Narrow Results

In this paper, we study the degeneration of hyperbolic surfaces along a ray given by the harmonic map parametrization of Teichmüller space. The direction of the ray is determined by a holomorphic quadratic differential on a punctured Riemann surface, which has poles of order $\ge 2$ at each puncture. We show that the rescaled distance functions of the universal covers of hyperbolic surfaces uniformly converge, on a certain non-compact region containing a fundamental domain, to the intersection number with the vertical measured foliation given by the holomorphic quadratic differential determining the direction of the ray.

This implies that hyperbolic surfaces along the ray converge to the dual $ℝ$-tree of the vertical measured foliation in the sense of Gromov–Hausdorff. As an application, we determine the limit of the hyperbolic surfaces in the Thurston boundary.

We give an up-to-date overview of geometric and topological properties of cosymplectic and coKähler manifolds. We also mention some of their applications to time-dependent mechanics.

In this note, we will show that the fundamental group of any negatively δ-pinched manifold cannot be the fundamental group of a quasi-compact Kähler manifold. As a consequence of our proof, we also show that any nonuniform lattice in F_4(-20) cannot be the fundamental group of a quasi-compact Kähler manifold. The corresponding result for uniform lattices was proved by Carlson and Hernández [3]. Finally, we follow Gromov and Thurston [6] to give some examples of negatively δ-pinched manifolds of finite volume which, as topological manifolds, admit no hyperbolic metric with finite volume under any smooth structure. This shows that our result for δ-pinched manifolds is a nontrivial generalization of the fact that no nonuniform lattice in SO(n,1)(n≥3) is the fundamental group of a quasi-compact Kähler manifold [21].

By simplifying the first and the second variation formulas of the energy functional and generalizing the Weitzenböck formula, we study the stability and the rigidity of harmonic maps between Finsler manifolds. It is proved that any nondegenerate harmonic map from a compact Einstein Riemannian manifold with nonnegative scalar curvature to a Berwald manifold with nonpositive flag curvature is totally geodesic and there is no nondegenerate stable harmonic map from a Riemannian unit sphere Sⁿ (n > 2) to any Finsler manifold.

We shall exploit the Grassmannian theoretic point of view introduced by Segal in order to study harmonic maps from a two-sphere into the symplectic group Sp(n). By using this methodology, we shall be able to deduce an "uniton factorization" of such maps and an alternative characterization of harmonic two-spheres in the quaternionic projective space ℍPⁿ.

The blow-up analysis for a sequence of exponentially harmonic maps from a closed surface is studied to reestablish an existence result of harmonic maps from a closed surface into a closed manifold whose 2-dimensional homotopy class vanishes.

This paper is to study further properties of harmonic maps between Finsler manifolds. It is proved that any conformal harmonic map from an n(>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.

Given a smooth map from a compact Riemann surface to a complex manifold equipped with a strongly pseudoconvex complex Finsler metric, we define the -energy of the map, whose absolute minimum is attained by a holomorphic map. A harmonic map is then defined to be a stationary map of the -energy functional. We prove that with each harmonic map is associated a holomorphic quadratic differential on the domain, which vanishes if the map is weakly conformal. Also, under the condition that the metric be weakly Kähler, we determine the second variation of the functional, and prove that any -energy minimizing harmonic map from the Riemann sphere to a weakly Kähler Finsler manifold of positive curvature is either holomorphic or anti-holomorphic.

In this paper, we study a coupled system of the Ricci–Bourguignon flow on a closed Riemannian manifold $M$ with the harmonic map flow. At the first, we will investigate the existence and uniqueness for solution of this flow on a closed Riemannian manifold and then we find evolution of some geometric structures of manifold along this flow.

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.

Given a compact connected Riemann surface $\mathrm{\Sigma }$ of genus $g_{\mathrm{\Sigma }}\ge 2$, and an effective divisor $D={\sum }_i{n}_i{x}_i$ on $\mathrm{\Sigma }$ with $\text{degree}(D)<2(g_{\mathrm{\Sigma }}-1)$, there is a unique cone metric on $\mathrm{\Sigma }$ of constant negative curvature $-4$ such that the cone angle at each point $x_i$ is $2\pi n_i$ [R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc.103 (1988) 222–224; M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821]. We describe the Higgs bundle on $\mathrm{\Sigma }$ corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on $\mathrm{\Sigma }$ parametrized by a nonempty open subset of $H^0(\mathrm{\Sigma },K_{\mathrm{\Sigma }}^{\otimes 2}\otimes {{𝒪}}_{\mathrm{\Sigma }}(-2D))$ that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin’s results in [N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59–126] for the case $D=0$.

