On the variation of the Einstein–Hilbert action in pseudohermitian geometry

Claudio Afeltra; Jih-Hsin Cheng; Andrea Malchiodi; Paul Yang; Xiaodong Wang

doi:10.1515/crelle-2024-0031

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On the variation of the Einstein–Hilbert action in pseudohermitian geometry

Claudio Afeltra , Jih-Hsin Cheng , Andrea Malchiodi , Paul Yang und Xiaodong Wang

Veröffentlicht/Copyright: 4. Juni 2024

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2024 Heft 813

Abstract

In this paper, we compute the first and second variation of the normalized Einstein–Hilbert functional on CR manifolds. We characterize critical points as pseudo-Einstein structures. We then turn to the second variation on standard spheres. While the situation is quite similar to the Riemannian case in dimension greater than or equal to five, in three dimensions, we observe a crucial difference, which mainly depends on the embeddable character of the perturbed CR structure.

Funding source: Ministry of Science and Technology, Taiwan

Award Identifier / Grant number: MOST 111-2115-M-001-005

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1509505

Funding statement: J.-H. Cheng is supported by the project MOST 111-2115-M-001-005 of Ministry of Science and Technology and NCTS of Taiwan. A. Malchiodi is supported by the project Geometric Problems with Loss of Compactness from Scuola Normale Superiore. He is also a member of GNAMPA as part of INdAM. P. Yang acknowledges support from the NSF for the grant DMS 1509505.

A Appendix

It is a well-known fact in Kahler geometry that a Kahler metric 𝜔 on a closed complex manifold 𝑀 with constant scalar curvature must be Kahler–Einstein if the Kahler class [ ω ] is proportional to the first Chern class c 1 ⁢ ( M ) . In this appendix, we discuss a CR analogue of this result.

We still follow [21] as our standard reference on CR geometry. Let 𝑀 be a CR manifold of dimension 2 ⁢ n + 1 . The first Chern class of the complex vector bundle T 1 , 0 ⁢ M will be simply denoted by c 1 ⁢ ( M ) . Given a pseudohermitian structure 𝜃, we always work with the Tanaka–Webster connection ∇, and 2 ⁢ π ⁢ c 1 ⁢ ( M ) is then represented by the closed 2-form

ρ θ = i ⁢ [ R μ ⁢ ν ̄ ⁢ θ μ ∧ θ ν ̄ + A α ⁢ γ , α ̄ ⁢ θ γ ∧ θ − A γ ̄ ⁢ α ̄ , α ⁢ θ ⁢ γ ̄ ∧ θ ] ,

where R μ ⁢ ν ̄ is the Ricci curvature and 𝐴 is the torsion of the Tanaka–Webster connection. Throughout this section, we always work with a local unitary frame { Z α ∣ α = 1 , … , n } .

Lemma A.1

Suppose ϕ = f μ ⁢ ν ̄ ⁢ θ μ ∧ θ ν ̄ is a ( 1 , 1 ) -form. Let Λ ⁢ ( ϕ ) = ∑ μ = 1 n f μ ⁢ μ ̄ be its trace. Then − d ∗ ⁢ ϕ = f α ⁢ ν ̄ , α ̄ ⁢ θ ν ̄ + f μ ⁢ α ̄ , α ⁢ θ μ + i ⁢ Λ ⁢ ( ϕ ) ⁢ θ , where d ∗ the dual of 𝑑 with respect to the adapted Riemannian metric.

Proof

By a standard formula, in terms of the Levi–Civita connection ∇ ̃ ,

− d ∗ ⁢ ϕ = T ⁢ ⌟ ⁢ ∇ ̃ T ⁢ ϕ + Z α ⁢ ⌟ ⁢ ∇ ̃ Z ̄ α ⁢ ϕ + Z ̄ α ⁢ ⌟ ⁢ ∇ ̃ Z α ⁢ ϕ .

We recall the relationship between the Levi–Civita connection ∇ ̃ and the Tanaka–Webster connection, that can be found in [14]: for X , Y horizontal,

∇ ̃ T ⁢ T = 0 , ∇ ̃ X ⁢ T = A ⁢ X + 1 2 ⁢ J ⁢ X , ∇ ̃ X ⁢ Y = ∇ X Y − [ ⟨ A ⁢ X , Y ⟩ + 1 2 ⁢ d ⁢ θ ⁢ ( X , Y ) ] ⁢ T .

