Computations of the orbifold Yamabe invariant

Abstract

We consider the Yamabe invariant of a compact orbifold with finitely many singular points. We prove a fundamental inequality for the estimate of the invariant from above, which also includes a criterion for the non-positivity of it. Moreover, we give a sufficient condition for the equality in the inequality. In order to prove it, we also solve the orbifold Yamabe problem under a certain condition. We use these results to give some exact computations of the Yamabe invariant of compact orbifolds.

The Weyl functional on 4-manifolds of positive Yamabe invariant

“…Also the Yamabe problem on an orbifold can be defined in the obvious way, and turns out to be almost parallel to the case of a smooth manifold. For details, the reader may consult K. Akutagawa [1].…”

“…Thus there exists ϕ ∈ L 2 1 (M) and a subsequence converging to ϕ weakly in L 2 1 , strongly in L 2 , and pointwisely almost everywhere. 1 By abuse of notation we let {ϕ i } be the subsequence.…”

The Yamabe invariant of a class of symplectic 4-manifolds

“…Let φ : [0, 4] → R be a non-negative smooth function such that φ is identically 1 on the interval [0, 1] and identically 0 on the interval [2,4]. We define the metric g t on the larger collar {z ∈ C 2 | t 2 < |z| < 4t} :…”

“…In [2], Akutagawa considers the connected sum of two orbifold 4−spaces M 1 and M 2 at two isolated orbifold points of type R 4 /Γ: On the other hand, M ′ admits a Kähler-Einstein metric g given by the positive solution of the Calabi conjecture [13]. This is the Yamabe minimizer in its conformal class, and satisfies:…”

A conformal integral invariant on Riemannian foliations

“…The project is supported by SFB/TR71 "Geometric partial differential equations" of DFG. 1 By (1.4), they found new functions which cannot be realized as the scalar curvature of a Riemmanian metric on S 2 . Remark that the proof given in [6] is different for dimension n ≥ 3 and for dimension n = 2.…”