Cycles of least mass in a Riemannian manifold, described through the ``phase transition'' energy of the sections of a line bundle
The authors study the convergence of a family of variational problems towards a singular limit in analogy to previous work of \textit{L. Modica} and \textit{S. Mortola} [Boll. Unione Mat. Ital., V. Ser., B 14, 285-299 (1977; Zbl 0356.49008)]. Let \(M\) be a compact \(n\)-dimensional Riemannian manifold and \(\xi\) a real line bundle over \(M\) with structure group \(O(1)\). For sections \(u\) of \(\xi\) in the Sobolev class \(W^{1,2}\) the family of energies \(F_\varepsilon\) is defined as \[ F_\varepsilon(u)= \int_M \Biggl(\varepsilon|\nabla u(x)|^2+{1\over \varepsilon} ||u(x)|- 1|^{{3\over 2}}\Biggr) d\text{ vol}_M(x). \] The authors show that, given minimizers \(u_\varepsilon\) of \(F_\varepsilon\), there is a sequence \(\varepsilon_h\to 0\) \((h\to\infty)\) such that \(u_{\varepsilon_h}\) converges in \(L^1\) towards a section \(u_0\) of modulus 1 and of bounded variation. Moreover, for any such limit section \(u_0\) the jump set of \(u_0\) (i.e., the set where \(u_0\) fails to be continuous) represents a class in the homology group \(H_{n-1}(M,\mathbb{Z}_2)\) which is Poincaré dual to the first Stiefel-Whitney class \(w_1(\xi)\in H^1(M, \mathbb{Z}_2)\) of the bundle \(\xi\) and minimizes the \(\text{mod }2\) mass in its homology class. The proof combines \(\Gamma\)-convergence techniques and geometric measure theory with some topological arguments.
scientific article; zbMATH DE number 1374308