WILLMORE SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD

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1 WILLMORE SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD Z. HU Department of Mathematcs, Zhengzhou Unversty, , Zhengzhou, People s Republc of CHINA E-mal: huz@zzu.edu.cn H. LI Department of Mathematcal Scences, Tsnghua Unversty , Beng, People s Republc of CHINA E-mal: hl@math.tsnghua.edu.cn Let N n+p be an n + p-dmensonal Remannan manfold and x : M N n+p an sometrc mmerson of an n-dmensonal Remannan manfold M. A mappng x : M N n+p s called Wllmore f t s an extremal submanfold of the followng Wllmore functonal: W x = S nh 2 n 2 dv, M where S = α,, hα 2 and H are respectvely the norm square of the second fundamental form and the mean curvature of the mmerson x, dv s the volume element of M. In ths survey paper, we calculate the Euler-Lagrangan equaton of W x for an n-dmensonal submanfold n an n + p-dmensonal Remannan manfold N n+p and gve many applcatons as well as many examples of Wllmore submanfolds. 1. Introducton Let N n+p, h be an orented smooth Remannan manfold of dmenson n+p. Let M n be a smooth manfold of dmenson n and x : M n N n+p be a dfferentable mmerson. By mposng on M n the nduced metrc we MSC 2000: 53C42, 53A10. Keywords : Wllmore functonal, Wllmore submanfolds, mnmal submanfolds. Partally supported by grants of CSC, NSFC and Outstandng Youth Foundaton of Henan, Chna. Partally supported by the Alexander von Humboldt Stftung and Zhongdan grant of NSFC. 275

2 276 can suppose M to be Remannan and x to be an sometrc mmerson. We wll agree on the followng range of ndces: 1,,, n; n + 1 α, β, γ, n + p; 1 A, B, C, n + p. Let {e A } be a local orthonormal bass for T N n+p wth dual bass {ω A } such that when restrcts to M n, {e } s a local orthonormal bass for T M and {e α } s a local orthonormal bass for the normal bundle of x : M n N n+p. We wrte the second fundamental form II and the mean curvature vector H of x by II = h α ω ω e α, H = 1 h α e α := H α e α. 1.1 n,,α α Let S = II 2 =,,α hα 2 and H = H be the norm square of II and the mean curvature of x : M N n+p, respectvely. We defne the followng non-negatve functon on M,α ρ 2 = S n H 2, 1.2 whch vanshes exactly at the umblc ponts of M. Wllmore functonal s the followng non-negatve functonal see [ 9 ], [ 34 ] or [ 43 ] W x := ρ n dv = S n H 2 n 2 dv. 1.3 M It s well nown cf. [ 9 ], [ 34 ] and [ 43 ] that ths functonal s nvarant under conformal transformatons of N n+p. We use the term Wllmore submanfolds to call crtcal ponts of W x, because when n = 2 and N n+p = S n+p, the functonal essentally concdes wth the well-nown Wllmore functonal W x and ts crtcal ponts are the Wllmore surfaces. The famous Wllmore conecture says that W x 4π 2 holds for all mmersed tor x : M S 3. The conecture was approached by Wllmore [ 46 ], L-Yau [ 25 ], Montel-Ros [ 28 ], Ros [ 38 ], Langer-Snger [ 17 ], Hertrch- Jeromn-Pnall [ 15 ], Ca M. [ 6 ], Kusner R. [ 16 ], Toppng P. [ 42 ] and many others see [ 45 ], [ 47 ] and references theren. Compared wth the fact that the Wllmore functonal has been extensvely studed when n = 2 and N 2+p s the Eucldean space R 2+p or the sphere S 2+p or the hyperbolc space H 2+p, because W s nvarant under conformal transformatons of ambent space, n hgher dmensonal case, n 3, the research for W x receved lttle attenton durng the same perod. We note that n [ 34 ], the authors stated the formula of Euler-Lagrangan equaton of W x for an n-dmensonal submanfold n an n + p-dmensonal M

