A general Weyl-type integration formula for isometric group actions

Magata, Frederick

doi:10.1007/s00031-010-9076-7

Cited by 3 publications

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“…We want to point out that there are also another way to deliver the mass of the restricted G-cycle f −1 (t) B G ǫ3 −k (p σ ). Since σp σ is the closure of the slice of σ at p σ (see Proposition 8.4 (iii)) and the slice representation is polar (see Definition A.2 and Proposition A.4(6)), one can use the Weyl-type integration formula for polar actions built by F.Magata in [19]. As a result, the mass of a G-cycle on σ would be represented by an integration on the slice σp σ .…”

Section: Proof Of Main Theoremmentioning

confidence: 99%

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Min-max theory for $G$-invariant minimal hypersurfaces

Wang

2020

Preprint

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In this paper, we consider a closed Riemannian manifold M n+1 with dimension 3 ≤ n + 1 ≤ 7, and a compact Lie group G acting as isometries on M with cohomogeneity at least 3. After adapting the Almgren-Pitts min-max theory to a G-invariant version, we show the existence of a nontrivial closed smooth embedded G-invariant minimal hypersurface generated by the min-max procedure. Moreover, we also build upper bounds as well as lower bounds of (G, p)-width which are analogs of the classical conclusions derived by Gromov[9] and Guth [10]. An application of our results combined with the work of Marques-Neves [22] shows the existence of infinitely many G-invariant minimal hypersurfaces when RicM > 0.

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Section: Proof Of Main Theoremmentioning

confidence: 99%

“…where B q is a slice of B G 2r 0 (q) at q, and Σ T t,q is the notation as we used in (19). Now, by the definition of the flat metric F, one can use the inequality above to see that then there exist q 1 , .…”

Section: Lower Bounds Of (G P)-widthmentioning

confidence: 99%

Min-max theory for $G$-invariant minimal hypersurfaces

Wang

2020

Preprint

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“…admitting a 0-section. If Σ is a fat section, then it is shown in [Ma08] that there is the fat Weyl group…”

Section: A Property Of the Quotientmentioning

confidence: 99%

Taut submanifolds and foliations

Wiesendorf¹

2014

J. Differential Geom.

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We give an equivalent description of taut submanifolds of complete Riemannian manifolds as exactly those submanifolds whose normal exponential map has integrable fibers. It turns out that every taut submanifold is also Z 2 -taut, so that tautness is essentially the same as Z 2 -tautness. In the case where the normal exponential map of a submanifold has integrable fibers, we explicitely construct generalized BottSamelson cycles for the critical points of the energy functionals on the path spaces which, generically, represent a basis for the Z 2 -cohomology. We also consider singular Riemannian foliations all of whose leaves are taut and discuss some of their main features. Using our characterization of taut submanifolds, we are able to show that tautness of a singular Riemannian foliation is actually a property of the quotient. Kurzzusammenfassung

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“…Situations in which the copolarity of an action is nontrivial and not equal to zero and where the minimal sections can be explicitly computed are described in Section 10 and [Mag08,Mag09]. To give some flavor:…”

Section: Fat Sections Fat Weyl Groups and The Copolarity Of Isometric...mentioning

confidence: 99%

Reductions, Resolutions and the Copolarity of Isometric Group Actions

Magata

2009

Preprint

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We present some results on reductions and the copolarity of isometric group actions, which we obtained in our thesis [Mag08]. We also describe a resolution construction for isometric actions with respect to a reduction and give examples.for 2 ≤ k ≤ n − 1 and a minimal section is given by R k 2 , which is embedded into R kn as block matrices with nonzero entries in the upper (k × k)-block only.Example 2.4. Consider the following action of T 2 × S(U(1) × U(2)) on SU(3). The first factor acts by matrix multiplication from the left and the second factor by matrix multiplication from the right by the inverted matrix. The copolarity in this case is equal to 1 and a minimal section is given by SO(3) ⊂ SU(3).The following three lemmas are frequently used throughout the paper. Lemma 2.6 and 2.7 are [GOT04, Lemma 5.1 and Lemma 5.2]. Originally, the second of these was stated for orthogonal representations only, but its proof also works in the general case.Lemma 2.5. Let (G, M) be an isometric action and suppose that M is connected and finite dimensional. If p ∈ M reg , then exp p (ν p (G • p)) intersects every G-orbit.Lemma 2.6. Let (G, M) be an isometric action and let q ∈ M be arbitrary. For v ∈ ν q (G • q) the following assertions are equivalent:(i) v is G q -regular.(ii) There exists ε > 0 such that exp q (tv) is G-regular for 0 < t < ε.(iii) exp q (t 0 v) is G-regular for some t 0 > 0.Lemma 2.7. Let Σ be a fat section of (G, M). For all q ∈ Σ there is a G q -regular v ∈ T q Σ ∩ ν q (G • q). Furthermore, v can be chosen such that p = exp q v is G-regular and arbitrarily close to q.

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A general Weyl-type integration formula for isometric group actions

Cited by 3 publications

References 15 publications

Min-max theory for $G$-invariant minimal hypersurfaces

Min-max theory for $G$-invariant minimal hypersurfaces

Taut submanifolds and foliations

Reductions, Resolutions and the Copolarity of Isometric Group Actions

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