Sisto Baldo

Fiber bundles and regular approximation of codimension-one cycles.

We derive an approximation of codimension-one integral cycles(and cycles modulo p) in a compact Riemannian manifold bymeans of piecewise regular cycles: we obtain both flat convergence andconvergence of the masses. The theorem is proved by using suitableprincipal bundles with a discrete group. As a byproduct, we give analternative proof of the main results, which does not use the regularitytheory for homology minimizers in a Riemannian manifold. This also givesa result of Γ-convergence.

Codimension one minimal cycles with coefficients inZ orZ p , and variational functionals on fibered spaces

Given a compact, oriented Riemannian manifold M, without boundary, and a codimension-one homology class in H* (M, Z) (or, respectively, in H* (M, Zp) with p an odd prime), we consider the problem of finding a cycle of least area in the given class: this is known as the homological Plateau’s problem.We propose an elliptic regularization of this problem, by constructing suitable fiber bundles ξ (resp. ζ) on M, and one-parameter families of functionals defined on the regular sections of ξ, (resp. ζ), depending on a small parameter ε.As ε → 0, the minimizers of these functionals are shown to converge to some limiting section, whose discontinuity set is exactly the minimal cycle desired.

Homotopy types for tamely approximable Sobolev maps between manifolds

We define a class ℐp(M,N) of Sobolev maps from a manifoldM into a manifoldN, in such a way that each mapu∈ ℐp(M, N) has a well defined [p]-homotopy type, providedN satisfies a topological hypothesis. Using this, we prove the existence of minimizers in [p]-homotopy classes for some polyconvex variational problems.

Approximation of the elastic deformation of a thin cylinder

The elastic displacement of a thin cylinder subject to given forces is approximated by means of a function constructed from the solutions of certain one-dimensional problems. Estimates are given for the error in terms of a decreasing function of the radius of the cylinder.