3 277 Remannan manfold, but they dd not wrte down the proof. In [ 43 ] C. P. Wang got the Euler-Lagrangan equaton of W x for submanfolds x : M n S n+p wthout umblc ponts n terms of Möbus geometry. In [ 13 ], the authors calculated the second varaton formula of W x for Wllmore submanfolds x : M n S n+p wthout umblc ponts n terms of Möbus geometry and gave many examples of Wllmore submanfolds. In [ 18 ], [ 20 ], the second author establshed the ntegral nequalty of Smons type for compact Wllmore submanfolds n x : M n S n+p and gave the characterzatons of Wllmore tor and Veronese surface see Theorem 4.1. We organze the paper as follows. In Secton 2, we gve a revew of n- dmensonal submanfolds n an n + p-dmensonal Remannan manfold. In Secton 3, we calculate the Euler-Lagrangan equaton for the crtcal ponts of W x for the most general case. For the purpose of later s applcaton, we also wrte out the equatons for some mportant specal cases. In Secton 4, we present some canoncal examples of Wllmore submanfolds n the n + p-dmensonal unt sphere S n+p. In Secton 5, we study Wllmore complex submanfolds n a complex space form. In Theorem 5.1, we prove that every complex submanfold wth constant scalar curvature n a complex space form s Wllmore. In Theorem 5.2, we prove that every complex curve n complex space form s Wllmore. These two theorems provde abundance of examples of Wllmore submanfolds, at least locally. In Secton 6, we study Lagrangan or totally real n the more general settng Wllmore submanfolds n complex space forms. After wrtng out the Euler-Lagrangan equaton for such cases see Theorem 6.1 and Theorem 6.2, we prove n Proposton 6.2 that every mnmal and Ensten totally real submanfold n a complex space form s Wllmore. 2. Submanfolds n Remannan manfold In ths secton, we gve a revew of submanfolds n the most general Remannan manfold, usng the method of movng frames whch we refer to [ 10 ] for more detals. We wll follow the notatons n the frst secton. Let {ω AB } be the connecton forms of N n+p, h, they are characterzed by the structure equatons dω A = B ω AB ω B, ω AB + ω BA = 0, 2.1 dω AB = C ω AC ω CB 1 R ABCD ω C ω D, C,D

4 278 where R ABCD are the components of the Remannan curvature tensor of N n+p, h. Now we restrct to a neghborhood of x : M N n+p. Let θ A, θ AB be the restrcton of ω A, ω AB to M. Then we have θ α = Tang ts exteror dervatve and mang use of 2.1, we get θ α θ = By Cartan s lemma we have θ α = h α θ, h α = h α, 2.5 from whch we can defne the second fundamental form II and the mean curvature vector H of x : M N n+p as stated n 1.1. We note that H s a normal vector feld over M. x : M N n+p s called a mnmal submanfold f H 0. If we denote by R l the Remannan curvature tensor of M, then from 2.2 and 2.5, we get the Gauss equaton R l = R l + α h α h α l h α l h α, 2.6 R = R + n α H α h α α, h α h α, 2.7 R =, R + n 2 H 2 S, 2.8 where R and R are the Rcc curvature and scalar curvature of M, respectvely. If we denote by R αβ the curvature tensor of the normal connecton n the normal bundle of x : M N n+p,.e., dθ αβ = θ αγ θ γβ 1 R αβ θ θ, γ then from 2.2, 2.5 and 2.9, we get the Rcc equaton R αβ = R αβ +, h α h β hα h β Tang exteror dervatve of 2.5 and mang use of 2.1, 2.2, we get the Codazz equaton of x : M N n+p h α h α = R α, 2.11

