Year: 2025
Type: article
Abstract: Drawing on results of Derdziński’s from the 1980s, we classify conformally Kähler, U(2)-invariant, Einstein metrics on the total space of $${{\mathcal {O}}}(-m)$$ O ( - m ) , for all $$m \in \mathbb {N}$$ m ∈ N . This yields infinitely many 1-parameter families of metrics exhibiting several different behaviours including asymptotically hyperbolic metrics (more specifically of Poincaré type), ALF metrics, and metrics which compactify to a Hirzebruch surface $$\mathbb {H}_m$$ H m with a cone singularity along the “divisor at infinity”. This allows us to investigate transitions between behaviours yielding interesting results. For instance, we show that a Ricci–flat ALF metric known as the Taub-bolt metric can be obtained as the limit of a family of cone angle Einstein metrics on $${\mathbb{C}\mathbb{P}}^2 \# \overline{{\mathbb{C}\mathbb{P}}}^2$$ C P 2 # C P ¯ 2 when the cone angle converges to zero. We also construct Einstein metrics which are asymptotically hyperbolic and conformal to a scalar-flat Kähler metric. Such metrics cannot be obtained by applying Derdziński’s theorem.
Source: Journal of Geometric Analysis
Authors: Gonçalo Oliveira, Rosa Sena-Dias
Institution: Instituto Superior Técnico
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Award: G384817180
OpenAlex ID: W4409163855