Square-integrability of solutions of the Yamabe equation

Ammann, Bernd; Dahl, Mattias; Humbert, Emmanuel

doi:10.4310/cag.2013.v21.n5.a2

Cited by 7 publications

(21 citation statements)

References 9 publications

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“…This problem can be avoided by restricting to 2-connected manifolds. Together with results from [2] we obtain an explicit positive number t n such that any compact ndimensional 2-connected manifold M with vanishing index, n = 4, satisfies σ(M ) ≥ t n , see Table 2 and Proposition 5.7. We thus see S n (2) ⊂ {0} ∪ [t n , σ(S n )] for all n = 4.…”

Section: Introduction and Resultsmentioning

confidence: 66%

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Low-dimensional surgery and the Yamabe invariant

Ammann¹,

Dahl²,

Humbert³

2015

J. Math. Soc. Japan

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Section: Introduction and Resultsmentioning

confidence: 66%

“…In dimensions n ≥ 7 an unsolved problem persists for surgeries of codimension 3, i.e. for n = k − 3, see [2] for details about this problem. This problem can be avoided by restricting to 2-connected manifolds.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Low-dimensional surgery and the Yamabe invariant

Ammann¹,

Dahl²,

Humbert³

2015

J. Math. Soc. Japan

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“…Assume that N m is obtained from M m by a surgery of dimension k. Then for any metric g on M a family of special metrics g ϑ , ϑ > 0, was constructed. It was proved in [5] in combination with estimates given in [6] that for all k ≤ m − 4 and all k = m − 3 ≤ 3 we have…”

Section: Gromov-hausdorff Convergencesmentioning

confidence: 92%

Relations Between Threshold Constants for Yamabe Type Bordism Invariants

Ammann

Große²

2015

J Geom Anal

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Abstract. In the work of Ammann, Dahl and Humbert it has turned out that the Yamabe invariant on closed manifolds is a bordism invariant below a certain threshold constant. A similar result holds for a spinorial analogon. These threshold constants are characterized through Yamabe-type equations on products of spheres with rescaled hyperbolic spaces. We give variational characterizations of these threshold constants, and our investigations lead to an explicit positive lower bound for the spinorial threshold constants.

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“…A proof of Schoen's conjecture (or even partial results) would be very helpful, as it would provide interesting conclusions about the Yamabe invariant of non-simply connected manifolds. For example, if we were able to obtain an upper bound on σ Γ (S n ) which is uniform in Γ, then the Yamabe invariant would define interesting subgroups of the spin bordism and oriented bordism groups, see [2].…”

Section: Overview Over the Classical Yamabe Invariantmentioning

confidence: 99%

“…Using surgery theory, Petean and Yun have proven that σ(M ) ≥ 0 for all simply-connected manifolds of dimension at least 5, see [26], [27]. Stronger results can be obtained with the surgery formula developed in [2]. For example, it now can be shown, see [4] and [3, 5], that simplyconnected manifolds of dimension 5 resp.…”

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confidence: 99%

The $\boldsymbol{S^1}$-Equivariant Yamabe Invariant of 3-Manifolds

Ammann

Madani

Pilca

2016

Int Math Res Notices

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We show that the S 1 -equivariant Yamabe invariant of the 3-sphere, endowed with the Hopf action, is equal to the (non-equivariant) Yamabe invariant of the 3-sphere. More generally, we establish a topological upper bound for the S 1 -equivariant Yamabe invariant of any closed oriented 3-manifold endowed with an S 1 -action. Furthermore, we prove a convergence result for the equivariant Yamabe constants of an accumulating sequence of subgroups of a compact Lie group acting on a closed manifold.Date: October 12, 2018. with σ(M ) ≤ 0, the value of σ(M ) is determined by the volume of the hyperblic pieces in the Thurston decomposition. We learned this from [17, Prop. 93.10 on page 2832], but ideas for this application go back to [8]. In the case σ(M ) > 0, n = 3, M is the connected sum of copies of quotients S 2 × S 1 and of quotients of S 3 . For connected sums of copies of S 2 ×S 1 we have σ(M ) = σ(S 3 ) but the precise value cannot be determined in most cases. Using inverse mean curvature flow, the Yamabe invariants of RP 3 and some related spaces were determined in [13] and [1], e.g. σ(RP 3 ) = 2 −2/3 σ(S 3 ). This is indeed a special case of Schoen's conjecture explained below.Also in higher dimensions the case of positive Yamabe invariant is notoriously difficult. In dimension n ≥ 5 one does not know any n-dimensional manifold M for which one can prove 0 < σ(M ) < σ(S n ). In dimensions n ≤ 4 there are some examples for which exact calculations can be carried out, even in the positive case. The values for CP 2 and some related spaces were calculated by LeBrun [20] using Seiberg-Witten theory. The calculation then was simplified considerably by Gursky and LeBrun [14]. This proof no longer uses Seiberg-Witten theory, but only the index theorem by Atiyah and Singer. See also [14,19, 21] for related results.Recently, surgery techniques known from the work of Gromov and Lawson could be refined to obtain explicit positive lower bounds for the Yamabe invariant. Such bounds are easily obtained for special manifolds, e. g. for manifolds with Einstein metrics or connected sum of such manifolds. Namely, a theorem by Obata [24] states that the Einstein-Hilbert functional of an Einstein metric g equals µ(M, [g]), thus providing a lower bound for σ(M ). For instance, if M is S n , T n , RP n or CP n , the canonical Einstein metrics provide lower bounds for σ(M ). However, obtaining a lower bound for σ(M ) is difficult in general if M carries a metric of positive scalar curvature but no Einstein metric. Using surgery theory, Petean and Yun have proven that σ(M ) ≥ 0 for all simply-connected manifolds of dimension at least 5, see [26], [27]. Stronger results can be obtained with the surgery formula developed in [2]. For example, it now can be shown, see [4] and [3, 5], that simplyconnected manifolds of dimension 5 resp. 6 satisfy σ(M ) ≥ 45.1 resp. σ(M ) ≥ 49.9.In order to find more manifolds with 0 < σ(M ) < σ(S n ), it would be helpful to prove the following conjecture by Schoen [29]: it states that if Γ is a finite grou...

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Square-integrability of solutions of the Yamabe equation

Cited by 7 publications

References 9 publications

Low-dimensional surgery and the Yamabe invariant

Low-dimensional surgery and the Yamabe invariant

Relations Between Threshold Constants for Yamabe Type Bordism Invariants

The $\boldsymbol{S^1}$-Equivariant Yamabe Invariant of 3-Manifolds

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