5 279 where the covarant dervatve h α s defned by h α θ = dh α + h α θ + h α θ + h β θ βα β Defne the second covarant dervatve h α l by h α l θ l = dh α + h α l θ l + h α l θ l l l l + h α l θ l + h β θ βα l β Tang exteror dervatve of 2.12, we have the followng Rcc denttes h α l h α l = m h α m R ml + m h α m R ml + β h β R βαl We defne the frst, second covarant dervatves and Laplacan of the mean curvature vector feld H = α Hα e α n the normal bundle of x : M N n+p as follows H, α θ = dh α + H β θ βα, 2.15 β H, α θ = dh, α + H, α θ + H β, θ βα, 2.16 β H α = H,, α H α = 1 h α n Let f be a smooth functon on M, we defne the frst, second covarant dervatves f, f, and Laplacan of f as follows df = f θ, f, θ = df + f θ, f = f, Euler-Lagrange equaton of the Wllmore functonal To calculate the frst varaton of the Wllmore functonal W x 0 for x 0 : M N n+p, we assume, wthout loss of generalty, that M s compact wth possbly empty boundary M. If otherwse, we wll consder the varaton wth compact support. Let x : M R N n+p be a smooth varaton of x 0 such that x, t = x 0 and dx t T M = dx 0 T M on M for each small t, where x t p = xp, t. We call such varaton an admssble varaton of x 0. We note that the two boundary condtons for an admssble varaton dsappear f M = φ.

6 280 Along x : M R N n+p, we choose a local orthonormal bass {e A } for T N n+p wth dual bass {ω A }, such that {e, t} forms a local orthonormal bass for x t : M {t} N n+p. Snce T M R = T M T R, the pullbac of {ω A } and {ω AB } on N n+p through x : M R N n+p have the decomposton x ω α = V α dt, x ω = θ + V dt, 3.1 x ω = θ + L dt, x ω α = θ α + M α dt, x ω αβ = θ αβ + N αβ dt, 3.2 where { V, V α, L, M α, N αβ } are local functons on M R wth L = L, N αβ = N βα and V = d x t = V dx o e + V α e α, 3.3 dt t=0 α s the varaton vector feld of x t : M N n+p. We note that the one forms {θ, θ, θ α, θ αβ } are defned on M {t}, for t = 0, they reduce to the forms wth the same notaton on M. We denote by d M the dfferental operator on T M, then d = d M + dt t on T M R. Lemma 3.1. Under the above notatons, we have θ t = V, + L θ,α h α V α θ, 3.4 M α = V α, + h α V, 3.5 θ α t = M α, + L h α β N βα h β R α V + β R αβ V β θ, 3.6 where h α and the covarant dervatves V,, V α, and M α, are defned on M {t} by θ α = h α θ, 3.7 V, θ = d M V + V θ, 3.8 V α, θ = d M V α + V β θ βα, 3.9 β

7 281 M α, θ = d M M α + M α θ + β M β θ βα Proof. These are drect calculatons. In fact, substtutng 3.1 and 3.2 nto the followng equatons, respectvely dx ω = x dω = x ω ω + ω α ω α, α dx ω α = x dω α = x ω α ω + ω αβ ω β, β dx ω α = x dω α = = x ω ω α + β ω β ω βα 1 R αcd ω C ω D, 2 and comparng the terms n T M dt for each equaton on the both sdes, we can get 3.4, 3.5 and 3.6, respectvely. C,D Lemma 3.2. h α t L h α + L h α + h α V = V α, + + N αβ h β + R αβ V β + h α h β V β β,β Proof. Dfferentatng 3.7 wth respect to t and usng 3.4, 3.6 we get h α t = M α, + L h α R α V N βα h β + R αβ V β β h α V, + h α L + h α h β V β.,β Covarant dfferentatng 3.5 over M {t} and usng the Codazz equaton 2.11 for x t : M N n+p, we get M α, = V α, + V, h α + V h α = V α, + V, h α + V h α + V Rα. Combnng the above two equatons, we prove Lemma 3.2.

