Monopole metrics and the orbifold Yamabe problem

Viaclovsky, Jeff A.

doi:10.5802/aif.2617

Cited by 27 publications

(29 citation statements)

References 38 publications

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“…In the study of Yamabe invariant, with certain geometric non-collapsing assumptions, we will often encounter Riemannian orbifolds (or Riemannian multi-folds more generally) as the limit spaces for sequences of Yamabe metrics (cf. [1,42,44]). For a compact n-orbifold M with an orbifold metric g, one can also define the corresponding Yamabe constant Y (M, [g] orb ) and Yamabe invariant Y orb (M ) (see Section 2 or [3] for details).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Computations of the orbifold Yamabe invariant

Akutagawa

2011

Math. Z.

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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.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Computations of the orbifold Yamabe invariant

Akutagawa

2011

Math. Z.

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“…This implies the existence of a conformal vector field on an admissible stratified space. We partially follow an argument of J. Viaclovsky (see the proof of Theorem 1.3 in [24]). We then give the alternative proof of the fact that an Einstein metric is a Yamabe metric: the main interest of the proof is that it shows the existence of an eigenfunction relative to the eigenvalue n. As a consequence we can conclude by applying the rigidity result of Theorem 3.1.…”

Section: Volg(m )mentioning

confidence: 99%

An Obata singular theorem for stratified spaces

Mondello

2017

Trans. Amer. Math. Soc.

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Abstract. Consider a stratified space with a positive Ricci lower bound on the regular set and no cone angle larger than 2π. For such stratified space we know that the first non-zero eigenvalue of the Laplacian is larger than or equal to the dimension. We prove here an Obata rigidity result when the equality is attained: the lower bound of the spectrum is attained if and only if the stratified space is isometric to a spherical suspension. Moreover, we show that the diameter is at most equal to π, and it is equivalent for the diameter to be equal to π and for the first non-zero eigenvalue of the Laplacian to be equal to the dimension. We finally give a consequence of these results related to the Yamabe problem. Consider an Einstein stratified space without cone angles larger than 2π: if there is a metric conformal to the Einstein metric and with constant scalar curvature, then it is an Einstein metric as well. Furthermore, if its conformal factor is not a constant, then the space is isometric to a spherical suspension. MSC2010: 53 Differential geometry, 58 Global analysis, analysis on manifolds.

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“…When all of the monopole points lie on a common line, these metrics are toric and, in fact, are exactly the Calderbank-Singer metrics for q = p − 1. This construction can also have a multiplicity at each point which results in orbifold metrics, see [Via10].…”

Section: 3mentioning

confidence: 99%

A smörgåsbord of scalar-flat Kähler ALE surfaces

Lock

Viaclovsky

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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There are many known examples of scalar-flat Kähler ALE surfaces, all of which have group at infinity either cyclic or contained in SU(2). The main result in this paper shows that for any non-cyclic finite subgroup Γ ⊂ U(2) containing no complex reflections, there exist scalar-flat Kähler ALE metrics on the minimal resolution of C 2 /Γ, for which Γ occurs as the group at infinity. Furthermore, we show that these metrics admit a holomorphic isometric circle action. It is also shown that there exist scalar-flat Kähler ALE metrics with respect to some small deformations of complex structure of the minimal resolution. Lastly, we show the existence of extremal Kähler metrics admitting holomorphic isometric circle actions in certain Kähler classes on the complex analytic compactifications of the minimal resolutions.

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Monopole metrics and the orbifold Yamabe problem

Cited by 27 publications

References 38 publications

Computations of the orbifold Yamabe invariant

Computations of the orbifold Yamabe invariant

An Obata singular theorem for stratified spaces

A smörgåsbord of scalar-flat Kähler ALE surfaces

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