Abstract
For a normal variation of a hypersurface Mn in a space form Q n+1 c by a normal vector field fN, R. Reilly proved:
where L r (0 < r < n − 1) is the linearized operator of the (r + 1)-mean curvature S r+1 of Mn given by L r = div(P r ∇); that is, L r = the divergence of the rth Newton transformation P r of the second fundamental form applied to the gradient ∇, and L0 = Δ the Laplacian of Mn.
From the Dirichlet integral formula for L r
new integral formulas are obtained by making different choices of f and g, generalizing known formulas for the Laplacian. The method gives a systematic process for proofs and a unified treatment for some Minkowski type formulas, via L r .
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References
Alencar, H., do Carmo, M. and Rosenberg, H.: On the first eigenvalue of the linearized operator of the r-th mean curvature of a hypersurface, Ann. Global Anal. Geom. 11 (1993), 387-391.
Alencar, H. and Frensel, K.: Hypersurfaces whose tangent geodesic omit a nonempty set, in Lawson, B. and Tenenblat, K. (eds), Differential Geometry, Pitman Monographs, Vol. 52, Longman, Essex, 1991, pp. 1-13.
Barbosa, L. and Colares, A. G.: Stability of hypersurfaces with constant r-mean curvature, Ann. Global Anal. Geom. 15 (1997), 277-297.
Barbosa, J. L. M., do Carmo, M. P. and Eschenburg, J.: Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Z. 197(1) (1988), 123-138.
Bivens, I.: Some integral formulas for hypersurfaces in a simply connected space form, Proc. Am. Math. Soc. 88(2) (1983), 113-118.
Chern, S. S.: Some formulas in theory of surfaces, Bol. Soc. Mat. Mexicana 10 (1953), 30-40.
Felipe, L. H. G.: On compact hypersurfaces immersed in a space form, Matemática Contemporânea 9 (1995), 75-90.
Fontenele, F.: On the Minkowski integrands, preprint, unpublished.
Gardner, R.: The Dirichlet integral in differential geometry, in Chern, S. S. and Smale, S. (eds), Global Analysis, Proc. Symp. Pure Math., Vol. 15, AMS, Providence, RI, 1970, pp. 237-245.
Heintz, E.: Extrinsic upper bounds for λ1, Math. Ann. 280 (1988), 389-402.
Hsiung, C.-C.: Some integral formulas for closed hypersurfaces, Math. Scand. 2 (1954), 286-294.
Hsiung, C.-C.: Some integral formulas for closed hypersurfaces in Riemannian space, Pacific J. Math. 6 (1956), 291-299.
Katsurada, Y.: Generalized Minkowski formulas for closed hypersurfaces in Riemann space, Ann. Mat. Pura Appl. 57 (1962), 283-293.
Kohlman, P.: Minkowski integral formulas for compact convex bodies in standard space forms, Math. Nachr. 166 (1994), 217-228.
Korevaar, N.: Sphere theorem via Alexandrov for constant Weingarten curvature hypersurfaces. Appendix to a note of A. Ros, J. Differential Geom. 27 (1988), 221-223.
Montiel, S. and Ros, A.: Compact hypersurfaces: The Alexandrov theorem for higher order mean curvatures, in Lawson, B. and Tenenblat, K. (eds), Differential Geometry, Pitman Monographs, Vol. 52, Longman, Essex, 1991, pp. 279-296.
Rosenberg, H.: Hypersurfaces of constant curvature in space forms, Bull. Sc. Math.2e série, 117 (1993), 211-239.
Rund, H.: Integral formulas on hypersurfaces in Riemannian manifolds, Ann. Mat. Pura Appl. 88 (1971), 99-122.
Reilly, R.: Extrinsic rigidity theorems for compact submanifolds of the sphere, J. Differential Geom. 4 (1970), 487-497.
Reilly, R.: Variational properties of functions of the mean curvature for hypersurfaces in space forms, J. Differential Geom. 8 (1973), 465-477.
Shahin, J. K.: Some integral formulas in Euclidean space, Proc. Am. Math. Soc. 19 (1968), 609-613.
Simon, U.: Minkowskische Integralformeln und ihre Anwendungen in der Differentialgeometrie im Großen, Math. Ann. 137 (1967), 307-321.
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Alencar, H., Colares, A.G. Integral Formulas for the r-Mean Curvature Linearized Operator of a Hypersurface. Annals of Global Analysis and Geometry 16, 203–220 (1998). https://doi.org/10.1023/A:1006555603714
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DOI: https://doi.org/10.1023/A:1006555603714