Conformal Harmonic Maps Research Articles

The fractional Hopf differential and a weak formulation of stationarity for the half Dirichlet energy

Abstract We obtain a weak formulation of the stationarity condition for the half Dirichlet energy, which can be expressed in terms of a fractional quantity, related to the trace of the Hopf differential of the harmonic extension of the original map. As an application we show that conformal harmonic maps from the disc are precisely the harmonic extensions of stationary points of the half Dirichlet energy on the circle. We also derive a Noether theorem and a Pohozaev identity for stationary points of the half Dirichlet energy.

A Schwarz lemma of harmonic maps into metric spaces

<p>We established a Schwarz lemma for harmonic maps from Riemannian manifolds to metric spaces of curvature bounded above in the sense of Alexandrov. We adopted the gradient estimate technique which was based on Zhang-Zhu's maximum principle. In particular, when the domain manifold was a hyperbolic surface, the energy of any conformal harmonic maps into $ \operatorname{CAT}(-1) $ spaces were bounded from above uniformly.</p>

Domains without parabolic minimal submanifolds and weakly hyperbolic domains

AbstractWe show that if is an ‐convex domain in for some whose boundary has a tubular neighbourhood of positive radius and is not ‐flat near infinity, then does not contain any immersed parabolic minimal submanifolds of dimension . In particular, if is a properly embedded non‐flat minimal hypersurface in with a tubular neighbourhood of positive radius, then every immersed parabolic hypersurface in intersects . In dimension , this holds if has bounded Gaussian curvature function. We also introduce the class of weakly hyperbolic domains in , characterised by the property that every conformal harmonic map is constant, and we elucidate their relationship with hyperbolic domains, and domains without parabolic minimal surfaces.

Novikov-Veselov Symmetries of the Two-Dimensional O(N) Sigma Model

We show that Novikov-Veselov hierarchy provides a complete family of commuting symmetries of two-dimensional O(N) sigma model. In the first part of the paper we use these symmetries to prove that the Fermi spectral curve for the double-periodic sigma model is algebraic. Thus, our previous construction of the complexified harmonic maps in the case of irreducible Fermi curves is complete. In the second part of the paper we generalize our construction to the case of reducible Fermi curves and show that it gives the conformal harmonic maps to even-dimensional spheres. Remarkably, the solutions are parameterized by spectral curves of turning points of the elliptic Calogero-Moser system.

Every smoothly bounded p-convex domain in [formula omitted] admits a p-plurisubharmonic defining function

We show that every bounded domain D in Rn with smooth p-convex boundary for 2≤p<n admits a smooth defining function ρ which is p-plurisubharmonic on D‾; if in addition bD has no p-flat points then ρ can be chosen strongly p-plurisubharmonic on D. If bD is 2-convex then for any open connected conformal surface M and conformal harmonic map f:M→D‾, either f(M)⊂D or f(M)⊂bD. In particular, every conformal harmonic map D⁎→D from the punctured disc extends to a conformal harmonic map D→D.

Harmonic branched coverings and uniformization of CAT(k) spheres

Abstract Let S be a surface with a metric d satisfying an upper curvature bound in the sense of Alexandrov (i.e. via triangle comparison). We show that an almost conformal harmonic map from a surface into ( S , d ) {(S,d)} is a branched covering. As a consequence, if ( S , d ) {(S,d)} is homeomorphically equivalent to the 2-sphere 𝕊 2 {\mathbb{S}^{2}} , then it is conformally equivalent to 𝕊 2 {\mathbb{S}^{2}} .

Bubble tree convergence for harmonic maps into compact locally CAT(1) spaces

We determine bubble tree convergence for a sequence of harmonic maps, with uniform energy bounds, from a compact Riemann surface into a compact locally CAT(1) space. In particular, we demonstrate energy quantization and the no-neck property for such a sequence. In the smooth setting, Jost (Two-dimensional geometric variational problems. Pure and applied mathematics. Wiley, New York, 1991) and Parker (J Differ Geom 44(3):595–633, 1996) respectively established these results by exploiting now classical arguments for harmonic maps. Our work demonstrates that these results can be reinterpreted geometrically. In the absence of a PDE, we take advantage of the local convexity properties of the target space. Included in this paper are an $$\epsilon $$ -regularity theorem, an energy gap theorem, and a removable singularity theorem for harmonic maps into metric spaces with upper curvature bounds. We also prove an isoperimetric inequality for conformal harmonic maps with small image.

