Equivariant Harmonic Maps Research Articles

Minimal surfaces for Hitchin representations

Given a reductive representation $\rho: \pi_1(S)\rightarrow G$, there exists a $\rho$-equivariant harmonic map $f$ from the universal cover of a fixed Riemann surface $\Sigma$ to the symmetric space $G/K$ associated to $G$. If the Hopf differential of $f$ vanishes, the harmonic map is then minimal. In this paper, we investigate the properties of immersed minimal surfaces inside symmetric space associated to a subloci of Hitchin component: $q_n$ and $q_{n-1}$ case. First, we show that the pullback metric of the minimal surface dominates a constant multiple of the hyperbolic metric in the same conformal class and has a strong rigidity property. Secondly, we show that the immersed minimal surface is never tangential to any flat inside the symmetric space. As a direct corollary, the pullback metric of the minimal surface is always strictly negatively curved. In the end, we find a fully decoupled system to approximate the coupled Hitchin system.

Harmonic maps for Hitchin representations

Let $(S,g_0)$ be a hyperbolic surface, $\rho$ be a Hitchin representation for $PSL(n,\mathbb R)$, and $f$ be the unique $\rho$-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 $e(f)\equiv 1$ and $\rho$ is the base $n$-Fuchsian representation of $(S,g_0)$. In particular, we show given a Hitchin representation $\rho$ for $PSL(n,\mathbb R)$, every $\rho$-equivariant minimal immersion $f$ from a hyperbolic plane $\mathbb H^2$ into the corresponding symmetric space $X$ is distance-increasing, i.e. $f^*(g_{X})\geq g_{\mathbb H^2}$. Equality holds at one point only if it holds everywhere and $\rho$ is an $n$-Fuchsian representation.

Stability of stationary equivariant wave maps from the hyperbolic plane

In this paper we initiate the study of equivariant wave maps from $2d$ hyperbolic space, ${\Bbb H}^2$, into rotationally symmetric surfaces. This problem exhibits markedly different phenomena than its Euclidean counterpart due to the exponential volume growth of concentric geodesic spheres on the domain. In particular, when the target is ${\Bbb S}^2$, we find a family of equivariant harmonic maps ${\Bbb H}^2\to{\Bbb S}^2$, indexed by a parameter that measures how far the image of each harmonic map wraps around the sphere. These maps have energies taking all values between zero and the energy of the unique co-rotational Euclidean harmonic map, $Q_{{\rm euc}}$, from ${\Bbb R}^2$ to ${\Bbb S}^2$, given by stereographic projection. We prove that the harmonic maps are asymptotically stable for values of the parameter smaller than a threshold that is large enough to allow for maps that wrap more than halfway around the sphere. Indeed, we prove Strichartz estimates for the operator obtained by linearizing around such a harmonic map. However, for harmonic maps with energies approaching the Euclidean energy of $Q_{{\rm euc}}$, asymptotic stability via a perturbative argument based on Strichartz estimates is precluded by the existence of gap eigenvalues in the spectrum of the linearized operator. When the target is ${\Bbb H}^2$, we find a continuous family of asymptotically stable equivariant harmonic maps ${ \Bbb H}^2\to{\Bbb H}^2$ with arbitrarily small and arbitrarily large energies. This stands in sharp contrast to the corresponding problem on Euclidean space, where all finite energy solutions scatter to zero as time tends to infinity.

Asymptotics of Higgs bundles in the Hitchin component

In this paper we pursue a more geometric approach to compactification of the Hitchin component. Our main motivation is Wolf's harmonic map interpretation of Thurston's compactification of Teichmüller space with measured foliations. Using Hitchin's parameterization of the Hitchin component by holomorphic differentials, we study asymptotics of certain rays of representations. More precisely, along these rays we solve the Hitchin equations asymptotically and use the solution to study the asymptotics of the parallel transport operator of the associated flat connection. The asymptotics of the corresponding family of equivariant harmonic maps to the symmetric space SL(n,R)/SO(n,R) proves a conjecture of Katzarkov, Noll, Pandit and Simpson [17] on the Hitchin WKB problem in our setting.

