Stability and Deformation Criteria in Free Boundary CMC Immersions

Abstract

Let ∑n and M n+1 be smooth manifolds with smooth boundary. Given a free boundary constant mean curvature (CMC) immersion φ: ∑ → M, we found results related to the existence and uniqueness of a deformation family of φ, {φt}t ∈I , composed by free boundary CMC immersions. In addition, we give to some criteria of stability and unstability for this type of deformations. These results are obtained from properties of the eigenvalues and eigenfunctions of the Jacobi operator Jφ associated to φ and establishing conditions for this operator such as Dim(Ker(Jφ)) = 0, or if Dim(Ker(Jφ)) = 1 and, for f ∈ Ker(Jφ); f ≠ 0, ∫∑ volφ*(g) ≠ 0. The deformation family is unique up to diffeomorphisms.