We compute, using the above identities,

T ⁢ ⌟ ⁢ ∇ ̃ T ⁢ ϕ = f μ ⁢ ν ̄ ⁢ ( ∇ ̃ T ⁢ θ μ ⁢ ( T ) ⁢ θ ν ̄ − ∇ ̃ T ⁢ θ ν ̄ ⁢ ( T ) ⁢ θ μ ) = f μ ⁢ ν ̄ ⁢ ( − θ μ ⁢ ( ∇ ̃ T ⁢ T ) ⁢ θ ν ̄ + θ ν ̄ ⁢ ( ∇ ̃ T ⁢ T ) ⁢ θ μ ) = 0 ,

Z α ⁢ ⌟ ⁢ ∇ ̃ Z ̄ α ⁢ ϕ = Z ̄ α ⁢ f α ⁢ ν ̄ ⁢ θ ν ̄ + f μ ⁢ ν ̄ ⁢ ∇ ̃ Z ̄ α ⁢ θ μ ⁢ ( Z α ) ⁢ θ ν ̄ − f μ ⁢ ν ̄ ⁢ ∇ ̃ Z ̄ α ⁢ θ ν ̄ ⁢ ( Z α ) ⁢ θ μ + f α ⁢ ν ̄ ⁢ ∇ ̃ Z ̄ α ⁢ θ ν ̄ = Z ̄ α ⁢ f α ⁢ ν ̄ ⁢ θ ν ̄ − f μ ⁢ ν ̄ ⁢ θ μ ⁢ ( ∇ ̃ X ̄ α ⁢ Z α ) ⁢ θ ν ̄ + f μ ⁢ ν ̄ ⁢ θ ν ̄ ⁢ ( ∇ ̃ Z ̄ α ⁢ Z α ) ⁢ θ μ − f α ⁢ ν ̄ ⁢ ( θ ν ̄ ⁢ ( ∇ ̃ Z ̄ α ⁢ Z β ) ⁢ θ β + θ ν ̄ ⁢ ( ∇ ̃ Z ̄ α ⁢ Z ̄ β ) ⁢ θ β ̄ + θ ν ̄ ⁢ ( ∇ ̃ Z ̄ α ⁢ T ) ⁢ θ ) = Z ̄ α ⁢ f α ⁢ ν ̄ ⁢ θ ν ̄ − f μ ⁢ ν ̄ ⁢ θ μ ⁢ ( ∇ Z ̄ α Z α ) ⁢ θ ν ̄ + f μ ⁢ ν ̄ ⁢ θ ν ̄ ⁢ ( ∇ Z ̄ α Z α ) ⁢ θ μ − f α ⁢ ν ̄ ( θ ν ̄ ( ∇ Z ̄ α Z β ) θ β + θ ν ̄ ( ∇ Z ̄ α Z β ̄ ) θ β ̄ + − i 2 δ α ν θ ) = f α ⁢ ν ̄ , α ̄ ⁢ θ ν ̄ + i 2 ⁢ Λ ⁢ ( ϕ ) ⁢ θ ,

