Hypersurfaces with constant scalar curvature in space forms. (English) Zbl 0864.53040

Li, Haizhong

Math. Ann. 305, No. 4, 665-672 (1996).

By the study of Cheng-Yau’s self-adjoint operator \(\square\), we prove two rigidity theorems for \(n\)-dimensional hypersurfaces with constant scalar curvature in the \((n+1)\)-dimensional unit sphere \(S^{n+1}\) and in \((n+1)\)-dimensional Euclidean space \(E^{n+1}\), respectively.

Reviewer: Li Haizhong (Beijing)

MSC:

53C40	Global submanifolds
53A07	Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces

Keywords:

rigidity theorems; hypersurfaces; constant scalar curvature; unit sphere; Euclidean space

References:

[1]	Barbasch, D.: The unitary dual of complex classical Lie groups. Invent. Math.96 (1989), 103-176 · Zbl 0692.22006
[2]	Cowling, M., Haagerup, U., Howe, R.: AlmostL 2 matrix coefficients. J. reine angew. Math.387 (1988), 97-110 · Zbl 0638.22004
[3]	Fell, J.: The dual spaces ofC *-algebras. Trans. Amer. Math. Soc.94 (1960), 364-403 · Zbl 0090.32803
[4]	Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York, 1978 · Zbl 0451.53038
[5]	Howe, R.: On a notion of rank for unitary representations of classical groups. Harmonic Analysis and Group Representations, Proc. C.I.M.E. (1980), 223-232
[6]	Howe, R., Moore, C.: Asymptotic properties of unitary representations. J. Fun. Anal.32 (1979), 72-96 · Zbl 0404.22015
[7]	Hausner, M., Schwartz, J.: Lie Groups, Lie Algebras. Gordon and Breach, New York, 1968 · Zbl 0192.35902
[8]	Kazhdan, D.: Connection of dual space of a group with the structure of its closed subgroups. Functional Anal. Appl.1 (1967), 63-65 · Zbl 0168.27602
[9]	Kazhdan, D., Savin, G.: Festschrift in honor of I. Piatetski-Shapiro. Israel Math Conference Proceedings, vol. 3, 1990 · Zbl 0737.22008
[10]	Li, J.: The minimal decay of matrix coefficients for classical groups. Preprint (1993)
[11]	Scaramuzzi, R.: A notion of rank for unitary representations of general linear groups. Trans. Amer. Math. Soc.319 (1990), 349-379 · Zbl 0704.22012
[12]	Shale, D.: Linear symmetries of free boson fields. Trans. Amer. Math. Soc.103 (1962), 149-167 · Zbl 0171.46901
[13]	Weil, A.: Sur certains groups d’operateurs unitaires. Acta Math.111 (1964), 143-211 · Zbl 0203.03305