Codimension one minimal cycles with coefficients in \(Z\) or \(Z_p\), and variational functionals on fibered spaces
Let \(M\) be a compact oriented Riemannian manifold without boundary. Consider a \(k\)-dimensional homology class in \(H_\ast(M, G)\) with coefficient group \(G\). The homological Plateau's problem consists of finding a representative cycle of this class having the smallest possible \(k\)-dimensional area. For cases \(G={\mathbb Z}\) and \({\mathbb Z}_p\) (\(p\) is an odd prime), the authors construct an elliptic regularization of this problem, by constructing a line bundle \(\xi\) on \(M\), and one-parameter family of ``phase transition-like'' functionals defined on the Sobolev space \(W^{1,2}(\xi)\) of sections of \(\xi\) of the type \[ F_\varepsilon(u)=\int_M\biggl[\varepsilon|\nabla^\xi u|^2 +\frac{1}{\varepsilon} \widetilde W(u)\biggr]d{\mathcal H}^n, \] where \(\widetilde W\) is a continuous function whose zeros correspond to \(G\). The authors prove the followings: (1) For every \(\varepsilon\), there exists a minimizer \(u_\varepsilon \in W^{1,2}(\xi)\) of \(F_\varepsilon\). (2) The family \(\{u_\varepsilon\}\) of minimizers is compact in \(L^1(\xi)\). (3) If \(u_0\) is a limit of this family, then its discontinuity set is a mass-minimizing cycle in the prescribed codimension-one homology class. The proofs are based on the former work of the authors for \(G={\mathbb Z}_2\) [Math. Z. 225, No. 4, 639-655 (1997; Zbl 0948.49024)], \(\Gamma\)-convergence, ideas of earlier works on phase transitions in fluids (especially, the work of \textit{L. Modica} and \textit{S. Mortola} [Boll. Unione Mat. Ital., V. Ser., B 14, 285-299 (1977; Zbl 0356.49008)]), and geometric measure theoretic techniques.
scientific article; zbMATH DE number 1542112
scientific article