Local curvature estimates for the Laplacian flow

Abstract

In this paper we give local curvature estimates for the Laplacian flow on closed $$G_{2}$$ G 2 -structures under the condition that the Ricci curvature is bounded along the flow. The main ingredient consists of the idea of Kotschwar et al. (J Funct Anal 271(9):2604–2630, 2016) who gave local curvature estimates for the Ricci flow on complete manifolds and then provided a new elementary proof of Sesum’s result (Sesum in Am J Math 127(6):1315–1324, 2005), and the particular structure of the Laplacian flow on closed $$G_{2}$$ G 2 -structures. As an immediate consequence, this estimates give a new proof of Lotay and Wei’s (Geom Funct Anal 27(1):165–233, 2017) result which is an analogue of Sesum’s theorem. The second result is about an interesting evolution equation for the scalar curvature of the Laplacian flow of closed $$G_{2}$$ G 2 -structures. Roughly speaking, we can prove that the time derivative of the scalar curvature $$R_{g(t)}$$ R g ( t ) is equal to the Laplacian of $$R_{g(t)}$$ R g ( t ) , plus an extra term which can be written as the difference of two nonnegative quantities.

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“…K ρ 2 e CKT U(m) + Vol g(t) B g (x 0 , ρ/ √ K) .As in[1,4,5], we can derive from (2.55) the follolwing estimateU(m) ≤ e ΞT U(m) t=0 + e CKT Vol g(t) B g (x 0 , ρ/ √ K) ≤ e ΞT U(m) t=0 + e CKT Vol g B g (x 0 , ρ/ Since M is closed, it follows that max M×[0,T max ) |Rm g(t) | g(t) ≤ C ′ (m, K, L, P, ρ, T max , Λ) < +∞which contradicts with (1.3). Hence the conclusion (1.4) holds.…”

“…Suppose now T max < +∞ and |Ric g(t) | g(t) ≤ K, |F(t)| g(t) ≤ L, |∇ g(t) F(t)| g(t) ≤ P on M × [0, T max ),for some constants K, L and P. We writef := Cu, u := 1 + |Rm g(t) | 2 g(t) + |F(t)| 2 g(t) + |∇ g(t) F(t)| 2 ≤ C(m, K, L, P, ρ, T max , Λ) < +∞for any integer m ≥ 1. If we apply Lemma 19.1 in[3] to (3.1) together with (3.2), as well as the proof in[1,4,5], we obtain particularly maxB g (x 0 ,ρ/4 √ K)×[0,T max )|Rm g(t) | g(t) ≤ C ′ (m, K, L, P, ρ, T max , Λ) < +∞…”

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“…In 2017, Lotay and Wei [34] proved the long time existence for the Laplacian G 2 flow (1.1) under bounded Riemann curvature or Ricci curvature. The second author [27] gave another proof that the Laplacian G 2 flow (1.1) will exist as long as the Ricci curvature remains bounded via local curvature estimates in 2021.…”

“…Hence, the question of Lotay and Wei is equivalent to Question 2: whether the Laplacian G 2 flow on closed G 2 -structures will exist as long as the scalar curvature remains uniformly bounded from below. The second author [27] computed the evolution equation for the scalar curvature, giving a weak bound for the scalar curvature which can not be used to answer Lotay-Wei's question.…”

“…Remark 1.3. From [27,34] or (2.15), we get R(g(t)) = −|T(t)| 2 g(t) ≤ 0, then we can use the torsion form T(t) to replace R(g(t)).…”