We give a simple procedure to estimate the smallest Lipshitz constant of a degree 1 map from a Riemannian 2-sphere to the unit 2-sphere, up to a factor of 10. Using this procedure, we are able to prove several inequalities involving this Lipshitz constant. For instance, if the smallest Lipshitz constant is at least 1, then the Riemannian 2-sphere has Uryson 1-Width less than 12 and contains a closed geodesic of length less than 160. Similarly, if a closed oriented Riemannian surface does not admit a degree 1 map to the unit 2-sphere with Lipshitz constant 1, then it contains a closed homologically non-trivial curve of length less than 4 pi. On the other hand, we give examples of high genus surfaces with arbitrarily large Uryson 1-Width which do not admit a map of non-zero degree to the unit sphere with Lipshitz constant 1.
We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed homotopy class and all hyperbolic metrics on the surface. We give explicit examples of such hyperbolic surfaces through a new interpretation of the Nielsen realization problem for the mapping class groups.
We present some results about harmonic maps with possibly infinite energy from punctured Riemann surfaces to CAT(0) spaces. In particular, we give precise estimates of their energy growth near the punctures and prove their uniqueness.
Demoulin surfaces in real projective $3$-space are investigated. Our result enable us to establish a generalized Weierstrass type representation for definite Demoulin surfaces by virtue of primitive maps into a certain semi-Riemannian $6$-symmetric space.
We study the blow-up analysis and qualitative behavior for a sequence of harmonic maps with free boundary from degenerating bordered Riemann surfaces with uniformly bounded energy. With the help of Pohozaev type constants associated to harmonic maps defined on degenerating collars, including vertical boundary collars and horizontal boundary collars, we establish a generalized energy identity.
We study the existence of harmonic maps and Dirac-harmonic maps from degenerating surfaces to non-positive curved manifold via the scheme of Sacks and Uhlenbeck. By choosing a suitable sequence of $α$-(Dirac-)harmonic maps from a sequence of suitable closed surfaces degenerating to a hyperbolic surface, we get the convergence and a cleaner energy identity under the uniformly bounded energy assumption. In this energy identity, there is no energy loss near the punctures. As an application, we obtain an existence result about (Dirac-)harmonic maps from degenerating (spin) surfaces. If the energie…
We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant Gaussian curvature surfaces correspond to harmonic maps, and we examine the relationship between the two types of maps and their singularities. Finally, we determine which finitely A-determined map-germs from the plane to the plane can be represented by harmonic maps.
We relate the existence problem of harmonic maps into $S^2$ to the convex geometry of $S^2$. On one hand, this allows us to construct new examples of harmonic maps of degree 0 from compact surfaces of arbitrary genus into $S^2$. On the other hand, we produce new example of regions that do not contain closed geodesics (that is, harmonic maps from $S^1$) but do contain images of harmonic maps from other domains. These regions can therefore not support a strictly convex function. Our construction builds upon an example of W. Kendall, and uses M. Struwe's heat flow approach for the existence of ha…
In this paper, we study the Gauss map of surfaces in 3-dimensional Heisenberg group using the Gans model of the hyperbolic plane. We establish a relationship between the tension field of the Gauss map and mean curvature of a surface in $\mathcal{H}_{3}$.
Wave maps (or Lorentzian-harmonic maps) from a $1+1$-dimensional Lorentz space into the $2$-sphere are associated to constant negative Gaussian curvature surfaces in Euclidean 3-space via the Gauss map, which is harmonic with respect to the metric induced by the second fundamental form. We give a method for constructing germs of Lorentzian-harmonic maps from their $k$-jets and use this construction to study the singularities of such maps. We also show how to construct pseudospherical surfaces with prescribed singularities using loop groups. We study the singularities of pseudospherical surface…
We propose a novel meshless method to compute harmonic maps and conformal maps for surfaces embedded in the Euclidean 3-space, using point cloud data only. Given a surface, or a point cloud approximation, we simply use the standard cubic lattice to approximate its $ε$-neighborhood. Then the harmonic map of the surface can be approximated by discrete harmonic maps on lattices. The conformal map, or the surface uniformization, is achieved by minimizing the Dirichlet energy of the harmonic map while deforming the target surface of constant curvature. We propose algorithms and numerical examples f…
We estimate the dimensions of the spaces of holomorphic sections of certain line bundles to give improved lower bounds on the index of complex isotropic harmonic maps to complex projective space from the sphere and torus, and in some cases from higher genus surfaces.
