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

For an arbitrary Dirac-harmonic map (φ, ψ) between compact oriented Riemannian surfaces, we shall study the zeros of |ψ|. With the aid of Bochner-type formulas, we explore the relationship between the order of the zeros of |ψ| and the genus of M and N . On the basis, we could clarify all of nontrivial Dirac-harmonic maps from S to S.

where (h) is the inverse of the matrix (hij), ∆M = ∑ i,j h ∂ij and Γ s tγ denote the Christoffel symbols of the Hermitian metric g on N . It follows from (1.1) that if u is holomorphic, then u must be harmonic. Thus, it is natural to ask under what circumstances a harmonic map is holomorphic or antiholomorphic. Under the assumption that both M and N are compact, Siu [31] demonstrated that if th...

Harmonic maps X:(S,h) --> N from a 2-manifold S with indefinite metric h to a semi-Riemannian manifold N are characterized, assuming that the induced metric I is nondegenerate. Except in one very special case, the characterizations involve a canonically determined holomorphic quadratic differential on a naturally chosen conformal structure. This is surprising because the Euler-Lagrange equation...

We consider piecewise monotone interval mappings which are topologically mixing and satisfy the Markov property. It has previously been shown that the invariant densities of the natural approximations converge exponentially fast in uniform pointwise topology to the invariant density of the given map provided its derivative is piecewise Lipshitz continuous. We provide an example of a map which i...