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Nathaniel Sagman

Publications and source records attributed to Nathaniel Sagman.

17 recordsLinked to original sources
arXiv2026

Holomorphicity of stable minimal surfaces of low genus

We prove that a (branched) minimal immersion from $\mathbb{C}$ to $\mathbb{R}^n$ is stable if and only if it lives in an even dimensional affine subspace and is holomorphic for some orthogonal complex structure on the subspace. More generally, we prove that the same result holds for a class of genus $0$ surfaces that can have infinite total curvature. This contributes to an inquiry initiated by Micallef, who previously proved the equivalence in genus $0$ assuming completeness and finite total curvature. As a corollary, we prove a holomorphicity result for covering stable minimal surfaces of genus $0$ and $1$, recovering a theorem of Fraser and Schoen as a particular case. Our approach is new, based on a method of constructing variations developed by the first named author and Markovi\'c. For unstable surfaces, we get explicit destabilizations and destabilization radii that can be read from the Weierstrass-Enneper data.

math.DG↗
arXiv2025

Domination between non-Fuchsian representations and anti-de Sitter geometry

Motivated by work of various authors on domination between surface group representations, harmonic maps, and $3$-dimensional anti-de Sitter geometry, we study a new domination problem between non-Fuchsian representations of closed surface groups. We solve the problem for representations that admit branched harmonic immersions, and we show that, outside of this case, the problem cannot always be solved. We then show that a dominating pair gives rise to an anti-de Sitter $3$-manifold with singularities, and we construct large families of branched anti-de Sitter $3$-manifolds.

math.DG↗
arXiv2025

Complex harmonic maps and rank 2 higher Teichm\"uller theory

We initiate and develop the theory of complex harmonic maps to holomorphic Riemannian symmetric spaces, which we make use of to study complex analytic aspects of higher Teichm\"uller theory, with a focus on rank $2$ Hitchin components. Complex harmonic maps lead to various generalizations of objects from the theory of Higgs bundles; for instance, the Hitchin fibration, cyclic Higgs bundles, and the affine Toda equations. Beyond such generalizations, we also find a relation between complex harmonic maps and opers. Within the realm of higher Teichm\"uller theory, for any rank $2$ Hitchin component, we prove a Bers-type theorem, which extends and improves our previous work on $\mathrm{SL}(3,\mathbb R)$, and we prove that Goldman's symplectic form is compatible with Labourie's complex structure, so that the two determine a mapping class group invariant pseudo-K\"ahler structure. We obtain partial generalizations in higher rank, and we construct K\"ahler structures on other spaces that are related to the Hitchin components.

math.DG↗
arXiv2025

Local asymptotics for Hitchin's equations and high energy harmonic maps

We find new estimates and a new asymptotic decoupling phenomenon for solutions to Hitchin's self-duality equations at high energy. These generalize previous results for generically regular semisimple Higgs bundles to arbitrary Higgs bundles. We apply our estimates to the Hitchin WKB problem and to high energy harmonic maps to symmetric spaces and buildings.

math.DG↗
arXiv2024

On Hitchin's equations for cyclic G-Higgs bundles

We develop a Lie-theoretic perspective on Hitchin's equations for cyclic $G$-Higgs bundles, which we use to study analytic and geometric properties of harmonic maps. Among other things, we prove Dai-Li's conjecture on the monotonicity of the energy density in the case of Coxeter cyclic $G$-Higgs bundles, for all $G$, and Dai-Li's negative curvature conjecture for Coxeter cyclic $G$-Higgs bundles, for all $G$ except those of type $\mathrm{E}_7$ and $\mathrm{E}_8.$

math.DG↗
arXiv2024

Holomorphic dependence for the Beltrami equation in Sobolev spaces

We prove that, given a path of Beltrami differentials on $\mathbb C$ that live in and vary holomorphically in the Sobolev space $W^{l,\infty}_{loc}(\Omega)$ of an open subset $\Omega\subset \mathbb C$, the canonical solutions to the Beltrami equation vary holomorphically in $W^{l+1,p}_{loc}(\Omega)$ for admissible $p > 2$. This extends a foundational result of Ahlfors and Bers (the case $l = 0$). As an application, we deduce that Bers metrics on surfaces depend holomorphically on their input data.

