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We show the existence of a solution to the Ricci flow with a compact length space of bounded curvature, i.e., a space that has curvature bounded above and below in the sense of Alexandrov, as its initial condition. We show that this flow converges in the \(C^{1,\alpha}\)-sense to a \(C^{1,\alpha}\)-continuous Riemannian manifold which is isometric to the original metric space. Moreover, we prove that the flow is uniquely determined by the initial condition, up to isometry.

Related project(s):
43Singular Riemannian foliations and collapse79Alexandrov geometry in the light of symmetry and topology

We prove that there exist ????????(3)-invariant metrics on Aloff-Wallach spaces W^7_{k1,k2}, as well as ????????(5)-invariant metrics on the Berger space B^{13}, which have positive sectional curvature and evolve under the Ricci flow to metrics with non-positively curved planes.

Related project(s):
79Alexandrov geometry in the light of symmetry and topology

We show that Ricci flow does not preserve a range of curvature conditions that interpolate between positive sectional and positive scalar curvature. 

Related project(s):
79Alexandrov geometry in the light of symmetry and topology

Related project(s):
43Singular Riemannian foliations and collapse

We introduce a new technique to the study and identification of submanifolds of simply-connected symmetric spaces of compact type based upon an approach computing k-positive Ricci curvature of the ambient manifolds and using this information in order to determine how highly connected the embeddings are. This provides codimension ranges in which the Cartan type of submanifolds satisfying certain conditions which generalize being totally geodesic necessarily equals the one of the ambient manifold. Using results by Guijarro-Wilhelm our approach partly generalizes recent work by Berndt-Olmos on the index conjecture.

Related project(s):
79Alexandrov geometry in the light of symmetry and topology

We give a characterization of those Alexandrov spaces admitting a cohomogeneity one action of a compact connected Lie group G for which the action is Cohen--Macaulay. This generalizes a similar result for manifolds to the singular setting of Alexandrov spaces where, in contrast to the manifold case, we find several actions which are not Cohen--Macaulay. In fact, we present results in a slightly more general context. We extend the methods in this field by a conceptual approach on equivariant cohomology via rational homotopy theory using an explicit rational model for a double mapping cylinder.

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

Alexandrov spaces are complete length spaces with a lower curvature bound in the triangle comparison sense. When they are equipped with an effective isometric action of a compact Lie group with one-dimensional orbit space they are said to be of cohomogeneity one. Well-known examples include cohomogeneity-one Riemannian manifolds with a uniform lower sectional curvature bound; such spaces are of interest in the context of non-negative and positive  sectional curvature.  In the present article we classify closed, simply-connected  cohomogeneity-one Alexandrov spaces in dimensions $5$, $6$ and  $7$. This yields, in combination with previous results for manifolds and Alexandrov spaces, a complete classification of  closed, simply-connected  cohomogeneity-one Alexandrov spaces in dimensions at most $7$.

Related project(s):
11Topological and equivariant rigidity in the presence of lower curvature bounds