8 282 Set = n 3.11 and sum over wth usng, L h α = 0, we get H α t = 1 n V a + H, α V + N αβ H β β + 1 h α h β n V β + 1 R αβ V β n,,β From 3.11 and the fact that S =,,αh α 2, N αβ h β hα = 0, we obtan,,α,β 1 S 2 t = h α V α, + 1 S, V 2,,α + R αβ h α V β +,,α,β,β,,,α,β,,,α From 3.12 and α,β N αβ H α H β = 0, we obtan n H 2 2 t L h α h α = 0, h α h α h β V β = H α V α + n H 2, V 2 α + H α h α h β V β + H α Rαβ V β. 3.14,,α,β,α,β For x t : M N n+p, we consder the Wllmore functonal W x t = S n H 2 n 2 dv M Notng ρ 2 = S n H 2, we can rewrte W x t as W x t = ρ n θ 1 θ n From 3.13 and 3.14, we get ρ n = n ρ2 ρn 2 t 2 t = n S 2 ρn 2 t n H2 t = n ρ n 2{ h α V α, + 1 ρ 2, V 2 +,,α,β +,,,α,β,,α M 3.17 R αβ h α V β H α V α α h α h α h β V β H α h α h β V β H α Rαβ V β }.,,α,β,α,β

9 283 From 3.4, we have t θ 1 θ n = θ 1 θ t θ n 3.18 V, + L h α V α θ1 θ n = = V, n H α V α θ1 θ n. α From , we get { W x t [ = n ρ n 2 h α V α, n ρ n 2 H α V α 3.19 t M,,α α + n 2 ρn 2 ρ 2, V + ρ ] n V, + n ρ [ n 2 R βα h β + h β hβ hα α,,β,,,β,,β H β h β hα n,β H β Rβα ρ 2 H α] V α } dv. We note that n 2 ρn 2 ρ 2, V + ρ n V, = ρ n V,, 3.20 and that our varaton s admssble, thus at each pont on M we have V = V α = 0 and 0 = t dx xt t = d = dv e + dv α e α, t α from whch we can deduce V α, = 0 on M. It follows from 3.19, 3.20 and Green s formula that W x t { = n ρ n 2[ h β t hβ hα + R βα h β 3.21 M α,,,β,,β H β h β hα H β Rβα ρ 2 H α],,β,β +, ρ n 2 h α, ρ n 2 H α} V α dv. From 3.3 and 3.21 wth restrcton to t = 0, we have proved

10 284 Theorem 3.1. The varaton of the Wllmore functonal depends only on the normal component of the varaton vector feld. A submanfold x : M N n+p s a Wllmore submanfold f and only f 0 = ρ n 2[ R βα h β H β h β hα 3.22,,β,β + ρ n 2 h α n + 1 α n + p.,,,β h β hβ hα +,,β H β Rβα ρ 2 H α] +, { 2 ρ n 2 hα + ρ n 2, hα } H α ρ n 2 ρ n 2 H α 2 ρ n 2 Hα,, Remar 3.1. As we have ponted out n the ntroducton, Pedt and Wllmore stated an equvalent verson of 3.22 n [ 34 ], where they dd not wrte down the proof and they used very dfferent notatons. We also note that for some specal cases, Theorem 3.1 was obtaned n [ 18 ], [ 20 ], [ 29 ], [ 37 ], [ 43 ], [ 44 ] and [ 2 ] also see [ 1 ], [ 41 ]. When the ambent space s of constant curvature,.e., N n+p = R n+p c s a space form of constant sectonal curvature c, the equaton 3.22 of Wllmore submanfold x : M R n+p c can be smplfed. Theorem 3.2. [ 20 ], [ 34 ] A submanfold x : M n R n+p c s a Wllmore submanfold f and only f 0 = ρ n 2[,,,β h β hβ hα,,β H β h β hα ρ 2 H α] 3.23 ρ n 2 Hα, + n 1 H α ρ n n 1 n + 1 α n + p. + n 1 ρ n 2 H α, Proof. In the present case, we have whch gves ρ n 2, n H α δ h α, R ABCD = c δ AC δ BD δ AD δ BC, 3.24 R βα = c δ αβ δ, R βα = n c δ βα, 3.25 and therefore t follows that R βα h β H β Rβα = ,,β,β

