We show that every closed, nonorientable surface can be minimally embedded in 𝕊^4, providing in particular the first known examples of embedded, nonorientable minimal surfaces in 𝕊^4 with negative Euler characteristic. All of the surfaces we obtain have area below 8π, which has applications to the existence of nonorientable surfaces minimizing the Willmore functional with prescribed topology in ℝ^n for n≥ 4. Moreover, the number of geometrically distinct embeddings of each nonorientable genus is shown to grow at least exponentially with respect to the genus. Among these surfaces, we identify a distinguished, highly symmetric family which converges in the large-genus limit to a union of four half-spheres, meeting along a great circle, whose poles are vertices of a regular tetrahedron. Rescalings of this family converge to a new singly-periodic nonorientable minimal surface in ℝ^4, which seems to provide the first example of a complete, embedded, nonorientable minimal surface in ℝ^4 without continuous symmetry group. The proofs further develop the equivariant eigenvalue optimization methodology from our earlier work, applied to carefully chosen families of symmetry groups acting on nonorientable surfaces. Interestingly, we also find natural pairs of surfaces and group actions for which there is no metric maximizing the first normalized eigenvalue of the Laplacian.

In recent work with Kusner, we developed a method, based on the equivariant optimization of Laplace and Steklov eigenvalues, for producing minimal surfaces of prescribed topology in low-dimensional balls and spheres. We used the method to construct many new minimal embeddings in S3 with area below 8 pi, and many new free boundary minimal embeddings in B3 with area below 2 pi. In this paper, we study the geometry of these surfaces in more detail, with an emphasis on studying sharp area estimates and varifold limits in the large Euler characteristic regime. This allows us to confirm some well-known conjectures regarding the space of low-area minimal surfaces in S3 in this class of examples and the special role played by Lawson's xi gamma,1 surfaces. We also confirm analogous statements in B3 and identify a family of free boundary minimal surfaces in B3 most closely resembling xi gamma,1.

We investigate the asymptotic behavior of the SU ( 2 ) \operatorname{SU}(2) -Yang-Mills-Higgs energy E ( Phi , A ) = integral M | d A Phi | 2 + | F A | 2 E(\Phi,A)=\int_{M}\lvert d_{A}\Phi vert{2}+\lvert F_{A} vert{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 is an element of Omega 0 ( M ; s u ( 2 ) ) \Phi_{m}\in\Omega{0}(M;\mathfrak{su}(2)) and A m is an element of Omega 1 ( M ; s u ( 2 ) ) A_{m}\in\Omega{1}(M;\mathfrak{su}(2)) indexed by a mass parameter m -> infinity m o\infty , satisfyingE ( Phi m , A m ) <= C m and lim m -> infinity 1 m integral M ( m - | Phi m | ) 2 = 0 , see text E(\Phi_{m},A_{m})\leq Cm\quad ext{and}\quad\lim_{m o\infty}\frac{1}{m}\int_{M}(m-\lvert\Phi_{m} vert){2}=0,we prove that the ( n - 3 ) (n-3) -currents dual to 1 2 pi m tr ( d A m Phi m boolean AND F A m ) \frac{1}{2\pi m}\operatorname{tr}(d_{A_{m}}\Phi_{m}\wedge F_{A_{m}}) converge subsequentially to a relative integral ( n - 3 ) (n-3) -cycle T of massM ( T ) <= lim inf m -> infinity 1 4 pi m E ( Phi m , A m ) , see text \mathbb{M}(T)\leq\liminf_{m o\infty}\frac{1}{4\pi m}E(\Phi_{m},A_{m}),and show conversely that any integral ( n - 3 ) (n-3) -current T with [ T ] = 0 is an element of H n - 3 ( M , partial derivative M ; Z ) [T]=0\in H_{n-3}(M,\partial M;\mathbb{Z}) admits such an approximation, with equality above. In the special case of pairs ( Phi m , A m ) (\Phi_{m},A_{m}) satisfying the generalized monopole equation & lowast; d A m Phi m = F A m boolean AND Theta *d_{A_{m}}\Phi_{m}=F_{A_{m}}\wedge\Theta for a calibration form Theta is an element of Omega n - 3 ( M ) \Theta\in\Omega{n-3}(M) , we deduce that the limit nu = lim m -> infinity 1 2 pi m | d A m Phi m | 2 u=\lim_{m o\infty}\frac{1}{2\pi m}\lvert d_{A_{m}}\Phi_{m} vert{2} of the Dirichlet energy measures satisfies nu <= | T | u\leq\lvert T vert , with equality if and only if T is calibrated by Theta, giving evidence for predictions of Donaldson-Segal in the settings of G 2 G_{2} -manifolds and Calabi-Yau 3-folds.