math.CV↗
arXiv2024

Complex affine spheres and a Bers theorem for SL(3,C)

For $S$ a closed surface of genus at least $2$, let $\mathrm{Hit}_3(S)$ be the Hitchin component of representations to $\mathrm{SL}(3,\mathbb{R}),$ equipped with the Labourie-Loftin complex structure. We construct a mapping class group equivariant holomorphic map from a large open subset of $\mathrm{Hit}_3(S)\times \overline{\mathrm{Hit}_3(S)}$ to the $\mathrm{SL}(3,\mathbb{C})$-character variety that restricts to the identity on the diagonal and to Bers' simultaneous uniformization on $\mathrm{T}(S)\times \overline{\mathrm{T}(S)}$. The open subset contains $\mathrm{Hit}_3(S)\times \overline{\mathrm{T}(S)}$ and $\mathrm{T}(S)\times \overline{\mathrm{Hit}_3(S)}$, and the image includes the holonomies of $\mathrm{SL}(3,\mathbb{C})$-opers. The map is realized by associating pairs of Hitchin representations to immersions into $\mathbb{C}^3$ that we call complex affine spheres, which are equivalent to certain conformal harmonic maps into $\mathrm{SL}(3,\mathbb{C})/\mathrm{SO}(3,\mathbb{C})$ and to new objects called bi-Higgs bundles. Complex affine spheres are obtained by solving a second-order complex elliptic PDE that resembles both the Beltrami and Tzitz\'eica equations. To study this equation we establish analytic results that should be of independent interest.

math.DG↗
arXiv2023

Non-convexity of extremal length

With respect to every Riemannian metric, the Teichmüller metric, and the Thurston metric on Teichmüller space, we show that there exist measured foliations on surfaces whose extremal length functions are not convex. The construction uses harmonic maps to $\mathbb{R}$-trees and minimal surfaces in $\mathbb{R}^n.$

math.GT↗
arXiv2023

Minimal diffeomorphisms with $L^1$ Hopf differentials

We prove that for any two Riemannian metrics $\sigma_1, \sigma_2$ on the unit disk, a homeomorphism $\partial\mathbb{D}\to\partial\mathbb{D}$ extends to at most one quasiconformal minimal diffeomorphism $(\mathbb{D},\sigma_1)\to (\mathbb{D},\sigma_2)$ with $L^1$ Hopf differential. For minimal Lagrangian diffeomorphisms between hyperbolic disks, the result is known, but this is the first proof that does not use anti-de Sitter geometry. We show that the result fails without the $L^1$ assumption in variable curvature. The key input for our proof is the uniqueness of solutions for a certain Plateau problem in a product of trees.

math.DG↗
arXiv2023

Minimal surfaces and the new main inequality

We establish the new main inequality as a minimizing criterion for minimal maps to products of $\mathbb{R}$-trees, and the infinitesimal new main inequality as a stability criterion for minimal maps to $\mathbb{R}^n$. Along the way, we develop a new perspective on destabilizing minimal surfaces in $\mathbb{R}^n$, and as a consequence we reprove the instability of some classical minimal surfaces; for example, the Enneper surface.

math.DG↗
arXiv2023

Spaces of harmonic surfaces in non-positive curvature

Let $\mathfrak{M}(Σ)$ be an open and connected subset of the space of hyperbolic metrics on a closed orientable surface, and $\mathfrak{M}(M)$ an open and connected subset of the space of metrics on an orientable manifold of dimension at least $3$. We impose conditions on $M$ and $\mathfrak{M}(M)$, which are often satisfied when the metrics in $\mathfrak{M}(M)$ have non-positive curvature. Under these conditions, the data of a homotopy class of maps from $Σ$ to $M$ gives $\mathfrak{M}(Σ)\times \mathfrak{M}(M)$ the structure of a space of harmonic maps. Using transversality theory for Banach manifolds, we prove that the set of somewhere injective harmonic maps is open, dense, and connected in the moduli space. We also prove some results concerning the distribution of harmonic immersions and embeddings in the moduli space.

