Hyper-Lagrangian Submanifolds of Hyperkähler Manifolds and ...

Hyper-Lagrangian Submanifolds of Hyperkähler Manifolds and Mean Curvature Flow 353
are complex structures orthogonal to J . Let’s denote these three combinations by J = ∑ α a αJ α ,
K = ∑ α b αJ α , JK = ∑ α c αJ α and write a = (a 1 ,a 2 ,a 3 ) and so on. Then the map J is
represented by
a(t,x) = (cos θ sin ϕ, sin θ sin ϕ, cos ϕ) .
By a straightforward calculation, we have
and
a(t o ,p)= (1, 0, 0), b(t o ,p)= (0, 1, 0), c(t o ,p)= (0, 0, 1) ,
∇b(t o ,p)= (−∇θ| (to ,p), 0, 0), ∇c(t o ,p)= (∇ϕ| (to ,p), 0, 0) ,
∂ t b(t o ,p)= (−∂ t θ| (to ,p), 0, 0), ∂ t c(t o ,p)= (∂ t ϕ| (to ,p), 0, 0) ,
b ×∇b| (to ,p) = (0, 0, ∇θ| (to ,p)), c ×∇c| (to ,p) = (0, ∇ϕ| (to ,p), 0) ,
△b(t o ,p)= ( −△θ,−|∇θ| 2 , 0 )∣ ∣
(to ,p) ,
△c(t o ,p)= ( △ ϕ,2∇θ ·∇ϕ,−|∇ϕ| 2)∣ ∣
(to ,p) .
In the above, ∇ and △ denote the gradient and the Laplacian with respect to the induced metric
on L to ; b ×∇b and c ×∇c the cross products by regarding b, ∇b, c, and ∇c as 3-vectors.
By assumption, both g(K·, ·) and g(JK·, ·) vanish on T x L t for any (t, x) in a neighborhood
of (t o ,p). In particular, ω 2 = ω 3 = 0at(t o ,p). Then by applying Proposition 3.1 to the 2-form
g(K·, ·), we have at the point (t o ,p),
0 = [(∂ t −△)b 1 ]ω 1,ij − 2∇ k b 1 ∇ k ω 1,ij
+ ∑ α
(h α,kkj ∇ i b α − h α,kki ∇ j b α ).
Note that we have used the fact that ∑ s Rs sji = 0 whenever g(K·, ·) ≡ 0 by the flatness of the
Ricci curvature, see [11]. On the other hand, Lemmas 3.3 and 3.2 imply
and
∇ω 1,ij = 0 ,
∑
h α,kkj ∇ i b α = 0 .
Hence, we conclude that
0 =[(∂ t −△ t )b 1 ]ω 1,ij .
Since ω 2 = ω 3 = 0, we have ω 1 ̸= 0. Therefore,
α
(∂ t −△ t )b 1 = 0 ,
which is equivalent to
(∂ t −△ t )θ = 0 at(t o ,p).
Similarly, by applying Proposition 3.1, and Lemmas 3.3 and 3.2 to the symplectic form g(JK·, ·),
we have
(∂ t −△ t )ϕ = 0 at(t o ,p).
All together, we have proved that J satisfies
as a mapping from [0,T)× L 0 to S 2 .
(∂ t −△ t )J = 0