We study the spinor flow on homogeneous spin manifolds. After providing the general setup we discuss the homogeneous spinor flow in dimension 3 and on almost abelian Lie groups in detail. As a further example the flag manifold in dimension 6 is treated.

A natural approach to the construction of nearly G2 manifolds lies in resolving nearly G2 spaces with isolated conical singularities by gluing in asymptotically conical G2 manifolds modelled on the same cone. If such a resolution exits, one expects there to be a family of nearly G2 manifolds, whose endpoint is the original nearly G2 conifold and whose parameter is the scale of the glued in asymptotically conical G2 manifold. We show that in many cases such a curve does not exist.The non-existence result is based on a topological result for asymptotically conical G2 manifolds: if the rate of the metric is below −7/2, then the G2 4-form is exact if and only if the manifold is Euclidean R 7 .A similar construction is possible in the nearly Kähler case, which we investigate in the same manner with similar results. In this case, the non-existence results is based on a topological result for asymptotically conical Calabi-Yau 6-manifolds: if the rate of the metric is below −3, then the square of the Kähler form and the complex volume form can only be simultaneously exact, if the manifold is Euclidean R 6 .

A natural approach to the construction of nearly $$G_2$$ G 2 manifolds lies in resolving nearly $$G_2$$ G 2 spaces with isolated conical singularities by gluing in asymptotically conical $$G_2$$ G 2 manifolds modelled on the same cone. If such a resolution exits, one expects there to be a family of nearly $$G_2$$ G 2 manifolds, whose endpoint is the original nearly $$G_2$$ G 2 conifold and whose parameter is the scale of the glued in asymptotically conical $$G_2$$ G 2 manifold. We show that in many cases such a curve does not exist. The non-existence result is based on a topological result for asymptotically conical $$G_2$$ G 2 manifolds: if the rate of the metric is below $$-3$$ - 3 , then the $$G_2$$ G 2 4-form is exact if and only if the manifold is Euclidean $$\mathbb R^7$$ R 7 . A similar construction is possible in the nearly Kähler case, which we investigate in the same manner with similar results. In this case, the non-existence results is based on a topological result for asymptotically conical Calabi–Yau 6-manifolds: if the rate of the metric is below $$-3$$ - 3 , then the square of the Kähler form and the complex volume form can only be simultaneously exact, if the manifold is Euclidean $$\mathbb R^6$$ R 6 .

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The regularity of limit spaces of Riemannian manifolds with L p curvature bounds, p > n/2, is investigated under no apriori non-collapsing assumption. A regular subset, defined by a local volume growth condition for a limit measure, is shown to carry the structure of a Riemannian manifold. One consequence of this is a compactness theorem for Riemannian manifolds with L p curvature bounds and an a priori volume growth assumption in the pointed Cheeger-Gromov topology.A different notion of convergence is also studied, which replaces the exhaustion by balls in the pointed Cheeger-Gromov topology with an exhaustion by volume non-collapsed regions. Assuming in addition a lower bound on the Ricci curvature, the compactness theorem is extended to this topology. Moreover, we study how a convergent sequence of manifolds disconnects topologically in the limit.In two dimensions, building on results of Shioya, the structure of limit spaces is described in detail: it is seen to be a union of an incomplete Riemannian surface and 1-dimensional length spaces.