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Integral Formulas for the r-Mean Curvature Linearized Operator of a Hypersurface

Authors:

Hilario Alencar A Gervasio Colares

Institution:

(1) Departamento de Matemática, Universidade Federal de Alagoas, 57072-970 Maceio – Al, Brazil

Abstract:

For a normal variation of a hypersurface Mⁿ in a space form Q _c ⁿ⁺¹ by a normal vector field fN, R. Reilly proved:

$\frac{d}{{dt}}S_{r + 1} (t)|_{t = 0} = L_r f + (S_1 S_{r + 1} - (r + 2)S_{r + 2} )f + c(n - r)S_r f,$

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 nabla

); that is, L _r = the divergence of the rth Newton transformation P_r of the second fundamental form applied to the gradient nabla

, and L₀ = Delta

the Laplacian of Mⁿ.From the Dirichlet integral formula for L _r

$\int {_{M^n } } (fL_r g + \left\langle {P_r \nabla f,\nabla g} \right\rangle ) = 0$

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.

Keywords:

integral formula linearized operator L _r r-mean curvature

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