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We present how to collapse a manifold equipped with a closed flat regular Riemannian foliation with leaves of positive dimension on a compact manifold, while keeping the sectional curvature uniformly bounded from above and below. From this deformation, we show that a closed flat regular Riemannian foliation with leaves of positive dimension on a compact simply-connected manifold is given by torus actions. This gives a geometric characterization of aspherical regular Riemannian foliations given by torus actions.

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

A singular foliation \(\mathcal{F}\) on a complete Riemannian manifold \(M\) is called Singular Riemannian foliation (SRF for short) if its leaves are locally equidistant, e.g., the partition of M into orbits of an isometric action. In this paper, we investigate variational problems in compact Riemannian manifolds equipped with SRF with special properties, e.g. isoparametric foliations, SRF on fibers bundles with Sasaki metric, and orbit-like foliations. More precisely, we prove two results analogous to Palais' Principle of Symmetric Criticality, one is a general principle for \(\mathcal{F}\) symmetric operators on the Hilbert space \(W^{1,2}(M)\), the other one is for \(\mathcal{F}\) symmetric integral operators on the Banach spaces \(W^{1,p}(M)\). These results together with a \(\mathcal{F}\) version of Rellich Kondrachov Hebey Vaugon Embedding Theorem allow us to circumvent difficulties with Sobolev's critical exponents when considering applications of Calculus of Variations to find solutions to PDEs. To exemplify this we prove the existence of weak solutions to a class of variational problems which includes \(p\)-Kirschoff problems.

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

We show that a singular Riemannian foliation of codimension two on a compact simply-connected Riemannian (????+2)-manifold, with regular leaves homeomorphic to the n-torus, is given by a smooth effective n-torus action. This solves in the negative for the codimension 2 case a question about the existence of foliations by exotic tori on simply-connected manifolds.

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

Using variational methods together with symmetries given by singular Riemannian foliations with positive dimensional leaves, we prove the existence of an infinite number of sign-changing solutions to Yamabe type problems, which are constant along the leaves of the foliation, and one positive solution of minimal energy among any other solution with these symmetries. In particular, we find sign-changing solutions to the Yamabe problem on the round sphere with new qualitative behavior when compared to previous results, that is, these solutions are constant along the leaves of a singular Riemannian foliation which is not induced neither by a group action nor by an isoparametric function. To prove the existence of these solutions, we prove a Sobolev embedding theorem for general singular Riemannian foliations, and a Principle of Symmetric Criticality for the associated energy functional to a Yamabe type problem.

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

We expand upon the notion of a pre-section for a singular Riemannian foliation \((M,\mathcal{F})\), i.e. a proper submanifold \(N\subset M\) retaining all the transverse geometry of the foliation. This generalization of a polar foliation provides a similar reduction, allowing one to recognize certain geometric or topological properties of \((M,\mathcal{F})\)  and the leaf space \(M/\mathcal{F}\). In particular, we show that if a foliated manifold  \(M\) has positive sectional curvature and contains a non-trivial pre-section, then the leaf space \(M/\mathcal{F}\) has nonempty boundary. We recover as corollaries the known result for the special case of polar foliations as well as the well-known analogue for isometric group actions.

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

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