Masterclass: Geometry of Phase Transitions

Course Title: Approximating the area functional in codimension two: Yang-Mills-Higgs energy on U(1) bundles

Abstract: Constructing minimal submanifolds in arbitrary Riemannian ambients has always been one of the most popular and important problems in the calculus of variations and geometric analysis, giving birth in the 60s to the vast field of geometric measure theory. For low dimensional submanifolds (curves and surfaces), a fruitful approach is to parametrize the submanifold, viewing it as the image of a map, typically relaxing area with the Dirichlet energy. In a dual way, in low codimension (one or two), it is useful to view the submanifold as the zero set of a map, and to seek an energy which resembles the area of the level set from a variational standpoint. In codimension one, this was successfully achieved using the Allen-Cahn energy, which models phase transitions between two pure phases. In codimension two, together with Daniel Stern, we showed that a successful candidate is the Yang-Mills-Higgs energy on U(1) bundles, coming from the Ginzburg-Landau model of superconductivity, in a special self-dual regime; additional evidence for this was obtained in later works with Davide Parise and Daniel Stern. The aim of this minicourse is to survey this “level set approach” and some of the functionals which have been proposed so far, with particular emphasis on Yang-Mills-Higgs and the relevant techniques.

1) Brief survey on Allen-Cahn, Ginzburg-Landau, abelian Yang-Mills-Higgs

2) Tools from geometric measure theory: rectifiable sets, currents, varifolds

3) Generalized varifolds and Ambrosio-Soner rectifiability criterion

4) Positive and negative results for Ginzburg-Landau (with no magnetic field)

5) Brief review of connections on vector bundles and their curvature. Abelian Yang-Mills-Higgs energy. Regularity of critical points in the Coulomb gauge

6) Analogue of Modica’s inequality. Monotonicity and clearing out

7) Exponential decay of energy density away from the zero set. Quantization and the vortex equations. Integrality of the limit varifold

8) Related results: Gamma-convergence of this energy to area, convergence of its gradient flow to mean curvature flow, analogue of Savin's theorem

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