11 285 Furthermore, the Codazz equaton 2.11 now reads h α = hα, applyng t we obtan that h α = n H, α, h α = n H α. 3.27, Puttng 3.26 and 3.27 nto 3.22, we obtan For the specal case n = 2, we have Corollary 3.1. [ 19 ], [ 44 ] A 2-dmensonal submanfold x : M 2 N 2+p s Wllmore f and only f 0 =, h α H α 2 H 2 H α +,,β R βα h β,β H β Rβα ,, R h α +,,β H β h β hα, R H α, 3 α 2 + p. If N 2+p = R 2+p c, 3.28 reduces to H α 2 H 2 H α +,,β H β h β hα = 0, 3 α 2 + p Proof. Now 3.22 taes the form 0 = h β hβ hα H β h β hα ρ 2 H α + h α 3.30,,,β,,β, H α + R βα h β H β Rβα, 3 α 2 + p.,,β,β From the Gauss equatons 2.7 and 2.8, we have h β hβ =,β ρ 2 = 2 H 2 +, R + 2 H β h β R, β R R Note that for 2-dmensonal case, we have R = R 2 δ Insertng 3.31 and 3.32 nto 3.30 we get 3.28.

12 286 If N 2+p = R 2+p c, we have R = c δ, R = 2 c,, h α = 2 H α. 3.33, From 3.28, 3.25 and 3.33, we get Ths proves Corollary 3.1. Another mportant applcaton of 3.28 s the case when p = n = 2 and N 4 = CP 2 4 s the complex proectve plane wth ts canoncal complex structure J and canoncal Fubn-Study metrc g of constant holomorphc sectonal curvature 4. Recall that for any sometrc mmerson x : M CP 2 4 of an orented surface M, we have a well-defned Kähler functon C on M defned by C = gje 1, e 2, where {e 1, e 2 } s any orthonormal bass of T M. Surfaces wth C = ±1 and C = 0 are called respectvely complex and Lagrangan cf. Secton 5 and Secton 6. For completeness, we conclude the Euler- Lagrangan equaton of W x for ths specal case, whch essentally s due to Montel and Urbano, cf. Proposton 5 of [ 29 ] where the Wllmore functonal was decomposed nto two modfed Wllmore functonal. To state the result by usng method of movng frames, we choose an orented local orthonormal bass {e A } 1 A 4 for T CP 2 such that {e } 1 2 s an orented orthonormal bass for T M and satsfy cf. [ 29 ] Je 1 = C e C 2 e 4, Je 2 = C e C 2 e 3, Je 3 = C e 4 1 C 2 e 2, Je 4 = C e 3 1 C 2 e Then we have Corollary 3.2. An sometrc mmerson x : M CP 2 4 s Wllmore f and only f H α C 2 2 H 2 H α + { 6 1 C 2 C H β h β,1, f α = 3; hα =,,β 6 1 C 2 C,2, f α = Proof. The Remannan curvature tensor R ABCD of CP 2 4 s gven by R ABCD = δ AC δ BD δ AD δ BC + gje A, e C gje B, e D 3.36 gje A, e D gje B, e C + 2 gje A, e B gje C, e D.

13 287 Usng 3.34 and 3.36, we have R = C 2 ; R h α = C 2 H α ;,,, R βα H β = 5 H α 3 C 2 H α,,β 3.37,,β R βα h β = { 2 H α C 2 h h 3 22, f α = 3; 2 H α C 2 h h 4 11, f α = Dfferentatng C = gje 1, e 2, we get C, ω = gjde 1, e 2 + gje 1, de = g J ω 1A e A, e 2 + g Je1, ω 2A e A A A = { } ω 2A gje 1, e A ω 1A gje 2, e A A = 1 C 2 ω 24 ω 13 = 1 C 2 h 4 2 h 3 1 ω, whch gves C, = 1 C 2 h 4 2 h 3 1, = 1, Puttng 3.40 nto 3.38, we get,,β R βα h β = { 8 6 C 2 H α C 2 C,1, f α = 3; 8 6 C 2 H α 3 1 C 2 C,2, f α = Fnally, we come to calculate, hα. By 2.11, we have h α = h α + R α = 2, H α + R α,, 3.42,, where R α = R α = 3 gje α, e gje, e can be calculated: R 31 = R 42 = 3 C 1 C 2, R 32 = R 41 = By usng 3.43 and the followng defnton of covarant dervatves R α, ω = d R α + R α ω + R β ω βα, 3.44 β

