Infinitely many new families of complete cohomogeneity one G_2-manifolds: G_2 analogues of the Taub-NUT and Eguchi-Hanson spaces

Understand

We construct infinitely many new 1-parameter families of simply connected complete noncompact G_2-manifolds with controlled geometry at infinity.

• The generic member of each family has so-called asymptotically locally conical (ALC) geometry.
• However, the nature of the asymptotic geometry changes at two special parameter values: at one special value we obtain a unique member of each family with asymptotically conical (AC) geometry; on approach to the other special parameter value the family of metrics collapses to an AC Calabi-Yau 3-fold.
• Our infinitely many new diffeomorphism types of AC G_2-manifolds are particularly noteworthy: previously the three examples constructed by Bryant and Salamon in 1989 furnished the only known simply connected AC G_2-manifolds.