math.DG↗
arXiv2022

Unstable minimal surfaces in symmetric spaces of non-compact type

We prove that if $\Sigma$ is a closed surface of genus at least 3 and $G$ is a split real semisimple Lie group of rank at least $3$ acting faithfully by isometries on a symmetric space $N$, then there exists a Hitchin representation $\rho:\pi_1(\Sigma)\to G$ and a $\rho$-equivariant unstable minimal map from the universal cover of $\Sigma$ to $N$. This follows from a new lower bound on the index of high energy minimal maps into an arbitrary symmetric space of non-compact type. Taking $G=\mathrm{PSL}(n,\mathbb{R})$, $n\geq 4$, this disproves the Labourie conjecture.

math.DG↗
arXiv2022

Unstable minimal surfaces in $\mathbb{R}^n$ and in products of hyperbolic surfaces

We prove that every unstable equivariant minimal surface in $\mathbb{R}^n$ produces a maximal representation of a surface group into $\prod_{i=1}^n\textrm{PSL}(2,\mathbb{R})$ together with an unstable minimal surface in the corresponding product of closed hyperbolic surfaces. To do so, we lift the surface in $\mathbb{R}^n$ to a surface in a product of $\mathbb{R}$-trees, then deform to a surface in a product of closed hyperbolic surfaces. We show that instability in one context implies instability in the other two.

math.DG↗
arXiv2021

Almost strict domination and anti-de Sitter 3-manifolds

We define a condition called almost strict domination for pairs of representations $\rho_1:\pi_1(S_{g,n})\to \textrm{PSL}(2,\mathbb{R})$, $\rho_2:\pi_1(S_{g,n})\to G$, where $G$ is the isometry group of a Hadamard manifold $(X,\nu)$, and prove it holds if and only if one can find a $(\rho_1,\rho_2)$-equivariant spacelike maximal surface in a certain pseudo-Riemannian manifold, unique up to fixing some parameters. The proof amounts to setting up and solving an interesting variational problem that involves infinite energy harmonic maps. Adapting a construction of Tholozan, we construct all such representations and parametrize the deformation space. When $(X,\nu)=(\mathbb{H},\sigma)$, an almost strictly dominating pair is equivalent to the data of an anti-de Sitter 3-manifold with specific properties. The results on maximal surfaces provide a parametrization of the deformation space of such $3$-manifolds as a union of components in a $\textrm{PSL}(2,\mathbb{R})\times \textrm{PSL}(2,\mathbb{R})$ relative representation variety.

math.DG↗
arXiv2021

Infinite energy equivariant harmonic maps, domination, and anti-de Sitter $3$-manifolds

We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our construction recovers a family of harmonic maps originally studied by Wolf. We employ these maps to solve a domination problem for representations. In particular, following ideas laid out by Deroin-Tholozan, we prove that any representation from a finitely generated free group to the isometry group of a CAT$(-1)$ Hadamard manifold is strictly dominated in length spectrum by a large collection of Fuchsian ones. As an intermediate step in the proof, we obtain a result of independent interest: parametrizations of certain Teichm{ü}ller spaces by holomorphic quadratic differentials. The main consequence of the domination result is the existence of a new collection of anti-de Sitter $3$-manifolds. We also present an application to the theory of minimal immersions into the Grassmanian of timelike planes in $\mathbb{R}^{2,2}$.

math.DG↗
arXiv2020

A factorization theorem for harmonic maps

Let $f$ be a harmonic map from a Riemann surface to a Riemannian $n$-manifold. We prove that if there is a holomorphic diffeomorphism $h$ between open subsets of the surface such that $f\circ h = f$, then $f$ factors through a holomorphic map onto another Riemann surface. If such $h$ is anti-holomorphic, we obtain an analogous statement. For minimal maps, this result is well known and is a consequence of the theory of branched immersions of surfaces due to Gulliver-Osserman-Royden. Our proof relies on various geometric properties of the Hopf differential.

math.DG↗