Using the fact that harmonic morphisms to a surface have minimal fibres, links between the volume-stability of the fibres and the energy-stability of the map are found of manifolds without boundary. A stability result for harmonic morphisms from a manifold with boundary to a Riemann surface is also established.

We use Sacks–Uhlenbeck's perturbation method to find critical points of the Yang–Mills–Higgs functional on fiber bundles with 2-dimensional base manifolds. Such critical points can be regarded as a generalization of harmonic maps from surfaces, and also a generalization of the so-called twisted holomorphic maps [15]. We prove an existence result analogous to the one for harmonic maps. In particular, we show that the so-called energy identity holds for the Yang–Mills–Higgs functional.

We investigate harmonic maps on almost contact metric manifolds which are locally conformal to almost cosymplectic manifolds. We obtain the necessary and sufficient conditions for the holomorphy to imply harmonicity and then we find obstructions to the existence of non-constant pluriharmonic maps. We also establish some results on the stability of the identity map on a locally conformal almost cosymplectic manifold of pointwise constant $\varphi$-holomorphic sectional curvature.

We study the strict convexity of the energy function of harmonic maps at their critical points from a Riemann surface to a Riemann surface, or to the product of negatively curved surfaces. When the target is a Riemann surface and when the map is of nonzero degree, we obtain a precise formula for the second derivative of the energy function along a Weil–Petersson geodesic, which implies that the energy function is strictly convex at its critical points. When the target is the product of two surfaces where each projection of the harmonic map is homotopic to a covering map, we also prove the strict convexity of the associated energy function. As an application we prove that the energy function has a unique critical point in these cases.

We survey results on infinitesimal deformations ("Jacobi fields") of harmonic maps, concentrating on (i) when they are integrable, i.e., arise from genuine deformations, and what this tells us, (ii) their relation with harmonic morphisms — maps which preserve Laplace's equation.

As a generalization of isometric immersions and Riemannian submersions, Riemannian maps were introduced by Fischer [Riemannian maps between Riemannian manifolds, Contemp. Math.132 (1992) 331–366]. It is known that a real valued Riemannian map satisfies the eikonal equation which provides a bridge between physical optics and geometrical optics. In this paper, we introduce invariant and anti-invariant Riemannian maps between Riemannian manifolds and almost Hermitian manifolds as a generalization of invariant immersions and totally real immersions, respectively. Then we give examples, present a characterization and obtain a geometric characterization of harmonic invariant Riemannian maps in terms of the distributions which are involved in the definition of such maps. We also give a decomposition theorem by using the existence of invariant Riemannian maps to Kähler manifolds. Moreover, we study anti-invariant Riemannian maps, give examples and obtain a classification theorem for umbilical anti-invariant Riemannian maps.

We present two transforms of non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct from one non-conformal harmonic map a sequence of non-conformal harmonic maps. We also discuss the correspondence between non-conformal harmonic maps into the 3-sphere, H-surfaces in Euclidean 3-space and almost complex surfaces in the nearly Kähler manifold S³ × S³.

In this paper, we study Riemannian submersions whose total manifolds admit a Ricci soliton. Here, we characterize any fiber of such a submersion is Ricci soliton or almost Ricci soliton. Indeed, we obtain necessary conditions for which the target manifold of Riemannian submersion is a Ricci soliton. Moreover, we study the harmonicity of Riemannian submersion from Ricci soliton and give a characterization for such a submersion to be harmonic.

In this paper, we study Clairaut Riemannian maps whose total manifolds admit a Ricci soliton and give a nontrivial example of such Clairaut Riemannian maps. First, we calculate Ricci tensors and scalar curvature of total manifolds of Clairaut Riemannian maps. Then we obtain necessary conditions for the fibers of such Clairaut Riemannian maps to be Einstein and almost Ricci solitons. We also obtain a necessary condition for vector field $\dot{\alpha }$ to be conformal, where $\alpha$ is a geodesic curve on total manifold of Clairaut Riemannian map. Further, we show that if total manifolds of Clairaut Riemannian maps admit a Ricci soliton with the potential mean curvature vector field of $\mathrm{\text{ker}}{F}_{\ast }$ then the total manifolds of Clairaut Riemannian maps also admit a gradient Ricci soliton and obtain a necessary and sufficient condition for such maps to be harmonic by solving Poisson equation.