The second variational formula for Willmore submanifolds

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Example 4.1. Sm � 2(m+1) ( S m ( � m ) → Sm+p with p = 1(m − 1)(m + 2). Let 2 2(m + 1) m ) = {(x0, x1, · · · , xm) ∈ R m+1 ; � x 2 i = 2(m + 1) m }. Let E be the space of (m + 1) × (m + 1) symmetric matrices (uij), (i, j = 1, · · · , m), such that � uii = 0; it is a vector space of dimension 1m(m + 3). We define a norm in E by 2 ||(uij)|| 2 = � u2 ij. Let Sm+p with p = 1(m − 1)(m + 2) be the unit hypersphere in E. The 2 mapping of Sm � 2(m+1) ) into Sm+p , defined by m uij = 1 � m 2 m + 1 (xixj − 2 m δij), is an isometric minimal immersion. (Actually, this gives an imbedding of the real projective space of m-dimension into Sm+p ). We know that Sm � 2(m+1) ( ) is an Einstein manifold, thus m from Theorem 4.2 this immersion is a Willmore submanifold. Example 4.2. CP m 2m/(m+1) → Sm(m+2)−1 . We can define a minimal immersion of the with holomorphic sectional curvature m-dimensional complex projective space CP m 2m/(m+1) 2m/(m+1) into a unit sphere Sm(m+2)−1 such that the usual coordinate functions of Rm(m+2) are all independent hermitian harmonic functions of degree 1 on CP m 2m/(m+1) (see Wallach [16]). We also know that CP m 2m/(m+1) is an Einstein manifold with Ricci curvature m. There- fore from Theorem 4.2 we conclude that this immersion is a Willmore submanifold. Proposition 4.1. Let M = Sm1 (a1) × · · · × Smr (ar) be the submanifold imbedded into Sm+r−1 , where m1 + · · · + mr = m. Then M is a Willmore submanifold if and only if Proof. Consider ai = � m − mi , i = 1, · · · , r. (4.14) m(r − 1) R m+r = R m1+1 r� mr+1 × · · · × R , m = i=1 S m1 (a1) × · · · × S mr (ar) = {(a1x1, · · · , arxr) ∈ R m+r : |xi| = 1, i = 1, · · · , r}, where Smi (ai) ⊂ Rmi+1 r� , (i = 1, · · · , r), a i=1 2 i = 1, is a m-dimensional submanifold in Sm+r−1 . Writing x = (a1x1, · · · , arxr) : M = Sm1 (a1) × · · · × Smr (ar) → Sm+r−1 , x1 · x1 = · · · = xr · xr = 1. Then r − 1 orthogonal normal vector fields of M are mi, en+λ = (aλ1x1, · · · , aλrxr), λ = 1, · · · , r − 1 where (aλ1, · · · , aλr) satisfies that (r × r) matrix ⎛ ⎜ A = ⎜ ⎝ a1 a11 . · · · · · · · · · ar a1r . ar−11 · · · ar−1r 18 ⎞ ⎟ ⎠
is an orthogonal matrix, thus r−1 � λ=1 aλiaλj = δij − aiaj. (4.15) Therefore the first fundamental form and the second fundamental form of M are given by I = dx · dx = a 2 1dx1 · dx1 + · · · + a 2 rdxr · dxr, r−1 � II = − [a1aλ1dx1 · dx1 + · · · + araλrdxr · dxr]em+λ. λ=1 We get the components of the second fundamental form ⎛ (h m+λ ⎜ ij ) = ⎜ ⎝ − aλ1 a1 E1 . .. aλr ar Er ⎞ ⎟ ⎠ , λ = 1, · · · , r − 1 (4.16) where Ei is a (mi × mi) unit matrix. It follows that ρ 2 = m(S − mH 2 )/(m − 1) is constant and ∇ ⊥ eα = 0, ∆H α = 0, α = m + λ. From (4.1), we know that M is a Willmore submanifold if and only if SH α + H β h β ijh α ij − h α ijh β ik hβ kj − m| � H| 2 H α = 0, n + 1 ≤ α = m + λ ≤ n + r − 1. (4.17) Putting H m+λ = 1 r� aλi mi, m i=1 ai 1 ≤ λ ≤ r − 1, S = � 1 − a mi i=1 2 i a2 , i − � h µ,i,j,k m+λ ij h m+µ ik hm+µ � mi kj = i a3 aλi(1 − a i 2 i ), � h i,j m+λ ij h m+µ ij = � aλiaµi mi i a2 i into (4.17), by use of (4.15) and (4.16) we can easily check that (4.17) can be reduced to � aλi{− i mi � mj(1 − mai j 1 a2 ) − j 1 � δij − aiaj mimj m j a2 i aj mi a3 (1 − a i 2 i ) + 1 m2 � δjk − ajak mimjmk } = 0, j,k aiajak where 1 ≤ λ ≤ r − 1. (4.18) is equivalent to Therefore (4.19) becomes −2m + � mj(1 − j 1 a2 ) − j 1 � δij − aiaj mimj m j a2 i aj mi a3 (1 − a i 2 i ) + 1 m2 � δjk − ajak mimjmk = 0. j,k aiajak − mi mai m − mi a 2 i + � j mj a 2 j (4.18) (4.19) (1 + mj ) = 0, i = 1, · · · , r. (4.20) m 19
Page 1 and 2: The second variational formula for
Page 3 and 4: and we shall agree that repeated in
Page 5 and 6: Moebius metric of xt. Let {Ei := ρ
Page 7 and 8: Multiplying B α ij to (2.29) and u
Page 9 and 10: tr(A) = 1 2m m + κ, (2.41) 2 where
Page 11 and 12: then we have where m� I = ω i=1
Page 13 and 14: Substituting (3.9), (3.11) and (3.1
Page 15 and 16: If we set A(i, j) = 2(m − k)(m 2
Page 17: Gauss equations of an m-dimensional
Page 21: [9] Li, P. and Yau, S.-T.: A new co

The second variational formula for Willmore submanifolds

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