Spinorial proofs of the Alexandrov Theorem for higher order mean curvatures in Rn+1Rn+1 and the Heintze–Karcher Inequality

We apply the key inequality (1) recently obtained by Hijazi and Montiel in [6] in the Euclidean space to some well-chosen spinor fields to obtain new proofs of the Alexandrov Theorem for higher order mean curvatures in Rn+1Rn+1 (Theorem 1) and the Heintze–Karcher Inequality (Theorem 2).