arXiv research

The study constructs equivariant harmonic maps into symmetric spaces with applications to Willmore surfaces.

Characterizes area-minimizing maps for surfaces of genus ≥ 2.

We present effective methods to compute equivariant harmonic maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and derive convergence of the discrete heat flow to the energy minimizer, with explic…

New minimal surface theory disproves a conjecture in symmetric spaces.

Minimal Lagrangian surfaces in complex hyperbolic quadric via loop group method.

A projection maps geodesic currents to Teichmüller space.

We show that the Korevaar-Schoen limit of the sequence of equivariant harmonic maps corresponding to a sequence of irreducible $SL_2({\mathbb C})$ representations of the fundamental group of a compact Riemannian manifold is an equivariant harmonic map to an ${\mathbb R}$-tree which is minimal and whose length function …

The paper studies minimal surfaces in complex hyperbolic spaces, showing moduli space connected components.

Maps on surfaces can be embedded into spheres with minimal dimensions.

The Hopf fibration is rigid among minimal maps between spheres.

We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen-Macaulay, which is a generalization of equivariant formality for torus actions wi…

Unified method for CNNs to approximate equivariant maps across various groups.

Given two Fuchsian representations $ρ_l$ and $ρ_r$ of the fundamental group of a closed oriented surface $S$ of genus $\geq 2$, we study the relation between Lagrangian submanifolds of $M_ρ=(\mathbb{H}^2/ρ_l(π_1(S)))\times (\mathbb{H}^2/ρ_r(π_1(S)))$ and $ρ$-equivariant embeddings $σ$ of $\widetilde S$ into Anti-de Sit…

Minimal surfaces with symmetries exist under certain group actions.

Equivariant neural networks use symmetry to interpret complex data.

The abstract applies waist inequality to dynamical systems and entropy.

A spine is constructed for a non-orientable surface's decorated Teichmüller space.

This paper gives a construction for all minimal immersions $f$ of the Poincaré disc into the complex hyperbolic plane $\mathbb{CH}^2$ which are equivariant with respect to an irreducible representation $ρ$ of a hyperbolic surface group into $PU(2,1)$. We exploit the fact that each such immersion is a twisted conformal …

Researchers found uncountable harmonic self-maps in complex projective spaces.

Constructs harmonic maps between special geometric shapes.

Proves unique maps from certain spaces to others.

This survey studies equivariant harmonic maps arising from Higgs bundles. We explain the non-abelian Hodge correspondence and focus on the role of equivariant harmonic maps in the correspondence. With the preparation, we review current progress towards some open problems in the study of equivariant harmonic maps.

Bound on equivariant index for min-max surfaces.

New minimal discs and annuli found in ellipsoids.

Minimal dimensions found for flag manifolds embeddings.

Unified rigidity theorem for cyclic and alternating surfaces.

Consider the equivariant wave map equation from Minkowski space to a rotationnally symmetric manifold which has an equator (example: the sphere). In dimension 3, this article gives a necessary and sufficient condition for the existence of a smooth self-similar blow up profile. More generally, we study the relation betw…

Projections from flats to maximal flats defined and studied.

Paper proves finiteness and Morse index estimates for equivariant min-max hypersurfaces.

Develops a new method for minimal Lagrangian surfaces in complex quadrics.

We present a general theory of Group equivariant Convolutional Neural Networks (G-CNNs) on homogeneous spaces such as Euclidean space and the sphere. Feature maps in these networks represent fields on a homogeneous base space, and layers are equivariant maps between spaces of fields. The theory enables a systematic cla…

We prove that a primitive harmonic map is equivariant if and only if it admits a holomorphic potential of degree one. We investigate when the equivariant harmonic map is periodic, and as an application discuss constant mean curvature cylinders with screw motion symmetries.

Proves minimization for Kähler manifolds with automorphisms.

Investigates harmonic self-maps' stability on cohomogeneity one manifolds.

We prove that for $k\ge 5$ there does not exist a continuous map $\partial CV(F_k)\to\mathbb PCurr(F_k)$ that is either $Out(F_k)$-equivariant or $Out(F_k)$-anti-equivariant. Here $\partial CV(F_k)$ is the "length-function" boundary of Culler-Vogtmann's Outer space $CV(F_k)$, and $\mathbb PCurr(F_k)$ is the space of pr…

Harmonic maps depend analytically on representations.

New minimal hypersphere found in 4-sphere solving Bernstein problem.

In this paper we continue our study of equivariant minimal Lagrangian surfaces in $\mathbb{C} P^2$, characterizing the rotationally equivariant cases and providing explicit formulae for relevant geometric quantities of translationally equivariant minimal Lagrangian surfaces in terms of Weierstrass elliptic functions.

The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a …

Normal forms for equivariant maps in infinite dimensions established.

Let $(S,g_0)$ be a hyperbolic surface, $ρ$ be a Hitchin representation for $PSL(n,\mathbb R)$, and $f$ be the unique $ρ$-equivariant harmonic map from $(\widetilde S, \widetilde g_0)$ to the corresponding symmetric space. We show its energy density satisfies $e(f)\geq 1$ and equality holds at one point only if $e(f)\eq…

New minimal surfaces found in ball with boundary constraints.

Minimal equivariant embedding found for flag manifolds.

Conditions for pre-quantizability of G-invariant forms are derived using moment maps.

Study the relationship between orbit braid group and equivariant mapping class group on surfaces.

The paper finds rotationally symmetric critical metrics for Laplace eigenvalues on tori.

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 constructio…

We show that if G is a discrete subgroup of the group of the isometries of the hyperbolic k-space H^k, and if R is a representation of G into the group of the isometries of H^n, then any R-equivariant map F from H^k to H^n extends to the boundary in a weak sense in the setting of Borel measures. As a consequence of thi…