On pseudo-harmonic maps in conformal geometry

We extend harmonic map techniques to the setting of more general differential equations in conformal geometry. We obtain an extension of Siu's rigidity to Kahler-Weyl geometry and apply the latter to Vaisman's conjecture. Other applications include topological obstructions to the existence of Kahler-Weyl structures. For example, we show that no co-compact lattice in SO(1,n), n>2, can be the fundamental group of a compact Kahler-Weyl manifold of certain type.Comment: errors corrected, revised versions of the results, the proof of the factorisation theorem is re-written, references updated, 32 page

An extremal eigenvalue problem in K\"ahler geometry

We study Laplace eigenvalues $\lambda_k$ on K\"ahler manifolds as functionals on the space of K\"ahler metrics with cohomologous K\"ahler forms. We introduce a natural notion of a $\lambda_k$-extremal K\"ahler metric and obtain necessary and sufficient conditions for it. A particular attention is paid to the $\lambda_1$-extremal properties of K\"ahler-Einstein metrics of positive scalar curvature on manifolds with non-trivial holomorphic vector fields.Comment: Added references and a number of minor corrections. To appear in special issue of the Journal of Geometry and Physic

An extremal eigenvalue problem in Kähler geometry

We study Laplace eigenvalues λk on Kähler manifolds as functionals on the space of Kähler metrics with cohomologous Kähler forms. We introduce a natural notion of a λk-extremal Kähler metric and obtain necessary and sufficient conditions for it. A particular attention is paid to the λ1-extremal properties of Kähler–Einstein metrics of positive scalar curvature on manifolds with non-trivial holomorphic vector fields

On geodesic homotopies of controlled width and conjugacies in isometry groups

We give an analytical proof of the Poincare-type inequalities for widths of geodesic homotopies between equivariant maps valued in Hadamard metric spaces. As an application we obtain a linear bound for the length of an element conjugating two finite lists in a group acting on an Hadamard space