Equivariant minimal surfaces in $\mathbb{CH}^2$ and their Higgs bundles

This paper gives a construction for all minimal immersions $f$ of the Poincar\'{e} disc into the complex hyperbolic plane $\mathbb{CH}^2$ which are equivariant with respect to an irreducible representation $\rho$ of a hyperbolic surface group into $PU(2,1)$. We exploit the fact that each such immersion is a twisted conformal harmonic map and therefore has a corresponding Higgs bundle. We identify the structure of these Higgs bundles and show how each is determined by properties of the map, including the induced metric and a holomorphic cubic differential on the surface. We show that the moduli space of pairs $(\rho,f)$ is a disjoint union of finitely many complex manifolds, whose structure we fully describe. The holomorphic (or anti-holomorphic) maps provide multiple components of this union, as do the non-holomorphic maps. Each of the latter components has the same dimension as the representation variety for $PU(2,1)$, and is indexed by the number of complex and anti-complex points of the immersion. These numbers determine the Toledo invariant and the Euler number of the normal bundle of the immersion. We show that there is an open set of quasi-Fuchsian representations of Toledo invariant zero for which the minimal surface is unique and Lagrangian.

Whitham deformations and the space of harmonic tori in S3

In this paper we investigate the space of harmonic maps from a 2-torus to $\mathbb{S}^3$ using the spectral curve correspondence and Whitham deformations. In an open and dense subset of a parameter space we find that the space of harmonic maps is smooth and has dimension two. We also show that the points that correspond to minimal tori (conformal harmonic maps) are either smooth points of dimension two or singular.

Generalized harmonic morphisms and horizontally weakly conformal biharmonic maps

Harmonic morphisms are maps between Riemannian manifolds that pull back harmonic functions to harmonic functions. These maps are characterized as horizontally weakly conformal harmonic maps and they have many interesting links and applications to several areas in mathematics (see the book [2] by Baird and Wood for details). In this paper, we study generalized harmonic morphisms which are defined to be maps between Riemannian manifolds that pull back harmonic functions to biharmonic functions. We obtain some characterizations of generalized harmonic morphisms into a Euclidean space and give two methods of constructions that can be used to produce many examples of generalized harmonic morphisms which are not harmonic morphisms. We also give a complete classification of generalized harmonic morphisms among the projections of a warped product space, which provides infinitely many examples of proper biharmonic Riemannian submersions and conformal submersions from a warped product manifold.

The regularity of conformal target harmonic maps

We establish that any weakly conformal $$W^{1,2}$$ map from a Riemann surface S into a closed oriented sub-manifold $$N^n$$ of an euclidian space $${\mathbb {R}}^m$$ realizes, for almost every sub-domain, a stationary varifold if and only if it is a smooth conformal harmonic map form S into $$N^n$$ .

Some rigidity theorems of harmonic maps between Finsler manifolds

This paper is to study further properties of harmonic maps between Finsler manifolds. It is proved that any conformal harmonic map from an n(&gt;2)-dimensional Finsler manifold to a Finsler manifold must be homothetic and some rigidity theorems for harmonic maps between Finsler manifolds are given, which improve some results in earlier papers and generalize Eells–Sampson's theorem and Sealey's theorem in Riemannian Geometry.

Superminimal surfaces in the 6-sphere

In this article, we use the harmonic sequence associated to a weakly conformal harmonic map f: S → S 6 in order to determine explicit examples of linearly full almost complex 2-spheres of S 6 with at most two singularities. We prove that the singularity type of these almost complex 2-spheres has an extra symmetry and this allows us to determine the moduli space of such curves with suitably small area. We also characterize projectively equivalent almost complex curves of S 6 in terms of G 2 ℂ -equivalence of their directrix curves.

A construction of conformal-harmonic maps

Conformal harmonic maps from a 4-dimensional conformal manifold to a Riemannian manifold are maps satisfying a certain conformally invariant fourth order equation. We prove a general existence result for conformal harmonic maps, analogous to the Eells–Sampson theorem for harmonic maps. The proof uses a geometric flow and relies on results of Gursky–Viaclovsky and Lamm.

