Integral formulas for the \(r\)-mean curvature linearized operator of a hypersurface
Let \(x:M^n \to Q^{n+1}_{c} \) be an isometric immersion of a compact oriented Riemannian manifold \(M^n\) into a simply connected space form \( Q^{n+1}_{c} \). We denote by \(S_{r}\) the \(r\)-mean curvature of \(M^n \) and by \(L_{r}\) \((0\leq r\leq n-1)\) the linearized operator of \(S_{r+1}\) arising from normal variations of \(x\) given by \(L_{r}= \text{div} (P_{r}\nabla)\). Here \(P_{r}\) is the \(r\)th Newton transformation, a polynomial in the second fundamental form \(B\) of \(x\) defined inductively by \(P_0=I, P_{r}=S_{r}I-BP_{r-1}\). The authors, by making different choices of the functions \(f\) and \(g\) on the Dirichlet integral formula for \(L_{r}\), \(\int_M (fL_r g +\langle P_r \nabla f, \nabla g\rangle)dM=0\), obtain new useful integral formulas which generalize known formulas for the Laplacian. The method gives a systematic process for proofs and a unified treatment for some Minkowski type formulas, via \(L_{r} \). As an application, the authors prove the following Theorem. Let \(M^n\) be as above with constant \((r+1)\)-mean curvature \((0\leq r\leq n-1)\). If \(c>0\), assume that \(x(M^n)\) is contained in an open hemisphere of \( Q^{n+1}_{c} \). Then, the set of points \(W= Q^{n+1}_{c}-\bigcup _{p\in M}( Q^n_{c})_p \) wich are omitted by the totally geodesic hypersurfaces \( (Q^n_{c})_p \) tangent to \(x(M^n)\) is non-empty if and only if \(x(M^n)\) is a geodesic sphere.
scientific article; zbMATH DE number 1172212