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We show that generic rank conditions on the second fundamental form of an isometric immersion f : M 2n → R 2n+p of a Kaehler manifold of complex dimension n ≥ 2 into Euclidean space with low codimension p imply that the submanifold has to be minimal. If M 2n if simply connected, this amounts to the existence of a one-parameter associated family of isometric minimal immersions unless f is holomorphic.

In this paper we classify Euclidean hypersurfaces f : M n → R n+1 with a principal curvature of multiplicity n − 2 that admit a genuine conformal deformationf :

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