Harmonic maps and constant mean curvature surfaces in H 2 x R

We introduce a hyperbolic Gauss map into the Poincar´e disk for any surface in H2×R with regular vertical projection, and prove that if the surface has constant mean curvature H = 1/2, this hyperbolic Gauss map is harmonic. Conversely, we show that every nowhere conformal harmonic map from an open simply connected Riemann surface Σ into the Poincaré disk is the hyperbolic Gauss map of a two-parameter family of such surfaces. As an application we obtain that any holomorphic quadratic differential on Σ can be realized as the Abresch-Rosenberg holomorphic differential of some, and generically infinitely many, complete surfaces with H = 1/2 in H2 x R. A similar result applies to minimal surfaces in the Heisenberg group Nil3. Finally, we classify all complete minimal vertical graphs in H2 × R.

Bumpy metrics and closed parametrized minimal surfaces in Riemannian manifolds

The purpose of this article is to study conformal harmonic maps f: Σ → M, where Σ is a closed Riemann surface and M is a compact Riemannian manifold of dimension at least four. Such maps define parametrized minimal surfaces, possibly with branch points. We show that when the ambient manifold M is given a generic metric, all prime closed parametrized minimal surfaces are free of branch points, and are as Morse nondegenerate as allowed by the group of automorphisms of E. They are Morse nondegenerate in the usual sense if Σ has genus at least two, lie on two-dimensional nondegenerate critical submanifolds if S has genus one, and on six-dimensional nondegenerate critical submanifolds if Σ has genus zero.

Harmonic maps between surfaces and Teichmüller spaces

We study harmonic maps between two distinct compact Riemann surfaces of the same genus. Our aim is to examine the role of the curvature in the relationship between energies of harmonic maps and conformal metrics on the target surface. We consider the energy of the energy minimizing map between the surfaces as a function of the metrics within the conformal class of the target and solve the variational problem of minimizing this functional when we restrict the upper curvature bound and normalize the area of the metrics. In doing so, we show that the upper curvature bound on the normalized target metric is an obstruction in obtaining a degree 1 conformal harmonic map between distinct Riemann surfaces.

The existence of embedded minimal surfaces and the problem of uniqueness

It was in the early nineteen thirties that Douglas and Radd solved the Plateau problem for a rectifiable Jordan curve in Euclidean space. The minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk. In 1948 Morrey generalized the theorem of Douglas and Radd to minimal surfaces in a homogeneously regular Riemannian manifold. In both cases the solutions are defective as they may possess branch points. It was not until 1968 that Osserman [-24] was able to prove that interior branch points do not exist on the Douglas-Radd-Morrey solution. (Osserman's theorem in this full generality was proved by Alt [-2, 3] and Gulliver [8].) If the Jordan curve is real analytic, Gulliver and Lesley [9] showed that the solution surface is free of branch points even on its boundary. Hence, at least in this case, we know that the Douglas-Radd-Morrey solution is an immersion and represents a classical surface. However, the Douglas-Radd-Morrey solution may in general have selfintersections. It was then a classical problem to find conditions on the Jordan curve to guarantee that the solution is an embedded disk. In 1932 Radd was the first to produce a general condition of this type. He proved that if the curve can be projected onto a convex curve in the plane, then it bounds a unique embedded minimal disk which is a graph over this plane. It was not until the seventies that there was further progress on the problem of embeddedness. Osserman had conjectured that if a Jordan curve is extreme, that is, lies on the boundary of its convex hull, then the Douglas-Rad6 solution is embedded. Gulliver and Spruck [10] proved this conjecture under the additional assumption that the absolute total curvature of the Jordan curve is less than 4~z. Right after this work, Tomi and Tromba [32] showed that an extreme curve must bound an embedded minimal disk which need not be stable. In 1976 Almgren and Simon [1] established the existence for such a curve of an embedded stable minimal disk (which need not be a Douglas-Radd solution). Finally in the same year, the authors [18] proved that any Douglas-Rad6 solution must be embedded.

Existence of harmonic maps into CAT(1) spaces

Let $\varphi\in C^0 \cap W^{1,2}(\Sigma, X)$ where $\Sigma$ is a compact Riemann surface, $X$ is a compact locally CAT(1) space, and $W^{1,2}(\Sigma,X)$ is defined as in Korevaar-Schoen. We use the technique of harmonic replacement to prove that either there exists a harmonic map $u:\Sigma \to X$ homotopic to $\varphi$ or there exists a conformal harmonic map $v:\mathbb S^2 \to X$. To complete the argument, we prove compactness for energy minimizers and a removable singularity theorem for conformal harmonic maps.