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An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures. (English) Zbl 1119.53004

This paper classifies those orientable hypersurfaces of Euclidean space which satisfy the equation \(L_kx= Ax+ b\) where \(x\) denotes the position vector, \(A\) denotes a constant matrix, \(b\) a constant vector, and \(L_k\) denotes the linearized operator for the \((k+ 1)\)-th mean curvature of the hypersurface. It is found that besides the ones with vanishing \((k+ 1)\)-th mean curvature there are only open pieces of round hyperspheres and spherical cylinders. This extends a theorem of T. Takahashi in [J. Math. Soc. Math. Japan 18, 380–385 (1966; Zbl 0145.18601)].

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53B25 Local submanifolds
53C40 Global submanifolds

Citations:

Zbl 0145.18601
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References:

[1] Alencar H., do Carmo M.P., 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
[2] Alías L.J., Ferrández A., Lucas P. (1992) Submanifolds in pseudo-Euclidean spaces satisfying the condition \(\Delta x=Ax+b\) . Geom. Dedicata 42, 345–354 · Zbl 0755.53012
[3] Alías L.J., Ferrández A., Lucas P. (1995) Hypersurfaces in space forms satisfying the condition \(\Delta x=Ax+b\) . Trans. Amer. Math. Soc. 347, 1793–1801 · Zbl 0829.53050
[4] Alías L.J., Malacarne J.M. (2004) On the first eigenvalue of the linearized operator of the higher order mean curvature for closed hypersurfaces in space forms. Illinois J. Math. 48, 219–240 · Zbl 1038.53061
[5] Barbosa J.L.M., Colares A.G. (1997) Stability of hypersurfaces with constant r-mean curvature. Ann. Global Anal. Geom. 15, 277–297 · Zbl 0891.53044
[6] Besse, A.L. Einstein Manifolds. Ergeb. Math. Grenzgeb. (3), 10. Springer-Verlag, Berlin (1987)
[7] do Carmo M.P., Elbert M.F. (2004) On stable complete hypersurfaces with vanishing r-mean curvature. Tohoku Math. J. 56(2): 155–162 · Zbl 1062.53052
[8] Chen B.-Y., Petrovic M. (1991) On spectral decomposition of immersions of finite type. Bull. Austral. Math. Soc. 44, 117–129 · Zbl 0771.53033
[9] Cheng S.Y., Yau S.T. (1977) Hypersurfaces with constant scalar curvature. Math. Ann. 225, 195–204 · Zbl 0349.53041
[10] Dillen F., Pas J., Verstraelen L. (1990) On surfaces of finite type in Euclidean 3-space. Kodai Math. J. 13, 10–21 · Zbl 0705.53027
[11] Garay O.J. (1988) On a certain class of finite type surfaces of revolution. Kodai Math. J. 11, 25–31 · Zbl 0644.53049
[12] Garay O.J. (1990) An extension of Takahashi’s theorem. Geom. Dedicata 34, 105–112 · Zbl 0704.53044
[13] Hartman P., Nirenberg L. (1959) On spherical image maps whose Jacobians do not change sign. Amer. J. Math. 81, 901–920 · Zbl 0094.16303
[14] Hasanis T., Vlachos T. (1991) Coordinate finite-type submanifolds. Geom. Dedicata 37, 155–165 · Zbl 0718.53044
[15] Hasanis T., Vlachos T. (1992) Hypersurfaces of E n+1 satisfying \(\Delta x=Ax+b\) . J. Austral. Math. Soc. Ser. A 53, 377–384 · Zbl 0767.53039
[16] Hounie J., Leite M.L. (1995) The maximum principle for hypersurfaces with vanishing curvature functions. J. Differential Geom. 41, 247–258 · Zbl 0821.53007
[17] Hounie J., Leite M.L. (1999) Uniqueness and nonexistence theorems for hypersurfaces with H r = 0. Ann. Global Anal. Geom. 17, 397–407 · Zbl 0938.53030
[18] Hounie J., Leite M.L. (1999) Two-ended hypersurfaces with zero scalar curvature. Indiana Univ. Math. J. 48, 867–882 · Zbl 0929.53033
[19] Levi-Civita T. (1937) Famiglie di superficie isoparametrische nell’ordinario spacio euclideo. Att. Accad. naz Lincie Rend. Cl. Sci. Fis. Mat. Nat. 26, 355–362 · Zbl 0018.08702
[20] Reilly R.C. (1970) Extrinsic rigidity theorems for compact submanifolds of the sphere. J. Differential Geom. 4, 487–497 · Zbl 0208.49503
[21] Reilly R.C. (1973) Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differential Geom. 8, 465–477 · Zbl 0277.53030
[22] Rosenberg H. (1993) Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117, 211–239 · Zbl 0787.53046
[23] Segre B. (1938) Famiglie di ipersuperficie isoparametrische negli spazi euclidei ad un qualunque numero di demensioni. Att. Accad. naz Lincie Rend. Cl. Sci. Fis. Mat. Natur 27, 203–207 · Zbl 0019.18403
[24] Takahashi T. (1966) Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan 18, 380–385 · Zbl 0145.18601
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