Search

Motivated by the recent interest in even-Clifford structures and in generalized complex and quaternionic geometries, we introduce the notion of generalized almost even-Clifford structure. We generalize the Arizmendi-Hadfield twistor space construction on even-Clifford manifolds to this setting and show that such a twistor space admits a generalized complex structure under certain conditions.

In this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection (M4, c, D). We show that there is an Eells-Salamon type correspondence between nonvertical 𝒥-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces. When (M, c, J) is conformally almost-Hermitian, there is a canonical Weyl connection. We show that for the canonical Weyl connection, branched Weyl-minimal surfaces satisfy the adjunction inequality

A compact compler surface with non-trivial canonical bundle and a Kähler metric ofzero scalar curvature must be a ruled surface. It is also known that not every ruled surface can admit such extremal Kähler metrics. In this paper we review recentjoint work with Kim and LeBrun in which deforma- tion theory of pairs of singular complex spaces it is used to show that any ruled surface (M, J) has blow-ups (M, J) which admit Kähler metrics of zero scalar curvature.

We construct complete noncompact Riemannian metrics with G 2-holonomy on noncompact orbifolds that are ℝ3-bundles with the twistor space [InlineMediaObject not available: see fulltext.] as a spherical fiber. [ABSTRACT FROM AUTHOR]