Equivariant Yamabe problem and Hebey–Vaugon conjecture

Madani, Farid

doi:10.1016/j.jfa.2009.10.002

Cited by 9 publications

(13 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, if µ > ω, Aubin, in [4], proved that ∂Br(P ) s g dv < 0 (see also [18]). The last inequality is sufficient to have the estimate (20), using the same test function φ ε , introduced above (see for instance in [16]). Now, we consider the case µ = ω. Thus,s is a homogeneous polynomial of degree ω, with zero average over the unit sphere, since r 1−n ∂Br(P ) s g dv g = O(r 2ω+2 ).…”

Section: The Test Functionsmentioning

confidence: 97%

See 1 more Smart Citation

The Equivariant Second Yamabe Constant

Henry

Madani

2018

J Geom Anal

Self Cite

View full text Add to dashboard Cite

show abstract

Section: The Test Functionsmentioning

confidence: 97%

“…Assume that the orbit O G (P 0 ) is finite. For δ small enough, let us consider the G−equivariant function φ ε defined as in(16), centered on the orbit of…”

mentioning

confidence: 99%

The Equivariant Second Yamabe Constant

Henry

Madani

2018

J Geom Anal

Self Cite

View full text Add to dashboard Cite

show abstract

“…More generally, it follows from Perelman's work on the Ricci flow that for our knowledge the first reference for the G-equivariant Yamabe constant µ(M, [g] G ) is Bérard Bergery [11]. In particular, he formulated a G-equivariant version of the Yamabe conjecture, which was the main subject of an article by Hebey and Vaugon [16] and by the second author [22,23]. In general neither…”

Section: Overview Over the Classical Yamabe Invariantmentioning

confidence: 99%

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

Ammann

Madani

Pilca

2016

Int Math Res Notices

Self Cite

View full text Add to dashboard Cite

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...

show abstract

“…Furthermore all metrics are assumed to be G-invariant and the Yamabe constant and Yamabe invariant are replaced by their equivariant analogues. The equivariant Yamabe problem is solved in many cases, in particular on spin manifolds or in the case that all orbits have positive dimension, see [27], [39,40]. An equivariant analogue of the Petean-Yun surgery formula was provided in [54].…”

Section: 5mentioning

confidence: 99%