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

##### MSC:
 53C40 Global submanifolds 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces
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##### 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
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