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We investigate the asymptotic behavior of the $\mathrm{SU}(2)$-Yang-Mills-Higgs energy $E(\Phi,A)=\int_M|d_A\Phi|^2+|F_A|^2$ in the large mass limit, proving convergence to the codimension-three area functional in the sense of De Giorgi's $\Gamma$-convergence. More precisely, for a compact manifold with boundary $M$ and any family of pairs $\Phi_m\in\Omega^0(M;\mathfrak{su}(2))$ and $A_m\in \Omega^1(M;\mathfrak{su}(2))$ indexed by a mass parameter $m\to\infty$, satisfying $$E(\Phi_m,A_m)\leq Cm\quad\text{and}\quad\lim_{m\to\infty}\frac{1}{m}\int_M(m-|\Phi_m|)^2=0,$$ we prove that the $(n-3)$-currents dual to $\frac{1}{2\pi m}\mathrm{tr}(d_{A_m}\Phi_m\wedge F_{A_m})$ converge subsequentially to a relative integral $(n-3)$-cycle $T$ of mass \begin{equation} \mathbb{M}(T)\leq \liminf_{m\to\infty}\frac{1}{4\pi m}E(\Phi_m,A_m), \end{equation} and show conversely that any integral $(n-3)$-current $T$ with $[T]=0\in H_{n-3}(M,\partial M;\mathbb{Z})$ admits such an approximation, with equality in the above inequality. In the special case of pairs $(\Phi_m,A_m)$ satisfying the generalized monopole equation $*d_{A_m}\Phi_m=F_{A_m}\wedge \Theta$ for a calibration form $\Theta\in \Omega^{n-3}(M)$, we deduce that the limit $\nu=\lim_{m\to\infty}\frac{1}{2\pi m}|d_{A_m}\Phi_m|^2$ of the Dirichlet energy measures satisfies $\nu\leq |T|$, with equality if and only if $T$ is calibrated by $\Theta$, giving evidence for predictions of Donaldson-Segal in the settings of $G_2$-manifolds and Calabi-Yau $3$-folds., Comment: 37 pages

We establish a direct log-epiperimetric inequality for Yang$-$Mills fields in arbitrary dimension and we leverage on it to prove uniqueness of tangent cones with isolated singularity for energy minimizing Yang$-$Mills fields and $\omega$-ASD connections (where $\omega$ is not necessarily closed) satisfying some suitable regularity assumptions. En route to this we establish a Luckhaus type lemma for Yang$-$Mills connections to exclude curvature concentration along blow-up sequences., Comment: Any comments are welcome

We prove that the singular set of an $m$-dimensional integral current $T$ in $\mathbb{R}^{n + m}$, semicalibrated by a $C^{2, \kappa_0}$ $m$-form $\omega$ is countably $(m - 2)$-rectifiable. Furthermore, we show that there is a unique tangent cone at $\mathcal{H}^{m - 2}$-a.e. point in the interior singular set of $T$. Our proof adapts techniques that were recently developed in [DLS23a, DLS23b, DLMS23] for area-minimizing currents to this setting., Comment: References updated

We study quantitative stability results for different classes of Sobolev inequalities on general compact Riemannian manifolds. We prove that, up to constants depending on the manifold, a function that nearly saturates a critical Sobolev inequality is quantitatively $W^{1,2}$-close to a non-empty set of extremal functions, provided that the corresponding optimal Sobolev constant satisfies a suitable strict bound. The case of sub-critical Sobolev inequalities is also covered. Finally, we discuss degenerate phenomena in our quantitative controls., Comment: Comments welcome!

We study the boundary behavior of any limit-interface arising from a sequence of general critical points of the Allen-Cahn energy functionals on a smooth bounded domain. Given any such sequence with uniform energy bounds, we prove that the limit-interface is a free boundary varifold which is integer rectifiable up to the boundary. This extends earlier work of Hutchinson and Tonegawa on the interior regularity of such limit-interface. A key novelty in our result is that no convexity assumption of the boundary is required and it is valid even when the limit-interface clusters near the boundary. Moreover, our arguments are local and thus works in the Riemannian setting. This work provides the first step towards the regularity theory for the Allen-Cahn min-max theory for free boundary minimal hypersurfaces, which was developed in the Almgren-Pitts setting by the first-named author and Zhou., Comment: 37 pages, no figure; all comments are welcome

We develop the asymptotic analysis as $\epsilon\to 0$ for the natural gradient flow of the self-dual $U(1)$-Higgs energies $$E_{\epsilon}(u,\nabla)=\int_M\left(|\nabla u|^2+\epsilon^2|F_{\nabla}|^2+\frac{(1-|u|^2)^2}{4\epsilon^2}\right)$$ on Hermitian line bundles over closed manifolds $(M^n,g)$ of dimension $n\ge 3$, showing that solutions converge in a measure-theoretic sense to codimension-two mean curvature flows -- i.e., integral $(n-2)$-Brakke flows -- generalizing results of the last two authors from the stationary case. Given any integral $(n-2)$-cycle $\Gamma_0$ in $M$, these results can be used together with the convergence theory developed in previous work of the authors to produce nontrivial integral Brakke flows starting at $\Gamma_0$ with additional structure, similar to those produced via Ilmanen's elliptic regularization., Comment: Comments welcome

Given a hermitian line bundle $L\to M$ 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_M\Big(|\nabla u|^2+\epsilon^2|F_\nabla|^2+\frac{(1-|u|^2)^2}{4\epsilon^2}\Big) \end{align*} defined for couples $(u,\nabla)$ consisting of a section $u\in\Gamma(L)$ 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 $\epsilon\to 0$ (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 ($2\pi$ 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 $2 \pi \mathbb{M}(\Gamma)\le\liminf_{\epsilon \rightarrow 0}E_\epsilon(u_\epsilon,\nabla_\epsilon)$. 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 $E_{\epsilon}$., Comment: 52 pages; comments welcome

For sequences of warped product metrics on a $3$-torus satisfying the scalar curvature bound $R_j \geq -\frac{1}{j}$, uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a subsequence which converges in both the Gromov-Hausdorff and the Sormani-Wenger Intrinsic Flat sense to a flat $3$-torus., Comment: 15 pages. We correct and simplify the argument for Theorem 0.3. The statements of the main theorems have not changed