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

Abstract

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...

Energy Minimizing Beltrami Fields on Sasakian 3-Manifolds

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Scalar curvature and the multiconformal class of a direct product Riemannian manifold

“…However, since an equivariant conformal class is smaller than the ordinary conformal class, one cannot expect an inequality like (2.3) for the equivariant ones (cf. [2,Example 3]).…”

“…As we are interested in the behavior of the scalar curvature of the metrics in a fixed multiconformal class [[g]] we calculate the scalar curvature of a multiconformal changẽ g = f 2 1 g 1 ⊕ • • • ⊕ f 2 l g l in terms of the scalar curvature of g i and the multiconformal factors f 1 , . .…”

“…Before providing an answer to his question, we generalize Yang's Theorem slightly to our setting as follows. Theorem 6.4 Let (M, g) = (M 1 , g 1 )ו • •×(M l , g l ) be a direct product of closed connected Riemannian manifolds and letg = f 2 1 g 1 ⊕ • • • f 2 l g l be a metric of constant scalar curvature. Ifg is of permutation type for a bijection σ with σ (i) = i for all 1 ≤ i ≤ l and the functions f 1 , .…”

“…for each f ∈ C ∞ (M) and X , Y ∈ T M and fix a multiconformal changeg = f 2 1 g 1 ⊕ • • • ⊕ f 2 l g l . Henceforth, we writeg = •, • and g = •, • .…”