The second variational formula for Willmore submanifolds

(3.8) and (3.16), we have
W ′′ (0) =
m − 1
2k(m − k)m3 � �
2k(m − k)(m − 1)(∆Mf)
M
2
+m 2 (k 2 + m 2 − km − 2m)f∆Mf
+2(m − 2) ((m − k)∆M1f − k∆M2f) 2
+m �
m(3km − k 2 − m 2 ) + 6(m − k) 2�
f∆M1f
+m �
m(m 2 − km − k 2 ) + 6k 2�
f∆M2f
+4k(m − k)m 2 f 2 ) dM.
(3.17)
Let λi, λ ′ i and µij be the eigenvalues of Laplace operators ∆M1, ∆M2 and ∆M, respectively,
then we have
λi = i(k + i − 1) n
n − k , λ′ j = j(n − k + j − 1) n
k , µij = λi + λ ′ j,
where i and j are nonnegative integers. Let fi, f ′ i be eigenfunctions corresponding to λi and
λ ′ i, respectively, then gij(p, q) = fi(p)f ′ j(q) ((p, q) ∈ M1×M2) is an eigenfunction corresponding
to µij. For any f ∈ C ∞ (M m k ) we have the decomposition of f into eigenfuctions
f = �
i+j�=0
where cij and c0 are constants. Thus we have
∆M1f = − �
λicijgij, ∆M2f = − �
i+j�=0
i+j�=0
Substituting (3.18) and (3.19) into formula (3.17), we get
W ′′ (0) ≥ m−1
2k(m−k)m3 � �
M i+j�=0
cijgij + c0, (3.18)
λ ′ jcijgij, ∆Mf = − �
�
2k(m − k)(m − 1)µ 2 ij
i+j�=0
−m2 (k2 + m2 − km − 2m)µij + 2(m − 2) �
(m − k)λi − kλ ′ �2 j
−m (m(3km − k 2 − m 2 ) + 6(m − k) 2 ) λi
−m (m(m 2 − km − k 2 ) + 6k 2 ) λ ′ j + 4k(m − k)k 2 ) c 2 ijg 2 ijdM
=
m−1
2k(m−k)m3 � �
i+j�=0 M {2(m − k)(m2 − 2m + k)λ2 i
−2(2m 3 + km 3 − 6km 2 + 3k 2 m)λi
+2k(m 2 − m − k)(λ ′ j) 2 − 2m(m 3 − m 2 − km 2 + 3k 2 )λ ′ j
+4k(m − k)λiλ ′ j + 4k(m − k)m 2 }c 2 ijg 2 ijdM.
14
µijcijgij. (3.19)
(3.20)
(3.21)