Curvature explosion in quotients and applications

Abstract

We prove that the quotient space of a variationally complete group action is a good Riemannian orbifold. The result is generalized to singular Riemannian foliations without horizontal conjugate points.2000 Mathematics Subject Classification. 53C20.

A slice theorem for singular Riemannian foliations, with applications

“…Since v and G(t, v) belong to the same leaf in νL, part c) follows directly from equifocality (see Proposition 5, or Proposition 4.3 in [LT10], or Theorem 2.9 in [Ale10]). …”

“…Now we specialize the discussion above to the case where the closed submanifold L ⊂ M is a leaf of a singular Riemannian foliation F. Recall that using the Homothetic Transformation Lemma (see Lemma 4, or [Mol88, Lemma 6.2]), there is a unique singular foliation on the normal bundle νL that is scaling invariant and which corresponds to F| U via the normal exponential map, where U is any small enough tubular neighbourhood around L (see also [LT10], section 4.3).…”

“…We will need the following fact, see Proposition 4.3 in [LT10], or Theorem 2.9 in [Ale10] for a proof.…”

Mean Curvature Flow of Singular Riemannian Foliations

“…The functions r Σ and f are constant along the leaves of F, and thus induce functions on the quotient, which we denote with the same letters. By Lytchak and Thorbergsson [8], the first focal point of a leaf L p corresponds to either a singular leaf, or to a conjugate point in M/F of the projection of L p . In the first case, f (p) = r Σ (p) and the proposition is proved.…”

“…[11, Theorem 6.1, Proposition 6.5], [2], [8]) that it is possible to find a neighbourhood P of q in L, a cylindrical neighbourhood O ǫ = T ub ǫ (P ) of q in M and diffeomorphism ϕ : O ǫ → V ⊆ T q M onto a neighbourhood V of the origin, such that:…”