We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem and its higher eigenvalue versions via the direct method of calculus of variations. The principal results include the general regularity properties of λ k -extremal metrics and the existence of a partially regular λ 1 -maximiser. (2000): 58J50, 58E11, 49R50. Mathematics Subject Classification

We extend harmonic map techniques to the setting of more general differential equations in conformal geometry. We discuss existence theorems and obtain an extension of Siu's strong rigidity to Kähler–Weyl geometry. Other applications include topological obstructions to the existence of Kähler–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 Kähler–Weyl manifold of certain type.

We prove two explicit bounds for the multiplicities of Steklov eigenvalues σ k on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index k of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given Riemannian surface with smooth boundary the multiplicities of Steklov eigenvalues σ k are uniformly bounded in k.

We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues λ k of conformal sub-Riemannian metrics that are asymptotically sharp as k → +∞. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for contact metric manifolds conformal to such Sasakian manifolds, we obtain eigenvalue inequalities that can be viewed as versions of the classical results by Korevaar and Buser in Riemannian geometry.

We study questions related to the orientability of the infinite-dimensional moduli spaces formed by solutions of elliptic equations for mappings of manifolds. The principal result states that the first Stiefel-Whitney class of such a moduli space is given by the Z 2 -spectral flow of the families of linearised operators. Under an additional compactness hypotheses, we develop elements of Morse-Bott theory and express the algebraic number of solutions of a non-homogeneous equation with a generic right-hand side in terms of the Euler characteristic of the space of solutions corresponding to the homogeneous equation. The applications of this include estimates for the number of homotopic maps with prescribed tension field and for the number of the perturbed pseudoholomorphic tori, sharpening some known results. (2000): 35J05, 58B15, 58E05, 58E20, 53D45 Mathematics Subject Classifications