Einstein Constants and Smooth Topology
Claude LeBrun
TL;DR
The paper investigates whether smooth closed manifolds can carry both Ricci-flat and positive Einstein metrics, surveying historical constructions of Einstein metrics with opposite signs and then proving new constraints. It establishes a no-go result in dimension $7$: a Sasaki-Einstein metric and a $G_2$-holonomy metric cannot coexist on a compact manifold, using a torsion property of $p_1$ in Sasaki-Einstein geometry and Joyce’s curvature formula. Focusing on dimension $6$, it narrows the search to Fano $3$-folds with $K$-polystability that could host a $ abla>0$ KE metric alongside a Calabi–Yau metric on the same underlying $6$-manifold, identifying four index-$2$ Fano $3$-fold deformation-types with $b_2=1$ and $b_3 eq 0$ as the only viable candidates. The work employs Wall’s classification of simply-connected six-manifolds to relate diffeomorphism types to invariants and shows that, although these four families are K-stable and admit KE metrics, no known Calabi–Yau realizations arise from complete intersections or branched covers of these types, leaving the coexistence problem open and guiding future construction in complex and differential geometry.
Abstract
It was first shown in (Catanese-LeBrun 1997) that certain high-dimensional smooth closed manifolds admit pairs of Einstein metrics with Ricci curvatures of opposite sign. After reviewing subsequent progress that has been made on this topic, we then prove various related results, with the ultimate goal of stimulating further research on associated questions.