Partial regularity for harmonic maps of evolution into spheres

We consider the global Cauchy problem for generalized Kirchhoff equations with small non-linear terms or small data. We solve this problem in the space of functions which are twice differentiable with respect to time coordinate and uniformly analytic with respect to other coordinates. We determine, in two different situations, estimates of lifespan of solutions for some problems with perturbations and we give stability result of the solution for small perturbations.     

Factoring periodic maps into Dehn twists

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\ge 1$. In this paper, we develop various methods for factoring periodic mapping classes into Dehn twists, up to conjugacy. As applications, we develop methods for factoring certain roots of Dehn twists as words in Dehn twists. We will also show the existence of conjugates of periodic maps of order 4g and $4g+2$, for $g\ge 2$, whose product is pseudo-Anosov.     

Constructions of harmonic polynomial maps between spheres

The complexity of _q -eigenmaps, i.e. homogeneous degreeq harmonic polynomial mapsf:S ^m S ⁿ, increases fast with the degreeq and the source dimensionm. Here we introduce a variety of methods of manufacturing new eigenmaps out of old ones. They include degree and source dimension raising operators. As a byproduct, we get estimates on the possible range dimensions of full eigenmaps and obtain a geometric insight of the harmonic product of ₂-eigenmaps.     

On the Regularity for Stationary Harmonic Maps

We prove a compactness theorem for k-indexed stationary harmonic maps, and show a regularity theorem for this kind of maps which says that the singular set of a k-indexed stationary harmonic map is of Hausdorff dimension at most m-3.     

Interior and boundary monotonicity formulas for stationary harmonic maps

We derive monotonicity formulas for stationary harmonic maps. The proofs are given in coordinates.     

Algebraic maps from spheres to spheres

Some homotopy classes of the unit spheres are explicitly represented by entire rational maps. Project supported by the National Natural Science Foundation of China (Grant No. 19531050), the Special Foundation of the Chinese Academy of Sciences, Hong Kong Qiu-Shi Foundation and the Education Foundation of Tsinghua University.     

On the singular set of stationary harmonic maps

LetM andN be compact riemannian manifolds, andu a stationary harmonic map fromM toN. We prove thatH ⁿ⁻² (Σ)=0, wheren=dimM and Σ is the singular set ofu. This is a generalization of a result of C. Evans [7], where this is proved in the special caseN is a sphere. We also prove that, ifu is a weakly harmonic map inW ^1,n (M, N), thenu is smooth. This extends results of F. Hélein for the casen=2, or the caseN is a sphere ([9], [10]).     

Approximation of maps into spheres by regulous maps

Let X be a compact real algebraic set of dimension n. We prove that every Euclidean continuous map from X into the unit n-sphere can be approximated by a regulous map. This strengthens and generalizes previously known results.     

Generalised twists,stationary loops,and the Dirichlet energy over a space of measure preserving maps

Let be a bounded Lipschitz domain and consider the Dirichlet energy functional

Remarks on biharmonic maps into spheres

We prove an apriori estimate in Morrey spaces for both intrinsic and extrinsic biharmonic maps into spheres. As applications, we prove an energy quantization theorem for biharmonic maps from 4-manifolds into spheres and a partial regularity for stationary intrinsic biharmonic maps into spheres.Received: 11 March 2003, Accepted: 18 November 2003, Published online: 25 February 2004     

Finite element approximations of harmonic map heat flows and wave maps into spheres of nonconstant radii

We prove the existence of weak solutions to the harmonic map heat flow, and wave maps into spheres of nonconstant radii. Weak solutions are constructed as proper limits of iterates from a fully practical scheme based on lowest order conforming finite elements, where discrete Lagrange multipliers are employed to exactly meet the sphere constraint at mesh-points. Computational studies are included to motivate interesting dynamics in two and three spatial dimensions.     

Hermitian harmonic maps from complete manifolds into convex balls

In this paper, we prove the existence and uniqueness of Hermitian harmonic maps from complete Hermitian manifolds into convex balls.     

Equivariant harmonic maps into the sphere via isoparametric maps

By using concrete isoparametric maps we obtain some new equivariant harmonic maps between spheres and solve equivariant boundary value problems for harmonic maps from unit open ballB ^m+1 intoS ⁿ. Research partially supported by NNSFC, SFECC and ICTP.