In recent work with Kusner, we developed a method, based on the equivariant optimization of Laplace and Steklov eigenvalues, for producing minimal surfaces of prescribed topology in low-dimensional balls and spheres. We used the method to construct many new minimal embeddings in 𝕊^3 with area below 8π, and many new free boundary minimal embeddings in 𝔹^3 with area below 2π. In this paper, we study the geometry of these surfaces in more detail, with an emphasis on studying sharp area estimates and varifold limits in the large Euler characteristic regime. This allows us to confirm some well-known conjectures regarding the space of low-area minimal surfaces in 𝕊^3 in this class of examples and the special role played by Lawson's ξ_γ,1 surfaces. We also confirm analogous statements in 𝔹^3 and identify a family of free boundary minimal surfaces in 𝔹^3 most closely resembling ξ_γ,1.

We prove stability estimates for the isoperimetric inequalities for the first and the second nonzero Laplace eigenvalues on surfaces, both globally and in a fixed conformal class. We employ the notion of eigenvalues of measures and show that if a normalized eigenvalue is close to its maximal value, the corresponding measure must be close in the Sobolev space $W^{-1,2}$ to the set of maximizing measures. In particular, this implies a qualitative stability result: metrics almost maximizing the normalized eigenvalue must be $W^{-1,2}$-close to a maximal metric. Following this approach, we prove sharp quantitative stability of the celebrated Hersch's inequality for the first eigenvalue on the sphere, as well as of its counterpart for the second eigenvalue. Similar results are also obtained for the precise isoperimetric eigenvalue inequalities on the projective plane, torus, and Klein bottle. The square of the $W^{-1,2}$ distance to a maximizing measure in these stability estimates is controlled by the difference between the normalized eigenvalue and its maximal value, indicating that the maxima are in a sense nondegenerate. We construct examples showing that the power of the distance can not be improved, and that the choice of the Sobolev space $W^{-1,2}$ is optimal.

We investigate the asymptotic behavior of the SU(2)-Yang-Mills-Higgs energy E(Φ,A)=∫_M|d_AΦ|^2+|F_A|^2 in the large mass limit, proving convergence to the codimension-three area functional in the sense of De Giorgi's Γ-convergence. More precisely, for a compact manifold with boundary M and any family of pairs Φ_m∈Ω^0(M;𝔰𝔲(2)) and A_m∈Ω^1(M;𝔰𝔲(2)) indexed by a mass parameter m→∞, satisfying E(Φ_m,A_m)≤ Cm and lim_m→∞1/m∫_M(m-|Φ_m|)^2=0, we prove that the (n-3)-currents dual to 1/2π mtr(d_A_mΦ_m∧ F_A_m) converge subsequentially to a relative integral (n-3)-cycle T of mass 𝕄(T)≤lim inf_m→∞1/4π mE(Φ_m,A_m), and show conversely that any integral (n-3)-current T with [T]=0∈ H_n-3(M,∂ M;ℤ) admits such an approximation, with equality in the above inequality. In the special case of pairs (Φ_m,A_m) satisfying the generalized monopole equation *d_A_mΦ_m=F_A_m∧Θ for a calibration form Θ∈Ω^n-3(M), we deduce that the limit ν=lim_m→∞1/2π m|d_A_mΦ_m|^2 of the Dirichlet energy measures satisfies ν≤ |T|, with equality if and only if T is calibrated by Θ, giving evidence for predictions of Donaldson-Segal in the settings of G_2-manifolds and Calabi-Yau 3-folds.

Twenty years ago, N. Kapouleas introduced a singular perturbation construction known as "doubling", which produces sequences of high-genus minimal surfaces converging to a given minimal surface with multiplicity two. Doubling constructions have since been implemented successfully in several settings, with deep work of Kapouleas-McGrath reducing their existence theory to the problem of finding suitable families of ansatz data on the initial minimal surface. In this paper, we introduce a variational approach to the existence of minimal doublings, relating the Kapouleas-McGrath construction to the study of nondegenerate critical points for a Coulomb-type interaction energy. By analyzing the minimizers of this energy, we prove that, in a generic closed 3-manifold, every two-sided, embedded minimal surface of index one admits a sequence of minimal doublings. As a corollary, we find that a generic 3-manifold contains an infinite sequence of embedded minimal surfaces with bounded area and arbitrarily large genus, whose geometry can be described with some precision.

Building on seminal work of Nadirashvili and previous work of the authors, we prove the existence of metrics maximizing the area-normalized first eigenvalue of the Laplacian on every closed nonorientable surface, and give a simple new proof of existence in the orientable case complementing that of [Pet24b], thus resolving the long-standing existence problem for λ_1-maximizing metrics on closed surfaces of any topology. Namely, we prove by contradiction that the supremum Λ_1(M) of the normalized first eigenvalue over all metrics on M obeys the strict monotonicity Λ_1(M#ℝℙ^2)>Λ_1(M) and Λ_1(M#𝕋^2)>Λ_1(M) under the attachment of cross-caps and handles, via a substantial refinement of techniques introduced in [KKMS24].

