×

Stability of area-preserving variations in space forms. (English) Zbl 1156.53037

The authors study compact hypersurfaces \(M\) of real space forms. They define inductively the functional \(\mathcal A_r = \int_M F_r dM\), where \(F_0=1\), \(F_1=S_1\), \(F_r = S_r+c(n-r+1)/(r-1) F_{r-2}\) and \(S_r\) denotes the r-th elementary symmetric function associated to the shape operator of the immersion. They consider the problem of minimizing \(\mathcal A_r\) while keeping the area of \(M\) constant and show that critical points of this variational problem are hypersurfaces for which the quotient of \(S_{r+1}\) and \(S_1\) is a constant. It is also shown that, in case \(S_1\) is positive and the surrounding space is the Euclidean space, a critical immersion is stable if and only if \(M\) is a sphere immersed in the Euclidean space as a totally umbilical hypersurface. Similar results are also obtained in case \(M\) is an hypersurface contained in the hyperbolic space or contained in an open hemisphere of the unit sphere.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A30 Conformal differential geometry (MSC2010)
53C43 Differential geometric aspects of harmonic maps
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Alencar H., do Carmo M. and Colares A.G. (1993). Stable hypersurfaces with constant scalar curvature. Math. Z. 213: 117–131 · Zbl 0792.53057
[2] Alencar H., do Carmo M. and Elbert M.F. (2003). Stability of hypersurfaces with vanishing r-mean curvatures in Euclidean spaces. J. Reine Angew. Math. 554: 201–216 · Zbl 1093.53063
[3] Alencar H., do Carmo M. and Rosenberg H. (1993). On the first eigenvalue of the linearized operator of the r-th mean curvature of a hypersurface. Ann. Global Anal. Geom. 11: 387–395 · Zbl 0816.53031
[4] Alencar H., Rosenberg H. and Santos W. (2004). On the Gauss map of hypersurfaces with constant scalar curvature in spheres. Proc. Am. Math. Soc. 132: 3731–3739 · Zbl 1061.53036
[5] Barbosa J.L. and Colares A.G. (1997). Stability of hypersurfaces with constant r-mean curvature. Ann. Global Anal. Geom. 15: 277–297 · Zbl 0891.53044
[6] Barbosa J.L.M. and do Carmo M. (2005). On stability of cones in \({\mathbb{R}}^{n+1}\) with zero scalar curvature. Ann. Global Anal. Geom. 28: 107–127 · Zbl 1082.53061
[7] Barbosa J.L. and do Carmo M. (1984). Stability of hypersurfaces with constant mean curvature. Math. Z. 185: 339–353 · Zbl 0529.53006
[8] Barbosa J.L., do Carmo M. and Eschenburg J. (1988). Stability of hypersurfaces of constant mean curvature in Riemannian manifolds. Math. Z. 197: 123–138 · Zbl 0653.53045
[9] Cao L.F. and Li H. (2007). r-Minimal submanifolds in space forms. Ann. Global Anal. Geom. 32: 311–341 · Zbl 1168.53029
[10] Cheng S.Y. and Yau S.T. (1977). Hypersurfaces with constant scalar curvature. Math. Ann. 225: 195–204 · Zbl 0349.53041
[11] Hardy G.H., Littlewood J.E. and Polya G. (1934). Inequalities. Cambridge University Press, London · Zbl 0010.10703
[12] He, Y.J., Li, H.: A new variational characterization of the Wulff shape. Diff. Geom. App. (to appear in) · Zbl 1146.53035
[13] Li H. (1996). Hypersurfaces with constant scalar curvature in space forms. Math. Ann. 305: 665–672 · Zbl 0864.53040
[14] Li H. (1997). Global rigidity theorems of hypersurfaces. Ark. Mat. 35: 327–351 · Zbl 0920.53028
[15] Reilly R. (1973). Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differ. Geom. 8: 465–477 · Zbl 0277.53030
[16] Rosenberg H. (1993). Hypersurfaces of constant curvature in space forms. Bull. Soc. Math., 26 Série 117: 211–239 · Zbl 0787.53046
[17] Yano K. (1970). Integral Formulas in Riemannian Geometry. Marcel Dekker, NY · Zbl 0213.23801
[18] Voss K. (1991). Variation of curvature integral. Results Math. 20: 789–796 · Zbl 0753.53004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.