14 288 we have R 32, ω = R 41, ω = 3 C 1 C 2 ω 12 + ω 34, 3.45 R 31, ω = R 42, ω = 3 C 1 C 2, ω On the other hand, by exteror dfferentatng both sde of the frst equaton of 3.34 and then mang nner product wth e 3, we get 1 C 2 ω 12 + ω 34 = C ω 14 + ω 23 = C h h 3 2 ω Now and 3.40 gve R 3, = 3 C 1 C 2,1 +3 C 2 h h 4 12 = 6C 2 H C 2 C,1. R 4, = 3 C 1 C 2,2 +3 C 2 h h 4 11 = 6C 2 H C 2 C,2. It then follows from 3.42 that { 2 H α + 6 C 2 H α C 2 C,1, f α = 3; h α = 2 H α + 6 C 2 H α 3 1 C 2 C,2, f α = 4., 3.48 Puttng 3.37, 3.41 and 3.48 nto 3.28, we obtan Ths proves Corollary 3.2. Remar 3.2. Let x : M CP 2 4 be a compact Lagrangan surface, n [ 29 ] Montel-Urbano defned the followng modfed Wllmore functonal W x = H dv M We can chec by use of 3.14, 3.18 and 3.37 that the Euler-Lagrangan equaton of W s H α C 2 2 H 2 H α +,,β H β h β hα = 0, 3 α

15 Examples of Wllmore submanfolds n S n+p 1 In ths secton, we wll present some examples of Wllmore submanfolds n the sphere S n+p 1. Before that we frst recall the followng well nown fact Example 4.1. [ 44 ] It follows from 3.29 that every mnmal surface n N 2+p c s automatcally Wllmore. From [ 3 ], we now that there exst many canoncal examples of mnmal surfaces n S n 1. In partcular, the followng Veronese surface s a mnmal surface n S 4 1, thus t s a Wllmore surface. Let x, y, z denote the canoncal coordnate system n R 3 and let u = u 1, u 2, u 3, u 4, u 5 the canoncal coordnates n R 5. Veronese mnmal surface u : S 2 3 S 4 1 s defned by u 1 = 1 3 y z, u 2 = 1 3 x z, u 3 = 1 3 x y, u 4 = x2 y 2, u 5 = 1 6 x2 + y 2 2 z 2, where x 2 + y 2 + z 2 = 3. We note that there are much more abundance of non-mnmal Wllmore surfaces n N 2+p c, see, e.g., [ 2, 8, 11, 12, 22, 27, 30, 35 ], among many others. In [ 5 ], [ 24 ], [ 23 ], [ 33 ] and [ 36 ], readers can fnd some related results about Wllmore surfaces. When n 3, a mnmal submanfold x : M n N n+p c may be not Wllmore, even t has constant scalar curvature. Tae for example, the Clfford mnmal tor C m,n m := S m m n S n m n m S n+1 1, 1 m n 1, n are not Wllmore, except for the case n = 2 m see [ 18 ] or [ 13 ]. From 3.23, we can chec the followng fact Proposton 4.1. [ 13 ], [ 34 ] Every mnmal and Ensten submanfold n N n+p c s Wllmore. Remar 4.1. The examples C m,n m n 2 m above show that the Ensten condton n Proposton 4.1 can not be weaen to the condton of constant scalar curvature. Applyng Proposton 4.1, Guo-L-Wang [ 13 ] constructed many mnmal Wllmore submanfolds n S n+p 1 see example 1 and example 2 n [ 13 ].