Given a hermitian line bundle L -> M on a closed Riemannian manifold (Mn, g), the self-dual Yang-Mills-Higgs energies are a natural family of functionals E-epsilon(u, del): = integral(M)(vertical bar del u vertical bar(2) + epsilon(2)vertical bar F-del vertical bar(2) + (1-vertical bar u vertical bar(2))(2)/4 epsilon(2)) defined for couples (u, del) consisting of a section u is an element of Gamma(L) and a hermitian connection del with curvature F-del. While the critical points of these functionals have been well-studied in dimension two by the gauge theory community, it was shown in [52] that critical points in higher dimension converge as... 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(c), del(c)) with sup(epsilon) E-c (u(c), del(c)) < infinity, we prove that a suitable gauge invariant Jacobian J(u(epsilon), del(epsilon)) converges to an integral (n - 2)-cycle G, in the homology class dual to the Euler class c(1)(L), with mass 2 pi M(Gamma) <= lim inf(epsilon -> 0) E-epsilon(u(epsilon), del(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.

Given a hermitian line bundle L → M $L\rightarrow M$ on a closed Riemannian manifold ( M n , g ) $(M^n,g)$ , the self-dual Yang–Mills–Higgs energies are a natural family of functionals

In the present paper, we study sharp isoperimetric inequalities for the first Steklov eigenvalue σ1 on surfaces with fixed genus and large number k of boundary components. We show that as k→∞ the free boundary minimal surfaces in the unit ball arising from the maximization of σ1 converge to a closed minimal surface in the boundary sphere arising from the maximization of the first Laplace eigenvalue on the corresponding closed surface. For some genera, we prove that the corresponding areas converge at the optimal rate logk k. This result appears to provide the first examples of free boundary minimal surfaces in a compact domain converging to closed minimal surfaces in the boundary, suggesting new directions in the study of free boundary minimal surfaces, with many open questions proposed in the present paper. A similar phenomenon is observed for free boundary harmonic maps associated to conformally constrained shape optimization problems.

Given a surface $M$ and a fixed conformal class $c$ one defines $\Lambda_k(M,c)$ to be the supremum of the $k$-th nontrivial Laplacian eigenvalue over all metrics $g\in c$ of unit volume. It has been observed by Nadirashvili that the metrics achieving $\Lambda_k(M,c)$ are closely related to harmonic maps to spheres. In the present paper, we identify $\Lambda_1(M,c)$ and $\Lambda_2(M,c)$ with min-max quantities associated to the energy functional for sphere-valued maps. As an application, we obtain several new eigenvalue bounds, including a sharp isoperimetric inequality for the first two Steklov eigenvalues. This characterization also yields an alternative proof of the existence of maximal metrics realizing $\Lambda_1(M,c)$, $\Lambda_2(M,c)$ and, moreover, allows us to obtain a regularity theorem for maximal Radon measures satisfying a natural compactness condition.

We prove the existence of nonconstant harmonic maps of optimal regularity from an arbitrary closed manifold (M^n,g) of dimension n>2 to any closed, non-aspherical manifold N containing no stable minimal two-spheres. In particular, this gives the first general existence result for harmonic maps from higher-dimensional manifolds to a large class of positively curved targets. In the special case of the round spheres N=𝕊^k , k⩾ 3 , we obtain a distinguished family of nonconstant harmonic maps M→𝕊^k of index at most k+1 , with singular set of codimension at least 7 for k sufficiently large. Furthermore, if 3⩽ n⩽ 5 , we show that these smooth harmonic maps stabilize as k becomes large, and correspond to the solutions of an eigenvalue optimization problem on M , generalizing the conformal maximization of the first Laplace eigenvalue on surfaces.

In 1970, Lawson solved the topological realization problem for minimal surfaces in the sphere, showing that any closed orientable surface can be minimally embedded in $\mathbb{S}^3$. The analogous problem for surfaces with boundary was posed by Fraser and Li in 2014, and it has attracted much attention in recent years, stimulating the development of many new constructions for free boundary minimal surfaces. In this paper, we resolve this problem by showing that any compact orientable surface with boundary can be embedded in $\mathbb{B}^3$ as a free boundary minimal surface with area below $2\pi$. Furthermore, we show that the number of minimal surfaces in $\mathbb{S}^3$ of prescribed topology and area below $8\pi$, and the number of free boundary minimal surfaces in $\mathbb{B}^3$ with prescribed topology and area below $2\pi$, grow at least linearly with the genus. This is achieved via a new method for producing minimal surfaces of prescribed topology in low-dimensional balls and spheres, based on the optimization of Laplace and Steklov eigenvalues in the presence of a discrete symmetry group. As a key ingredient, we develop new techniques for proving the existence of maximizing metrics, which can be used to resolve the existence problem in many symmetric situations and provide at least partial existence results for classical eigenvalue optimization problems.

