In [Mo], [Sa], [Mo-Sa], variational characterizations of $m$-dimensional submanifolds with prescribed (possibly constant) mean curvature are described in higher codimension by introducing several notions of "enclosed volume", generalizing the well known classical case of codimension one, due to Barbosa, do Carmo and Eschenburg. In these settings, these submanifolds are critical points for the area functional under the enclosed volume constraint and we discuss when minimizers exist, that is, they solve an isoperimetric problem, and their uniqueness. When the ambient space is endowed by a rank $m+1$ calibration, with consequent definition of enclosed volume, we use the second variation of the area and volume functionals to define stability. Then we ask if spheres uniquely solve the isoperimetric problem and if are stable ([Sa1][Mo-Sa][Sa2]).

References:

• [Mo] Frank Morgan, "Perimeter-minimizing curves and surfaces in $\mathbb{R}^n$ enclosing prescribed multi-volume", Asian J. Math. 4 (2000), 373-382.
• [Sa1] I.Salavessa, "Stability of submanifolds with parallel mean curvature in calibrated manifolds", Bull. Braz. Math. Soc. 41 (2010), 495-530.
• [Mo-Sa], F. Morgan and I.Salavessa. "The ispoperimetric problem in higher codimension".
• [Sa2] I.Salavessa. "Stable 3-spheres in C3", J. Mathematics Research 4 , no.2 (April) 2012

I will recall the notion of the spectrum of a Riemannian manifold M and some of its properties related to curvature. Then I will specify for the case M is a complete minimal immersed submanifold of a Riemannian manifold, namely when it is bounded (in a certain sense), as the Martin-Morales surfaces case treated by Bessa Jorge and Montenegro, and also for the unbounded case as the graphic minimal submanifolds.