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- W3148889202 abstract "Given a hermitian line bundle $Lto M$ on a closed Riemannian manifold $(M^n,g)$, the self-dual Yang-Mills-Higgs energies are a natural family of functionals begin{align*} &E_epsilon(u,nabla):=int_MBig(|nabla u|^2+epsilon^2|F_nabla|^2+frac{(1-|u|^2)^2}{4epsilon^2}Big) end{align*} defined for couples $(u,nabla)$ consisting of a section $uinGamma(L)$ and a hermitian connection $nabla$ with curvature $F_nabla$. While the critical points of these functionals have been well-studied in dimension two by the gauge theory community, it was shown in previous work of the second- and third-named authors that critical points in higher dimension converge as $epsilonto 0$ (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 $E_epsilon$ to ($2pi$ times) the codimension two area: more precisely, given a family of couples $(u_epsilon,nabla_epsilon)$ with $sup_epsilon E_epsilon(u_epsilon,nabla_epsilon)<infty$, we prove that a suitable gauge invariant Jacobian $J(u_epsilon,nabla_epsilon)$ converges to an integral $(n-2)$-cycle $Gamma$, in the homology class dual to the Euler class $c_1(L)$, with mass $2 pi mathbb{M}(Gamma)leliminf_{epsilon rightarrow 0}E_epsilon(u_epsilon,nabla_epsilon)$. 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 $E_{epsilon}$." @default.
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- W3148889202 date "2021-03-26" @default.
- W3148889202 modified "2023-09-27" @default.
- W3148889202 title "Convergence of the self-dual $U(1)$-Yang-Mills-Higgs energies to the $(n-2)$-area functional" @default.
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