Annals of Global Analysis and Geometry

, Volume 16, Issue 3, pp 203–220

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

  • Hilario Alencar
  • A. Gervasio Colares

DOI: 10.1023/A:1006555603714

Cite this article as:
Alencar, H. & Colares, A.G. Annals of Global Analysis and Geometry (1998) 16: 203. doi:10.1023/A:1006555603714


For a normal variation of a hypersurface Mn in a space form Qcn+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 Lr (0 < r < n − 1) is the linearized operator of the (r + 1)-mean curvature Sr+1 of Mn given by Lr = div(Pr∇); that is, Lr = the divergence of the rth Newton transformation Pr of the second fundamental form applied to the gradient ∇, and L0 = Δ the Laplacian of Mn.

From the Dirichlet integral formula for Lr

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

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 Lr.

integral formula linearized operator Lr r-mean curvature 

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Hilario Alencar
    • 1
  • A. Gervasio Colares
    • 1
  1. 1.Departamento de MatemáticaUniversidade Federal de AlagoasMaceio – AlBrazil

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