In this paper, we consider a closed Riemannian manifold $$M^{n+1}$$ with dimension $$3\le n+1\le 7$$ , and compact Lie group G acting as isometries on M cohomogeneity at least 3. After adapting the Almgren–Pitts min–max theory to G-equivariant version, show existence of non-trivial smooth embedded G-invariant minimal hypersurface $$\Sigma \subset M$$ provided that union non-principal orbits for...

A Kähler metric is said to be Bochner-Kähler if its Bochner curvature vanishes. This is a nontrivial condition when the complex dimension of the underlying manifold is at least 2. In this article it will be shown that, in a certain well-defined sense, the space of Bochner-Kähler metrics in complex dimension n has real dimension n+1 and a recipe for an explicit formula for any Bochner-Kähler met...

We construct a new (singular) cohomogeneity-three metric of G 2 holonomy. The solution can be viewed as a triple intersection of smeared Taub-NUTs. The metric comprises three non-compact radial-type coordinate, with the principal orbits being a T 3 bundle over S 1. We consider an M-theory vacuum (Minkowski) 4 × M 7 where M 7 is the G 2 manifold. Upon reduction on a circle in the T 3 , we obtain...

We find general first-order equations for G 2 metrics of cohomogeneity one with S 3 × S 3 principal orbits. These reduce in two special cases to previously-known systems of first-order equations that describe regular asymptotically locally conical (ALC) metrics B 7 and D 7 , which have weak-coupling limits that are S 1 times the deformed conifold and the resolved conifold respectively. Our more...

Any sphere S admits a metric of constant sectional curvature. These canonical metrics are homogeneous and Einstein, that is the Ricci curvature is a constant multiple of the metric. The spheres S, m > 1 are known to have another Sp(m + 1)-homogeneous Einstein metric discovered by Jensen [Jen73]. In addition, S has a third Spin(9)-invariant homogeneous Einstein metric discovered by Bourguignon a...