and similarly,

Z ̄ α ⁢ ⌟ ⁢ ∇ ̃ Z α ⁢ ϕ = Z α ⁢ f μ ⁢ α ̄ ⁢ θ μ − f μ ⁢ ν ̄ ⁢ ∇ ̃ Z α ⁢ θ ν ̄ ⁢ ( X ̄ α ) ⁢ θ μ + f μ ⁢ ν ̄ ⁢ ∇ ̃ Z α ⁢ θ μ ⁢ ( Z ̄ α ) ⁢ θ ν ̄ − f μ ⁢ α ̄ ⁢ ∇ ̃ Z α ⁢ θ μ = Z α ⁢ f μ ⁢ α ̄ ⁢ θ μ + f μ ⁢ ν ̄ ⁢ θ ν ̄ ⁢ ( ∇ ̃ Z α ⁢ X ̄ α ) ⁢ θ μ − f μ ⁢ ν ̄ ⁢ θ μ ⁢ ( ∇ ̃ Z α ⁢ Z ̄ α ) ⁢ θ ν ̄ + f μ ⁢ α ̄ ⁢ ( θ μ ⁢ ( ∇ ̃ Z α ⁢ Z β ) ⁢ θ β + θ μ ⁢ ( ∇ ̃ Z α ⁢ Z ̄ β ) ⁢ θ ̄ β + θ μ ⁢ ( ∇ ̃ Z α ⁢ T ) ⁢ θ ) = Z α ⁢ f μ ⁢ α ̄ ⁢ θ μ + f μ ⁢ ν ̄ ⁢ θ ν ̄ ⁢ ( ∇ Z α Z ̄ α ) ⁢ θ μ − f μ ⁢ ν ̄ ⁢ θ μ ⁢ ( ∇ Z α Z ̄ α ) ⁢ θ ν ̄ + f μ ⁢ α ̄ ⁢ ( θ μ ⁢ ( ∇ Z α Z β ) ⁢ θ β + θ μ ⁢ ( ∇ Z α Z ̄ β ) ⁢ θ ̄ β + i 2 ⁢ δ α μ ⁢ θ ) = f μ ⁢ α ̄ , α ⁢ θ μ + i 2 ⁢ Λ ⁢ ( ϕ ) ⁢ θ .

We remark that, in the calculations, the torsion 𝐴 does not appear because it anti-commutes with 𝐽 and therefore maps a ( 1 , 0 ) -vector to a ( 0 , 1 ) -vector and vice versa. Combining these results yields − d ∗ ⁢ ϕ = f α ⁢ ν ̄ , α ̄ ⁢ θ ν ̄ + f μ ⁢ α ̄ , α ⁢ θ μ + i ⁢ Λ ⁢ ( ϕ ) ⁢ θ , concluding the proof. ∎

Recall that a pseudohermitian structure 𝜃 is called pseudo-Einstein if R μ ⁢ ν ̄ = W n ⁢ δ μ ⁢ ν ̄ , where 𝑊 is the scalar curvature. When n = 1 , this is always true. From now on, we assume n ≥ 2 . Lee showed in [21] that a necessary condition for the existence of a pseudo-Einstein structure is c 1 ⁢ ( M ) = 0 .

Proposition A.2

Suppose 𝑀 is a closed CR manifold of dimension 2 ⁢ n + 1 ≥ 5 with c 1 ⁢ ( M ) = 0 . If 𝜃 is a pseudohermitian structure with zero torsion and constant scalar curvature, then it is pseudo-Einstein.

Proof

Let ϕ = ρ θ − W n ⁢ d ⁢ θ . Since A = 0 , we have

ϕ = i ⁢ R μ ⁢ ν ̄ ⁢ θ μ ∧ θ ν ̄ − W n ⁢ d ⁢ θ = i ⁢ ( R μ ⁢ ν ̄ − W n ⁢ δ μ ⁢ ν ̄ ) ⁢ θ μ ∧ θ ν ̄ .

It is a real ( 1 , 1 ) -form with Λ ⁢ ( ϕ ) = 0 . As A = 0 , we have R α ⁢ ν ̄ , α = R μ ⁢ α ̄ , α = 0 by the Bianchi identities. Together with 𝑊 being constant, we see that we have d ∗ ⁢ ϕ = 0 by Lemma A.1. As c 1 ⁢ ( T 1 , 0 ⁢ M ) = 0 , ρ θ is exact, i.e. there is a real 1-form 𝜒 such that ρ θ = d ⁢ χ . Then ϕ = d ⁢ χ ̃ , where χ ̃ = χ − W n ⁢ θ . It follows

‖ ϕ ‖ 2 = ⟨ ϕ , d ⁢ χ ̃ ⟩ = ⟨ d ∗ ⁢ ϕ , χ ̃ ⟩ = 0 .

Therefore, ϕ = 0 , i.e. 𝜃 is pseudo-Einstein. ∎

Acknowledgements

J.-H. Cheng (P. Yang, resp.) is grateful to Scuola Normale Superiore and Princeton University (Academia Sinica in Taiwan, resp.) for the kind hospitality. A. Malchiodi would like to thank Academia Sinica in Taiwan and Princeton University for arranging some collaboration visits.

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Received: 2023-09-05

Revised: 2024-04-08

Published Online: 2024-06-04

Published in Print: 2024-08-01

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