Integral Formulas for the r-Mean Curvature Linearized Operator of a Hypersurface
- Cite this article as:
- Alencar, H. & Colares, A.G. Annals of Global Analysis and Geometry (1998) 16: 203. doi:10.1023/A:1006555603714
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For a normal variation of a hypersurface Mn in a space form Qcn+1 by a normal vector field fN, R. Reilly proved:
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
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.