16 290 However, the followng example shows that a Wllmore submanfold n N n+p c can be nether mnmal nor Ensten. Example 4.2. [ 13 ], [ 20 ] W n1,,n p+1 := S n1 a 1 S np+1 a p+1 s an n-dmensonal Wllmore submanfold n S n+p 1, where n = n 1 + +n p+1 and a are defned by n n a =, = 1,, p + 1. np Furthermore, W n1,,n p+1 s a mnmal submanfold n S n+p 1 f and only f t s Ensten wth n 1 = = n p+1 = n 1 p + 1, a =, = 1, 2,, p + 1. p + 1 When p = 1, we have the followng mportant example Example 4.3. [ 18 ], [ 13 ] Wllmore tor. n m m W m,n m := S m n Sn m, 1 m n 1, n are Wllmore hypersurfaces n S n+1 1. Accordng to [ 18 ], [ 13 ], we call them Wllmore tor. In [ 18 ] and [ 20 ], the second author proved the followng ntegral nequalty of Smons type for n-dmensonal compact Wllmore submanfolds n S n+p cf. [ 39 ] Theorem 4.1. [ 18 ], [ 20 ] Let M be a n-dmensonal n 2 compact Wllmore submanfold n a n + p-dmensonal unt sphere S n+p. Then we have ρ n n 2 1/p ρ2 dv In partcular, f M 0 ρ 2 n 2 1/p, 4.2 then ether ρ 2 0 and M s totally umblc, or ρ 2 n 2 1/p. In the latter case, ether p = 1 and M s a Wllmore torus W m,n m ; or n = 2, p = 2 and M s the Veronese surface.

17 291 For n = 2, the followng result was proved by L [ 19 ] also see L-Smon [ 21 ] Theorem 4.2. [ 19 ], [ 21 ] Let M be a compact Wllmore surface n an 2 + p-dmensonal unt sphere S 2+p. Then we have ρ ρ2 dv In partcular, f M 0 ρ 2 4 3, 4.4 then ether ρ 2 = 0 and M s totally umblc, or ρ 2 = 4/3. In the latter case, p = 2 and M s the Veronese surface. In the followng, we gve some examples of Wllmore hypersurface n S n+1 1, whch are also soparametrc but n general are nether mnmal nor Ensten. For an soparametrc hypersurface n S n+1 1, ρ 2 and the mean curvature H are both constant, by droppng the upper ndex α = n + 1 the Wllmore condton 3.23 becomes h h h 2 S H + n H 3 = ,, Example 4.4. Cartan s soparametrc hypersurfaces. For n = 3, 6, 12, 24, let x n : M n S n+1 1 be the n-dmensonal compact mnmal soparametrc hypersurface constructed by E. Cartan [ 7 ]. Each of these hypersurface has three dstnct prncpal curvature λ 1 = 3, λ 2 = 0, λ 3 = 3 wth the same multplctes. We can show by usng 4.5 see [ 18 ] that all of these are Wllmore hypersurfaces and they are not Ensten. Example 4.5. Nomzu s soparametrc hypersurfaces. Let S n+1 1 = { x 1, x 2,, x 2r+1, x 2r+2 R n+2 n+2 =1 x2 = 1 }, for n = 2 r 4. We defne a functon on S n+1 1 by F x = r+1 x x 2 2 r x 2 1 x =1 Accordng to Nomzu [ 31 ], { Mt n = x S n+1 1 } F x = cos 2 2 t, 0 < t < π 4, 4.7 =1

