# Submanifolds with parallel mean curvature in calibrated manifolds: A variational approach

by Isabel Salavessa (Centro de Física das Interacções Fundamentais, IST)

It is well known that m-spheres are the unique smooth solutions for the isoperimetric problem in *R ^{m+1}*. This can be proved by showing that spheres are the unique stable hypersurfaces with constant mean curvature for the Area functional acting on hypersurfaces with a fixed enclosed volume. This was proved by Barbosa and do Carmo (1980) and extended to geodesic spheres in space forms in a jonit work of the same autors with Eschenburg (1988). I show how to extend this variational problem to m-submanifolds in a

*(m+n)*-dimensional Riemannian manifold

*N*possessing a calibration Ω of rank (m+1), by defining an enclosed Ω-volume for one-parameter variations. The Jacobi operator arising from the second variation is now the usual one plus a first-order differential operator depending on the calibration, conditioning the stability. I study the stability of geodesic

*m*-spheres on Hopf fibrations of

*S*or on fibrations of

^{m+n}*R*and of

^{m+n}*H*with totally geodesic fibres. If

^{m+n}*N=R*necessary and sufficient conditions are given on the calibration for m-spheres to be the unique stable solutions. I study the case

^{m+n}*S*in

^{2}*R*with the associative 3-calibration coming from the octonions, and related to this variational problem, and using the Hodge and spectral theory of

^{7}*S*, I derive some Cauchy-Riemann type inequalities for pairs, or more generally, for 4-tuples of functions in

^{2}*S*. Some other examples coming from special geometries are described. Finally, I propose a forced mean curvature flow related to this problem.

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