat.ufc.br
C. CRUZ
Affiliation:
Department of Mathematics, Universidade Federal do Ceará, 60455-760 - Fortaleza-CE, Brazil. e-mail: abbarros@mat.ufc.br, tiarlos@mat.ufc.br
R. BATISTA
Affiliation:
Department of Mathematics, Universidade Federal do Piauí, 64049-550 - Teresina-PI, Brazil. e-mail: marcolino@ufpi.edu.br, e-mail: paulosousa@ufpi.edu.br
P. SOUSA
Affiliation:
Department of Mathematics, Universidade Federal do Piauí, 64049-550 - Teresina-PI, Brazil. e-mail: marcolino@ufpi.edu.br, e-mail: paulosousa@ufpi.edu.br
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Abstract

The aim of this paper is to prove a sharp inequality for the area of a four dimensional compact Einstein manifold (Σ, gΣ) embedded into a complete five dimensional manifold (M5, g) with positive scalar curvature R and nonnegative Ricci curvature. Under a suitable choice, we have $area(\Sigma)^{\frac{1}{2}}\inf_{M}R \leq 8\sqrt{6}\pi$. Moreover, occurring equality we deduce that (Σ, gΣ) is isometric to a standard sphere ($\mathbb{S}$4, gcan) and in a neighbourhood of Σ, (M5, g) splits as ((-ϵ, ϵ) × $\mathbb{S}$4, dt2 + gcan) and the Riemannian covering of (M5, g) is isometric to $\Bbb{R}$ × $\mathbb{S}$4.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 158 , Issue 2 , March 2015 , pp. 355 - 363
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

REFERENCES

[1]Anderson, M. T.On Area-Minimising hypersurfaces in manifolds of non-negative curvature. Indiana Univ. Math. J. 32 (1983), 745–760.Google Scholar
[2]Avez, A.Application de la formule de Gauss–Bonnet–Chern aux variétés à quatre dimensions. C.R.A.S. 256 (1963), 5488–5490.Google Scholar
[3]Bray, H., Brendle, S. and Neves, A.Rigidity of area-minimising two-spheres in three-manifolds. Comm. Anal. Geom. 18 (2010), 821–830.Google Scholar
[4]Bray, H., Brendle, S., Eichmair, M. and Neves, A.Area-minimising projective planes in three-manifolds. Comm. Pure. Appl. Math. 63 (1980), 1237–1247.Google Scholar
[5]Berger, M.Sur les variétés d'Einstein compactes. C. R. de la III^ Réunion du Groupemnt des Mathématiques D'expression Latine (Namur, 1965), pp. 35–55.Google Scholar
[6]Bombieri, E. and de Giorgi, E.Minimal cones and the Bernstein problem. Invent. Math. 7 (1968), 243–269.Google Scholar
[7]Cai, M. and Galloway, G.Rigidity of area minimising tori in 3-manifolds of nonnegative scalar curvature. Comm. Anal. Geom. 8 (2000), 565–573.Google Scholar
[8]Cheeger, J. and Ebin, D.Comparison Theorems in Riemannian Geometry. vol 365. (American Mathematical Society, Chelsea Publishing, Providence, RI, 2008), 161 pp.Google Scholar
[9]Cheeger, J. and Gromoll, D.The splitting theorem for manifolds of non-negative Ricci curvature. J. Differential Geom. 6 (1971), 119–128.Google Scholar
[10]Gursky, M. J.Locally conformally flat four and six-manifolds of positive scalar curvature and positive Euler characteristic. Indiana Univ. Math. J. 43 (1994), 747–774.Google Scholar
[11]Huisken, G. and Ilmanen, T.The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differential Geom. 59 (2001), 353–437.CrossRefGoogle Scholar
[12]Huisken, G. and Yau, S.T.Deffinition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Invent. Math. 124 (1996), 281–311.CrossRefGoogle Scholar
[13]Leite, M. L.Rotational hypersurfaces of space forms with constant scalar curvature. Manuscripta Math. 67 (1990), 285–304.Google Scholar
[14]Mazzeo, R., Pollack, D. and Uhlenbeck, K.Moduli spaces of singular Yamabe metrics. J. Amer. Math. Soc. 9 (1996), 303–344.Google Scholar
[15]Micallef, M. and Moraru, V. Splitting of 3-manifolds and rigidity of area-minimising surfaces, arXiv:1107.5346.Google Scholar
[16]Nabonnand, P.Sur les variétés Riemanniennes complètes à courbure de Ricci positive. C.R. Acad. Sci. Paris. 291 (1980), 591–593.Google Scholar
[17]Nunes, I. Rigidity of Area-Minimising hyperbolic surfaces in three-manifolds. PhD. thesis. IMPA (2011).Google Scholar
[18]Nunes, I.Rigidity of Area-Minimising hyperbolic surfaces in three-manifolds. J. Geom. Anal. 23 (2013), 1290–1302.Google Scholar
[19]Seshadri, H.Weyl curvature and the Euler characteristic in dimension four. Differential Geom. App. 24 (2006), 172–177.Google Scholar
[20]Schoen, R.Variational theory for the total scalar curvature funcional for Riemannian metrics and related topics, Topics in calculus of variations (Montecatini Terme, 1987), Lecture Notes in Math. vol. 1365 (Springer-Verlag, 1989), pp. 120–154.Google Scholar
[21]Schoen, R.Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom. 20 (1984), 479–495.Google Scholar
[22]Santos, A.A construction of constant scalar curvature manifolds with Delaunay-type ends. Ann. Henri Poincaré. 10 (2010), 1487–1535.CrossRefGoogle Scholar
[23]Toponogov, V.Evaluation of the length of a closed geodesic on a convex surface. Dokl. Akad. Nauk. SSSR. 124 (1959), 282–284.Google Scholar