A gap theorem for positive Einstein metrics on the four-sphere

Abstract

We show that there exists a universal positive constant ε 0 > 0 with the following property: Let g be a positive Einstein metric on S 4 . If the Yamabe constant of the conformal class [g] satisfieswhere g S denotes the standard round metric on S 4 , then, up to rescaling, g is isometric to g S . This is an extension of Gursky's gap theorem for positive Einstein metrics on the four-sphere. √ 3 Y (S 4 , [g S ])) : Date

Compactness of conformally compact Einstein manifolds in dimension 4

“…In the earlier version of this paper, the authors have stated both these two theorems without the additional assumption that the boundary of the four manifold X is S 3 as is in the current version of the paper; in the proof we had quoted a result of M. Anderson (the claim after Proposition 3.10 in [2], see also the result of M. Anderson in another paper [3] Lemma 6.3) to establish the argument of no interior blow up. It was pointed out to us by the referee that this result of Anderson was questioned in the recent work of Akutagawa-Endo-Seshadri [1]. Inspired by the proof in the paper of Akutagawa-Endo-Seshadri, we have in this version applied a result of Chrisp-Hillman [23] to establish the argument of no interior blow up; under the additional assumption that the boundary is topologically S 3 .…”

“…Step 4. We claim that there exists some c v > 0 such that for any p ∈ X ∞ and for any r < 1 2 ρ ∞ (p) (4.5)…”

“…We assume the boundary Yamabe metric ĝi in conformal infinity is of nonnegative type and denote g i be the corresponding FG compactification. Assume (1) The boundary Yamabe metrics ĝi form a compact family in C k+3 norm with k ≥ 2;…”

“…The authors are also grateful to the referee for pointing out the question raised in the paper of Akutagawa-Endo-Seshadri [1]; for suggesting the elliptic iteration argument (used in section 4.4 of this paper) to improve the order of the regularity and also for making many other useful comments concerning the presentation of the paper.…”

Thermodynamic ensembles with cosmological horizons

“…In D = 4 spacetime dimensions, the only known classical solution to the Einstein equation with positive cosmological constant is the round 4-sphere, i.e., Euclidean dS, but it is not known whether this is the unique solution. In dimensions between 5 and 9 there are infinitely many solutions [78]. The Euclidean action for a solution to the Einstein equation with cosmological constant Λ is proportional to −ΛV , where V is the volume of the manifold.…”