$α$-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $α$-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For $α>1$, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in constructing $α$-harmonic maps for $α>1$ and then letting $α\to 1$. The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed $α$-Dirac-harmonic maps when …
We show that every one-harmonic map, in the sense of Trapani and Valli, from a Hadamard surface of pinched negative curvature to $\mathbb{H}^2$ has image the interior of the convex hull of a subset of $\partial_\infty\mathbb{H}^2$. The proof relies on Minkowski geometry, by interpreting one-harmonic maps as the Gauss maps of convex surfaces.
We exhibit the duality between best Lipschitz (infinity harmonic) maps and least gradient maps in the case of maps from surfaces to the circle. We show that given a homotopy class of a map from a surface to the circle the infinity harmonic map defines a geodesic lamination on the surface and the dual least gradient map defines a transverse measure on the lamination. This is the initial step towards an analytic approach to Thurston's work on best Lipschitz maps between hyperbolic surfaces and Thurston's asymmetric metric on Teichmueller space.
We construct a family of Hermitian metrics on the Hopf surface $ \mathbb{S}^3\times \mathbb{S}^1$, whose fundamental classes represent distinct cohomology classes in the Aeppli cohomology group. These metrics are locally conformally Kähler. Among the toric fibres of $π:\mathbb{S}^{3} \times \mathbb{S}^1\to\mathbb{C} P^1$ two of them are stable minimal surfaces and each of the two has a neighbourhood so that fibres therein are given by stable harmonic maps from 2-torus and outside, far away from the two tori, there are unstable harmonic ones that are also unstable minimal surfaces.
In this paper, we determine the type numbers of the pseudo-hyperbolic Gauss maps of all oriented Lorentzian surfaces of constant mean and Gaussian curvatures and non-diagonalizable shape operator in the $3$-dimensional anti-de Sitter space. Also, we investigate the behavior of type numbers of the pseudo-hyperbolic Gauss map along the parallel family of such oriented Lorentzian surfaces in the $3$-dimensional anti-de Sitter space. Furthermore, we investigate the type number of the pseudo-hyperbolic Gauss map of one of Lorentzian hypersurfaces of B-scroll type in a general dimensional anti-de Si…
In this paper, we study the behavior of a sequence of harmonic maps from surfaces with uniformly bounded energy on the generalized neck domain. The generalized neck domain is a union of ghost bubbles and annular neck domains, which connects non-trivial bubbles. An upper bound of the energy density is proved and we use it to study the limit of the nullity and index of the sequence.
For a sequence of coupled fields $\{(ϕ_n,ψ_n)\}$ from a compact Riemann surface $M$ with smooth boundary to a general compact Riemannian manifold with uniformly bounded energy and satisfying the Dirac-harmonic system up to some uniformly controlled error terms, we show that the energy identity holds during a blow-up process near the boundary. As an application to the heat flow of Dirac-harmonic maps from surfaces with boundary, when such a flow blows up at infinite time, we obtain an energy identity.
We consider codimension 2 sphere congruences in pseudo-conformal geometry that are harmonic with respect to the conformal structure of an orthogonal surface. We characterise the orthogonal surfaces of such congruences as either $S$-Willmore surfaces, quasi-umbilical surfaces, constant mean curvature surfaces in 3-dimensional space forms or surfaces of constant lightlike mean curvature in 3-dimensional lightcones. We then investigate Bryant's quartic differential in this context and show that generically this is divergence free if and only if the surface under consideration is either superconfo…
A curvature estimate for holomophic maps on open Riemann surfaces
Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Department of Mathematics, University of Education, Hue University（顺化大学教育学院数学系）
We consider harmonic diffeomorphisms to a fixed hyperbolic target $Y$, from a family of domain Riemann surfaces degenerating along a Teichmüller ray. We use the work of Minsky to show that there is a limiting harmonic map from the conformal limit of the Teichmüller ray, to a crowned hyperbolic surface. The target surface is the metric completion of the complement of a geodesic lamination on $Y$. The conformal limit is obtained by attaching half-planes and cylinders to the critical graph of the holomorphic quadratic differential determining the ray. As an application, we provide a new proof of …
We extend the well-known Sacks-Uhlenbeck energy gap result (1981) for harmonic maps from closed Riemann surfaces into closed Riemannian manifolds from the case of maps with small energy (thus near a constant map), to the case of harmonic maps with high absolute energy but small energy relative to a reference harmonic map.
