Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds

Abstract

We study compact complex 3-manifolds M admitting a (locally homogeneous) holomorphic Riemannian metric g. We prove the following: (i) If the Killing Lie algebra of g has a non trivial semi-simple part, then it preserves some holomorphic Riemannian metric on M with constant sectional curvature; (ii) If the Killing Lie algebra of g is solvable, then, up to a finite unramified cover, M is a quotient \G, where is a lattice in G and G is either the complex Heisenberg group, or the complex SO L group.

Holomorphic G-structures and foliated Cartan geometries on compact complex manifolds

“…O(n, C)-structure. Giving an O(n, C)-structure on M is equivalent to giving a pointwise nondegenerate (i.e., of rank n) holomorphic section g of the bundle S 2 (T * M) of symmetric complex quadratic forms in the holomorphic tangent bundle of M. This structure is also known as a holomorphic Riemannian metric [Le,Gh,Du,DZ,BD4]. Notice that the complexification of any real-analytic Riemannian or pseudo-Riemannian metric defines a local holomorphic Riemannian metric which coincides with the initial (pseudo-)Riemannian metric on the real locus.…”

“…. + dz 2 n defines a flat holomorphic O(n, C)-structure in the sense of Section 2 and hence a flat holomorphic Riemannian metric (see [Le,Gh,Du,DZ,BD3]). Any flat holomorphic Riemannian metric is locally isomorphic with the complex Euclidean space.…”

Infinite energy equivariant harmonic maps, domination, and anti-de Sitter $3$-manifolds

“…Domination is essential to understanding complete manifolds locally modeled on G = PO(n, 1) 0 : a geometrically finite representation ρ 1 : Γ → G strictly dominates ρ 2 : Γ → G if and only if the (ρ 1 , ρ 2 )-action on G by left and right multiplication is properly discontinous [GK17]. For n = 2 these are the anti-de Sitter (AdS) 3-manifolds, and for n = 3 we have the 3-dimensional complex holomorphic-Riemannian 3-manifolds of constant non-zero curvature (see [DZ09] for details).…”

Strict contractions and exotic SO0(d,1) quotients

“…In this case, the Killing form is a bi‐invariant holomorphic Riemannian metric of constant negative curvature (see Ghys ). Therefore, the quotients inherit such holomorphic Riemannian metrics (see the work of Dumitrescu and Dumitrescu‐Zeghib for classification results for these structures in low dimensions). Ghys studied such quotients in the case that is a small deformation of the trivial representation, proving that these were precisely the small deformations of the complex structure.…”