Maximal $${\mathsf {Sp}}(4,{\mathbb {R}})$$ surface group representations, minimal immersions and cyclic surfaces

Let S be a closed surface of genus at least 2. For each maximal representation $$\rho : \pi _1(S){\rightarrow }{\mathsf {Sp}}(4,{\mathbb {R}})$$ in one of the $$2g-3$$ exceptional connected components, we prove there is a unique conformal structure on the surface in which the corresponding equivariant harmonic map to the symmetric space $${\mathsf {Sp}}(4,{\mathbb {R}})/{\mathsf {U}}(2)$$ is a minimal immersion. Using a Higgs bundle parameterization of these components, we give a mapping class group invariant parameterization of such components as fiber bundles over Teichmuller space. Unlike Labourie’s recent results on Hitchin components, these bundles are not vector bundles.

Loop groups in Yang–Mills theory

We consider the Yang–Mills equations with a matrix gauge group G on the de Sitter dS4, anti-de Sitter AdS4 and Minkowski R3,1 spaces. On all these spaces one can introduce a doubly warped metric in the form ds2=−du2+f2dv2+h2dsH22, where f and h are the functions of u and dsH22 is the metric on the two-dimensional hyperbolic space H2. We show that in the adiabatic limit, when the metric on H2 is scaled down, the Yang–Mills equations become the sigma-model equations describing harmonic maps from a two-dimensional manifold (dS2, AdS2 or R1,1, respectively) into the based loop group ΩG=C∞(S1,G)/G of smooth maps from the boundary circle S1=∂H2 of H2 into the gauge group G. For compact groups G these harmonic map equations are reduced to equations of geodesics on ΩG, solutions of which yield magnetic-type configurations of Yang–Mills fields. The group ΩG naturally acts on their moduli space.

Harmonic maps and hypersymplectic geometry

We study the hypersymplectic geometry of the moduli space of solutions to Hitchin’s harmonic map equations on a G-bundle. This is the split-signature analogue of Hitchin’s Higgs bundle moduli space. Due to the lack of definiteness, this moduli space is globally not well-behaved. However, we are able to construct a smooth open set consisting of solutions with small Higgs field, on which we can investigate the hypersymplectic geometry. Finally, we reinterpret our results in terms of the Riemannian geometry of the moduli space of G-connections.

ENERGY CONCENTRATION FOR 2-DIMENSIONAL RADIALLY SYMMETRIC EQUIVARIANT HARMONIC MAP HEAT FLOWS

We give a description of singularity formation in terms of energy quanta for 2-dimensional radially symmetric equivariant harmonic map heat flows. Adapting Struwe's energy method we first establish a finite bubble tree result with a discrete multiple of energy quanta disappearing in the singularity. We then use intersection-comparison arguments to show that the bubble tree consists of a single bubble only and that there is a well defined scale RBHK(t) ↓ 0 in which the solution converges to the standard harmonic map.

An Overview on Numerical Analyses of Nematic Liquid Crystal Flows

The purpose of this work is to provide an overview of the most recent numerical developments in the field of nematic liquid crystals. The Ericksen-Leslie equations govern the motion of a nematic liquid crystal. This system, in its simplest form, consists of the Navier-Stokes equations coupled with an extra anisotropic stress tensor, which represents the effect of the nematic liquid crystal on the fluid, and a convective harmonic map equation. The sphere constraint must be enforced almost everywhere in order to obtain an energy estimate. Since an almost everywhere satisfaction of this restriction is not appropriate at a numerical level, two alternative approaches have been introduced: a penalty method and a saddle-point method. These approaches are suitable for their numerical approximation by finite elements, since a discrete version of the restriction is enough to prove the desired energy estimate.The Ginzburg-Landau penalty function is usually used to enforce the sphere constraint. Finite element methods of mixed type will play an important role when designing numerical approximations for the penalty method in order to preserve the intrinsic energy estimate.The inf-sup condition that makes the saddle-point method well-posed is not clear yet. The only inf-sup condition for the Lagrange multiplier is obtained in the dual space of H 1(Ω). But such an inf-sup condition requires more regularity for the director vector than the one provided by the energy estimate. Herein, we will present an alternative inf-sup condition whose proof for its discrete counterpart with finite elements is still open.

Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres

Let \({{\bf X} \subset {\mathbb R}^n}\) be a generalised annulus and consider the Dirichlet energy functional $$ {\mathbb E}[u; {\bf X}]:=\frac{1}{2} \int\nolimits_{\bf X} |\nabla u (x)|^2 \, dx, $$ on the space of admissible maps $$ {\mathcal A}_\varphi({\bf X}) = \bigg\{u \in W^{1,2} ({\bf X}, {\mathbb S}^{n-1}) : u|_{\partial {\bf X}} = \varphi \bigg\}. $$ Here \({\varphi \in {\bf C}(\partial {\bf X}, {\mathbb S}^{n-1})}\) is fixed and \({{\mathcal A}_\varphi({\bf X})}\) is non-empty. In this paper we introduce a class of maps referred to as spherical twists and examine them in connection with the Euler–Lagrange equation associated with \({{\mathbb E}[\cdot, {\bf X}]}\) on \({{\mathcal A}_\varphi({\bf X})}\) [the so-called harmonic map equation on X]. The main result here is an interesting discrepancy between even and odd dimensions. Indeed for even n subject to a compatibility condition on φ the latter system admits infinitely many smooth solutions modulo isometries whereas for odd n this number reduces to one or none. We discuss qualitative features of the solutions in view of their novel and explicit representation through the exponential map of the compact Lie group SO(n).

Spacelike surfaces in De Sitter 3-space and their twistor lifts

We deal here with the geometry of the so-called twistor fibration Z → S 1 3 over the De Sitter 3-space, where the total space Z is a five-dimensional reductive homogeneous space with two canonical invariant almost CR structures. Fixed the normal metric on Z we study the harmonic map equation for smooth maps of Riemann surfaces into Z . A characterization of spacelike surfaces with harmonic twistor lifts to Z is obtained. Also it is shown that the harmonic map equation for twistor lifts can be formulated as the curvature vanishing of an S 1 -loop of connections i.e. harmonic twistor lifts exist within S 1 -families. Special harmonic maps such as holomorphic twistor lifts are also considered and some remarks concerning (compact) vacua of the twistor energy are given.

Bergman approximations of harmonic maps into the space of Kahler metrics on toric varieties

We generalize the results of Song–Zelditch on geodesics in spaces of Kahler metrics on toric varieties to harmonic maps of any compact Riemannian manifold with boundary into the space of Kahler metrics on a toric variety. We show that the harmonic map equation can always be solved and that such maps may be approximated in the $C2$ topology by harmonic maps into the spaces of Bergman metrics. In particular, Wess–Zumino–Witten (WZW) maps, or equivalently solutions of a homogeneous Monge–Ampere equation on the product of the manifold with a Riemann surface with $S1$ boundary admit such approximations. We also show that the Eells–Sampson flow on the space of Kahler potentials is transformed to the usual heat flow on the space of symplectic potentials under the Legendre transform, and hence it exists for all time and converges.

Note on equivariant harmonic maps in complex projective spaces

We give a simple criterion for equivariant harmonic maps into complex projective spaces CP n . As an application of the criterion, we give examples of equivariant harmonic cylinders. We also give examples of non-equivariant harmonic cylinders as perturbations of equivariant harmonic cylinders.

