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Stability and Deformation Criteria in Free Boundary CMC Immersions
Criterios de estabilidad y deformación en inmersiones con CMC y frontera libre
DOI:
https://doi.org/10.15446/recolma.v57nSupl.112445Keywords:
Free boundary constant mean curvature hypersurfaces, Deformation, Stability, Jacobi operator (en)Hipersuperficies con curvatura media constante y frontera libre, deformación, estabilidad, operador de Jacobi (es)
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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.
Sean ∑n y M n+1 variedades suaves con frontera suave. Dada una inmersión φ: ∑ → M, con curvatura media constante (CMC) y frontera libre, encontramos resultados relacionados con la existencia y unicidad de una familia de deformación de φ, {φt}t ∈I , compuesta por inmersiones con curvatura media constante y frontera libre. Adicionalmente, damos algunos criterios de estabilidad e inestabilidad para este tipo de deformaciones. Estos resultados son obtenidos a partir de las propiedades de los valores propios y las funciones propias del operador de Jacobi Jφ asociado a φ, y condiciones de estabilidad para este operador, tales como, Dim(Ker(Jφ)) = 0, o si Dim(Ker(Jφ)) = 1 para f ∈ Ker(Jφ); f ≠ 0, ∫∑ volφ*(g) ≠ 0. La familia de deformación es única, salvo difeomorsmos.
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