Given a hermitian line bundle L on a closed Riemannian manifold (M, g), the self-dual Yang-Mills-Higgs energies are a natural family of functionals defined for couples consisting of a section and a hermitian connection. While the critical points of these functionals have been well-studied in dimension two by the gauge theory community, it was shown by Pigati-Stern that critical points in higher dimension converge (in an appropriate sense) to minimal submanifolds of codimension two, with strong parallels to the correspondence between the Allen-Cahn equations and minimal hypersurfaces. In this paper, we complement this idea by showing the Gamma-convergence of these energies to (a multiple of) the codimension two area: more precisely, given a family of couples with bounded energy, we prove that a suitable gauge invariant Jacobian converges to an integral cycle, in the homology class dual to the Euler class, with mass bounded by (the liminf of) the energy. We also obtain a recovery sequence, for any integral cycle in this homology class. Finally, we apply these techniques to compare min-max values for the (n-2)-area from the Almgren-Pitts theory with those obtained from the Yang-Mills-Higgs framework, showing that the former values always provide a lower bound for the latter. As an ingredient, we also establish a Huisken-type monotonicity result along the gradient flow of the energy.

Convergence of the self-dual U(1)-Yang-Mills-Higgs energies to the (n-2)-area functional

Pigati, Alessandro;
2024

Abstract

Given a hermitian line bundle L on a closed Riemannian manifold (M, g), the self-dual Yang-Mills-Higgs energies are a natural family of functionals defined for couples consisting of a section and a hermitian connection. While the critical points of these functionals have been well-studied in dimension two by the gauge theory community, it was shown by Pigati-Stern that critical points in higher dimension converge (in an appropriate sense) to minimal submanifolds of codimension two, with strong parallels to the correspondence between the Allen-Cahn equations and minimal hypersurfaces. In this paper, we complement this idea by showing the Gamma-convergence of these energies to (a multiple of) the codimension two area: more precisely, given a family of couples with bounded energy, we prove that a suitable gauge invariant Jacobian converges to an integral cycle, in the homology class dual to the Euler class, with mass bounded by (the liminf of) the energy. We also obtain a recovery sequence, for any integral cycle in this homology class. Finally, we apply these techniques to compare min-max values for the (n-2)-area from the Almgren-Pitts theory with those obtained from the Yang-Mills-Higgs framework, showing that the former values always provide a lower bound for the latter. As an ingredient, we also establish a Huisken-type monotonicity result along the gradient flow of the energy.
2024
2023
Parise, Davide; Pigati, Alessandro; Stern, Daniel
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11565/4061144
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