The Radius of Vanishing Bubbles in Equivariant Harmonic Map Flow from $D^2$ to $S^2$

We derive an upper bound for the radius $R(t)$ of a vanishing bubble in a family of equivariant maps $F_t:D^2\to S^2$ which evolve by the harmonic map flow. The self-similar “type 1” radius would be $R(t)=C\sqrt{T-t}$. We prove that $R(t)=o(T-t)$.

Asymptotic stability of harmonic maps under the Schrödinger flow

For Schr ¨ odinger maps from R 2 × R + to the 2-sphere S 2 , it is not known if finite energy solutions can form singularities (blow up) in finite time. We consider equivariant solutions with energy near the energy of the two-parameter family of equivariant harmonic maps. We prove that if the topological degree of the map is at least four, blowup does not occur, and global solutions converge (in a dispersive sense, i.e., scatter) to a fixed harmonic map as time tends to infinity. The proof uses, among other things, a time-dependent splitting of the solution, the generalized Hasimoto transform, and Strichartz (dispersive) estimates for a certain two space–dimensional linear Schr¨ odinger equation whose potential has critical power spatial singularity and decay. Along the way, we establish an energy-space local well-posedness result for which the existence time is determined by the length scale of a nearby harmonic map.

The higher flows of harmonic maps and an application to the Virasoro action

One characteristic of soliton equations is their infinite number of conservation laws. These conservation laws give rise to a hierarchy of partial differential equations. One example is the AKNS hierarchy. Terng and Uhlenbeck, Poisson Actions and Scattering Theory for Integrable Systems, Surveys in Differential Geometry: Integrable Systems (A supplement to J. Diff. Geom.), 4 (1998) described the conservation laws that give rise to the AKNS hierarchy. This hierarchy can also be realized in a loop group setting, where the equations are called the positive flows. The name arises from the power of λ which generates each flow. Terng (Loop Groups and Integrable Systems, Preliminary notes) also showed that in some sense, the harmonic map equation arises as a −1-flow in this hierarchy. The harmonic map equation also has a hierarchy of negative flows. However, these are not differential equations like the positive flows. It remained to be asked how the positive and negative flows relate and if they commute. The answer to the latter question is in the positive. As for the first question, the two hierarchies are part of one larger hierarchy. This was solved by using a more complex loop group and explicitly determining the loop algebra splitting. This splitting has applications to physics and is applied to an example of Schwarz involving the Virasoro action in Sec. VIII.

Global existence and blow-up for harmonic map heat flow

For degree-one equivariant maps on bounded domains, the question of finite-time blow-up vs. global existence of solutions to the harmonic map heat flow has been well studied. In this paper we study the Cauchy problem for degree- m equivariant harmonic map heat flow from ( 2 + 1 ) -dimensional space–time into the 2-sphere with initial energy close to the energy of harmonic maps. It is proved that solutions are globally smooth for m ⩾ 4 , whereas for m = 1 , we show that finite-time singularities can form for this class of data.

Geometric evolution equations in critical dimensions

We make a qualitative comparison of phenomena occurring in two different geometric flows: the harmonic map heat flow in two space dimensions and the Yang–Mills heat flow in four space dimensions. Our results are a regularity result for the degree-2 equivariant harmonic map flow, and a blow-up result for an equivariant Yang–Mills-like flow. The results show that qualitatively differing behaviours observed in the two flows can be attributed to the degree of the equivariance.

Existence and regularity for generalised harmonic maps associated to a nonlocal polyconvex energy of Skyrme type

We prove existence and regularity of critical points of arbitrary degree for a generalised harmonic map problem, in which there is an additional nonlocal polyconvex term in the energy, heuristically of the same order as the Dirichlet term. The proof of regularity hinges upon a special nonlinear structure in the Euler–Lagrange equation similar to that possessed by the harmonic map equation. The functional is of a type appearing in certain models of the quantum Hall effect describing nonlocal Skyrmions.

On a class of mappings between Riemannian manifolds

Effects of geometric constraints on a steady flow potential are described by an elliptic-hyperbolic generalization of the harmonic map equations. Sufficient conditions are given for global triviality.