The second variational formula for Willmore submanifolds

Since T ∗ (M × R) = T ∗ M ⊕ T ∗ R we can write
Ψi = ψi + aidt, Φα = φα + bαdt, (2.17)
Ωij = ωij + Pijdt, Ωiα = ωiα + Liαdt, Ωαβ = ωαβ + Qαβdt, (2.18)
where {ai, bα, Pij, Qαβ} are local functions with Pij = −Pji and Qαβ = −Qβα. Let {Bα ij, Aij, Cα i }
be Moebius invariants for xt defined in §1. If we denote by dM the exterior differential operator
on T ∗M, then we have d = dM + dt ∧ ∂
∂t on T ∗ (M × R). It follows from (2.6), (2.16)
and (2.17) (comparing the terms in T ∗ (M) ∧ dt ) that
ai = −v,i + vαC α i , (2.19)
where v,i := Ei(v). Similarly we get from (2.8), (2.16), (2.17), (2.18) that
and get from (2.7), (2.16), (2.18) that
Liα = vα,i
(2.20)
∂ωi
∂t = (Pij + vδij − vαB α ij)ωj. (2.21)
By a direct calculation in a similar way, we obtain from (2.9)∼(2.13) that
∂ωij
∂t = (Pij,k + B α ikvα,j − B α jkvα,i − aiδkj + ajδik)ωk, (2.22)
∂ψi
∂t = (ai,j + PikAkj + vα,iC α j − bαB α ij − vAij)ωj (2.23)
∂φα
∂t = (bα,i + QαβC β
i − Aijvα,j + ajB α ji − vC α i )ωi, (2.24)
∂ωiα
∂t = (vα,ij + PikB α kj − B β
ijQβα + Aijvα + bαδij)ωj
(2.25)
∂ωαβ
∂t = (Qαβ,i + vβ,jB α ji − vα,jB β
ji + vβC α i − vαC β
i )ωi, (2.26)
where {vα,i} are covariant derivatives of {vα}. Since φα = Cα i ωi and ψi = Aijωj, from (2.21),
(2.23) and (2.24) we have
∂C α i
∂t = bα,i + QαβC β
i + ajB α ij − Aijvα,j + PijC α j + B β
ijC α j vβ − 2vC α i , (2.27)
∂Aij
∂t = ai,j + PikAkj − PkjAki + vα,iC α j − B α ijbα + AikB α kjvα − 2vAij. (2.28)
Since ωiα = B α ijωj, from (2.21) and (2.25) we have
∂B α ij
∂t = vα,ij − vB α ij + PikB α kj − PkjB α ki − B β
ijQβα + vβB α ikB β
kj + Aijvα + bαδij. (2.29)
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