We develop the asymptotic analysis as ε→0 for the natural gradient flow of the self-dual U(1)-Higgs energies E_ε(u,∇ )=∫ _M (|∇ u|^2+ ε ^2|F_∇|^2+ (1-|u|^2)^2/4ε ^2 ) on Hermitian line bundles over closed manifolds (Mn,g) of dimension n≥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 (Pigati and Stern in Invent. Math. 223:1027–1095, 2021) from the stationary case. Given any integral (n−2)-cycle Γ0 in M, these results can be used together with the convergence theory developed in (Parise et al. in Convergence of the self-dual U(1)-Yang–Mills–Higgs energies to the (n−2)-area functional, 2021, arXiv: 2103.14615 ) to produce nontrivial integral Brakke flows starting at Γ0 with additional structure, similar to those produced via Ilmanen’s elliptic regularization.

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.

Given a family of critical points $u_{\epsilon}:M^n\to\mathbb{C}$ for the complex Ginzburg--Landau energies \begin{align*} &E_\epsilon(u)=\int_{M}\left(\frac{|du|^2}{2}+\frac{(1-|u|^2)^2}{4\epsilon^2}\right), \end{align*} on a manifold $M$, with natural energy growth $E_{\epsilon}(u_{\epsilon})=O(|\log\epsilon| )$, it is known that the vorticity sets $\{|u_\epsilon|\leq \frac{1}{2}\}$ converge subsequentially to the support of a stationary, rectifiable $(n-2)$-varifold $V$ in the interior, characterized as the concentrated portion of the limit $\lim_{\epsilon\to 0} \frac{e_\epsilon(u_\epsilon)}{\pi|\log\epsilon| }$ of the normalized energy measures. When $n=2$ or the solutions $u_{\epsilon}$ are energy-minimizing, it is known moreover that this varifold $V$ is integral; i.e., the $(n-2)$-density $\Theta_{n-2}(|V|,x)$ of $|V|$ takes values in $\mathbb{N}$ at $|V|$-a.e. $x\in M$. In the present paper, we show that for a general family of critical points with $E_{\epsilon}(u_{\epsilon})=O(|\log\epsilon| )$ in dimension $n\geq 3$, this energy quantization phenomenon only holds where the density is less than $2$: namely, we prove that the density $\Theta_{n-2}(|V|,x)$ of the limit varifold takes values in $\{1\}\cup [2,\infty)$ at $|V|$-a.e. $x\in M$, and show that this is sharp, in the sense that for any $n\geq 3$ and $\theta\in \{1\}\cup [2,\infty)$, there exists a family of critical points $u_{\epsilon}$ for $E_{\epsilon}$ in the ball $B_1^n(0)$ with concentration varifold $V$ given by an $(n-2)$-plane with density $\theta$.

We use a natural two-parameter min-max construction to produce critical points of the Ginzburg–Landau functionals on a compact Riemannian manifold of dimension $\geq 2$. We investigate the limiting behavior of these critical points as $\varepsilon \to 0$, and show in particular that some of the energy concentrates on a nontrivial stationary, rectifiable $(n-2)$-varifold as $\varepsilon \to 0$, suggesting connections to the min-max construction of minimal $(n-2)$-submanifolds.

We study the limiting behavior as $$p\uparrow 2$$ of the singular sets $$Sing(u_p)$$ and p-energy measures $$\mu _p:=(2-p)|du_p|^pdvol$$ for families of stationary p-harmonic maps $$u_p\in W^{1,p}(M,S^1)$$ from a closed, oriented manifold M to the circle. When the measures $$\mu _p$$ have uniformly bounded mass, we show that—up to subsequences—the singular sets $$Sing(u_p)$$ converge in the Hausdorff sense to the support of a stationary, rectifiable varifold V of codimension 2, and the measures $$\mu _p$$ converge weakly in $$(C^0(M))^*$$ to a limit of the form $$\begin{aligned} \mu =\Vert V\Vert +|h|^2dvol, \end{aligned}$$where h is a harmonic one-form. For solutions on two-dimensional domains, we show moreover that the density of V takes values in $$2\pi {\mathbb {N}}$$. Finally, we observe that nontrivial families $$u_p$$ of such maps arise naturally on any closed Riemannian manifold of dimension $$n\ge 2$$, via variational methods.