A remark on the Laplacian flow and the modified Laplacian co-flow in $${\mathrm{G}}_2$$ G 2 -geometry
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A remark on the Laplacian flow and the modified Laplacian co‑flow in G2‑geometry Lucio Bedulli1 · Luigi Vezzoni2 Received: 15 May 2020 / Accepted: 7 July 2020 © The Author(s) 2020
Abstract We give a shorter proof of the well-posedness of the Laplacian flow in G2-geometry. This is based on the observation that the DeTurck–Laplacian flow of G2-structures introduced by Bryant and Xu as a gauge fixing of the Laplacian flow can be regarded as a flow of (not necessarily closed) G2-structures, which fits in the general framework introduced by Hamilton in J Differ Geom 17(2):255–306, 1982. A similar application is given for the modified Laplacian co-flow. Keywords Laplacian flow · G2-geometry · Short-time existence
1 Introduction In[1] Bryant introduced a geometric flow in G2-geometry which evolves an initial closed G2-structure 𝜑0 in the direction of its Laplacian. Given a compact seven-dimensional manifold with a closed G2-structure (M, 𝜑0 ) , a Laplacian flow is a solution to the evolution equation 𝜕 𝜑 𝜕t t
= Δ𝜑t 𝜑t ,
d𝜑t = 0,
(1)
𝜑|t=0 = 𝜑0 .
The well-posedness of Eq. (1) is proved in[2] by applying the Nash–Moser theorem to the gauge fixing 𝜕 𝜑 𝜕t t
= Δ𝜑t 𝜑t + LV(𝜑t ) 𝜑t ,
d𝜑t = 0,
(2)
𝜑|t=0 = 𝜑0 ,
where L is the Lie derivative and V ∶ C∞ (M, Λ3+ ) → C∞ (M, TM) is a first-order differential operator which depends on the choice of a connection on M. Here, Λ3+ denotes the open * Luigi Vezzoni [email protected] Lucio Bedulli [email protected] 1
Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università dell’Aquila, Via Vetoio 67100, L’Aquila, Italy
2
Dipartimento di Matematica G. Peano, Università di Torino, Via Carlo Alberto 10, 10123 Turin, Italy
13
Vol.:(0123456789)
Annals of Global Analysis and Geometry
subbundle of Λ3 of G2-structures on M. A solution to (2) is usually called a DeTurck–Laplacian flow. A DeTurck–Laplacian flow 𝜑t is also a solution to 𝜕 𝜑 𝜕t t
= dd𝜑∗ 𝜑t + d𝜄V(𝜑t ) 𝜑t , t
(3)
𝜑|t=0 = 𝜑0 .
In the present note, we observe that Eq. (3) fits in the general framework introduced by Hamilton in[4]. As a direct consequence, we have the following theorem which in particular implies the well-posedness of (2)
Theorem 1.1 Let (M, 𝜑0 ) be a compact seven-dimensional manifold with a G2-structure. Then, Eq. (3) has a unique short-time solution. In[5] Karigiannis, McKay and Tsui introduced the Laplacian co-flow as the solution to the evolution equation 𝜕 (∗ 𝜕t 𝜑t
𝜑t ) = −Δ𝜑t ∗𝜑t 𝜑t ,
d ∗𝜑t 𝜑t = 0,
(4)
𝜑|t=0 = 𝜑0 ,
where in this case 𝜑0 is supposed to be co-closed with respect to the metric induced by itself. The well-posedness of this last equation is still an open problem and Grigorian introduced in[3] the following modification 𝜕 (∗ 𝜕t 𝜑t
𝜑t ) = Δ𝜑t ∗𝜑t 𝜑t + 2d((A − Tr(T(𝜑t ))𝜑t ),
d ∗𝜑t 𝜑t = 0,
𝜑|t=0 = 𝜑0 ,
(5)
where A is a constant and T(𝜑t ) is the torsion of 𝜑t . In[3], the well-posedness of (5) is proved following the same approach of Bryant in[1] by applying the Nash–Moser theorem to the gauge fixing 𝜕 (∗ 𝜕t 𝜑t
𝜑t ) = Δ𝜑t ∗𝜑t 𝜑t + 2d(
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