Bubbling of the heat flows for harmonic maps from surfaces

In this article we prove that any Palais-Smale sequence of the energy functional on surfaces with uniformly L²-bounded tension fields converges pointwise, by taking a subsequence if necessary, to a map from connected (possibly singular) surfaces, which consist of the original surfaces and finitely many bubble trees. We therefore get the corresponding results about how the solutions of heat flow for harmonic maps from surfaces form singularities at infinite time. © 1997 John Wiley & Sons, Inc.     

Harmonic maps from degenerating Riemann surfaces

We study harmonic maps from degenerating Riemann surfaces with uniformly bounded energy and show the so-called generalized energy identity. We find conditions that are both necessary and sufficient for the compactness in W ^1,2 and C ⁰ modulo bubbles of sequences of such maps.     

Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians

Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the Grothendieck Quot Scheme. The latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by Bradlow. This gives an effective way of computing certain intersection numbers (known as ``Gromov invariants') on the space of holomorphic maps into Grassmannians. We carry out these computations in the case where the Riemann surface has genus one.

     

The qualitative behavior for approximate Dirac-harmonic maps into stationary Lorentzian manifolds

For a sequence of approximate Dirac-harmonic maps from a closed spin Riemann surface into a stationary Lorentzian manifold with uniformly bounded energy, we study the blow-up analysis and show that the Lorentzian energy identity holds. Moreover, when the targets are static Lorentzian manifolds, we prove the positive energy identity and the no neck property.

     

General expansion for period maps of Riemann surfaces

In this paper,we get the full expansion for period map from the moduli space Mg of curves to the coarse moduli space Ag of g-dimensional principally polarized abelian varieties in Bers coordinates.This generalizes fully the famous Rauch's variational formula.As applications,we compute the curvature of Siegel metric at point [X] with Π([X]) =√ -1 Ig and the Christoffel symbols of L2-induced Bergman metric on Mg.     

Regularity of harmonic maps into a static Lorentzian manifold

We study the regularity of harmonic maps from Riemannian manifold into a static Lorentzian manifold. We show that when the domain manifold is two-dimensional, any weakly harmonic map is smooth. We also show that when dimension n of the domain manifold is greater than two, there exists a weakly harmonic map for the Dirichlet problem which is smooth except for a closed set whose (n − 2)-dimensional Hausdorff measure is zero.     

Fourth order approximation of harmonic maps from surfaces

Let be a compact Riemannian surface and let be a compact Riemannian manifold, both without boundary, and assume that N is isometrically embedded into some ℝ ^l . We consider a sequence of critical points of the functional with uniformly bounded energy. We show that this sequence converges weakly in and strongly away from finitely many points to a smooth harmonic map. One can perform a blow-up to show that there separate at most finitely many non-trivial harmonic two-spheres at these finitely many points. Finally we prove the so called energy identity for this approximation in the case that ↪ ℝ ^l . Mathematics Subject Classification (2000) 58E20, 35J60, 53C43     

Harmonic maps between annuli on Riemann surfaces

Let ρ _Σ = h(|z|²) be a metric in a Riemann surface Σ, where h is a positive real function. Let H _{r 1} = {w = f(z)} be the family of a univalent ρ _Σ harmonic mapping of the Euclidean annulus A(r ₁, 1):= {z: r ₁ < |z| < 1} onto a proper annulus A _Σ of the Riemann surface Σ, which is subject to some geometric restrictions. It is shown that if A _Σ is fixed, then sup{r ₁: ℋ _{r 1} ≠ ∅} < 1. This generalizes similar results from the Euclidean case. The cases of Riemann and of hyperbolic harmonic mappings are treated in detail. Using the fact that the Gauss map of a surface with constant mean curvature (CMC) is a Riemann harmonic mapping, an application to the CMC surfaces is given (see Corollary 3.2). In addition, some new examples of hyperbolic and Riemann radial harmonic diffeomorphisms are given, which have inspired some new J. C. C. Nitsche-type conjectures for the class of these mappings.     

Bubbling analysis for extrinsic biharmonic maps from general Riemannian 4-manifolds

In this paper, we study the blow-up analysis of extrinsic biharmonic maps from a general Riemannian4-manifold into a compact Riemannian manifold. We prove that the energy identity and the no neck property hold with the aid of a Pohozaev identity over general Riemannian manifolds.     

Pseudo-hyperbolic Gauss maps of Lorentzian surfaces in anti-de Sitter space

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 Sitter space.     

Energy identity for a class of approximate Dirac-harmonic maps from surfaces with boundary

For a sequence of coupled fields $\{({?}_n,{\psi }_n)\}$ 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.     

A note on non-univalent harmonic maps between surfaces

We show that a decomposition theorem of Duren-Hengartner about planar harmonic maps can be generalized to give a necessary and sufficient condition for a harmonic map between smooth surfaces to be decomposable as a holomorphic map followed by a univalent harmonic embedding.

     

Approximation by harmonic functions on closed subsets of Riemann surfaces

Given continuous functionsu and ∈ on a closed subsetF of a Riemann surface, we seek a harmonic functionv on the surface (possibly with logarithmic singularities) such that |u−v|<∈ onF. Research supported in part by NSERC—Canada and FCAR—Quebec.