18 292 defnes an soparametrc famly of hypersurfaces n S n+1 1. The prncpal curvatures of the hypersurface Mt n for a fxed t can be computed: λ 1 = = λ r 1 = cot t; λ r = cot π 4 t ; 4.8 λ r+1 = = λ n 1 = cot π 2 t ; λ n = cot 3 π 4 t. For ths famly of soparametrc hypersurfaces, a straghtforward calculaton from 4.5 and 4.8 shows that Mt n s a Wllmore hypersurface n S n+1 1 f and only f t = t w := cot 1 n 2 + 5n + n 8n 2 +17n 34 n 2 + n + 2 n 4 + n 8n 2 +17n 34 n 2 We also note that from 4.8 we can see that Mt n s mnmal f and only f t = t m := cot 1 n+ 2 n 2. A further computaton shows that t w = t m := t 0 f and only f n = 4, then the correspondng four prncpal curvatures are gven by λ 1 = 2 1, λ 2 = 2 + 1, λ 3 = 2 1, λ 4 = Therefore Mt 4 0 s a mnmal, Wllmore and non-ensten hypersurface n S 5 1. We note that, when n = 4, example 4.5 s due to Cartan.. 5. Wllmore complex submanfolds n a complex space form Let M m 4c, g, J be a complex m-dmensonal complex space form of constant holomorphc sectonal curvature 4c wth assocated Kähler metrc g and complex structure J. Let {e A }={e 1,, e m ; e 1 =Je 1,, e m =Je m } be a local orthonormal bass for T M wth dual bass {ω A }. We mae use the notaton of e A = e A, δ A B = δ AB and the ndces range conventon A, B, C, D = 1,, m, 1,, m. Then the Remannan curvature tensor of M m 4c satsfes R ABCD = c δ AC δ BD δ AD δ BC δ A C δ B D δ A D δ B C + 2 δ A B δ C D. Assume x : M n M m 4c s an sometrc mmersed submanfold of real dmenson n. For p M, we have a drect sum decomposton T p M = Tp M Tp M. Recall that M s called a complex resp. totally real submanfold of M f J Tp M T p M resp. J T p M Tp M for each p M. A totally real submanfold M of M s called Lagrangan f m = n. We note that a complex submanfold s mnmal and has even dmenson.

19 293 For complex submanfolds n a complex space form, we can strengthen Proposton 4.1 by provng Theorem 5.1. Let x : M M m 4c be a complex submanfold wth constant scalar curvature. Then M s Wllmore. Proof. Snce M n s a complex submanfold, n = 2 r s even. We can choose {e A } such that, restrcted to M, {e 1,, e r, e 1 = Je 1,, e r = Je r } s a local orthonormal bass for T M. Durng the proof of Theorem 5.1, we further use the followng conventon on the range of ndces:,,, l = 1,, r, 1,, r ; a, b, c, d = 1,, r; α, β = r + 1,, m, r + 1,, m ; λ, µ = r + 1,, m. Then we have [ 48. p. 181 ] h α a b = hα ab = h α ab, hα a b = h α ab = hα ab. 5.2 Now t follows from the mnmalty of M n and Theorem 3.1. that M n s a Wllmore submanfold n M m 4c f and only f 0 = ρ n 2[ h β hβ hα + ] R βα h β 5.3,,,β,,β + { ρ n 2, hα + ρ n 2 hα + ρ n 2 h} α, α., From the Codazz equaton 2.11, the mnmalty of M n and 5.1, 5.2, a straghtforward calculaton gves h α = 0, R βα h β = 0,,,β,,,β h β hβ hα = 0, α. 5.4 Combnng 5.3 and 5.4, we have proved the followng: M m 4c s Wll- Proposton 5.1. A complex submanfold x : M n more f and only f for each α, ρ n 2, hα = 0., Note that n the present stuaton, 2.8 gves that ρ 2 = S =, R R = n n + 2 c R. 5.5 Theorem 5.1 then follows from 5.5 and Proposton 5.1. When n = 2, a complex submanfold s called a complex curve. Prevous proposton mples

20 294 Theorem 5.2. Every complex curve n a complex space form s Wllmore. Example 5.1. [ 48 ] Let CP n+1 be the n + 1-dmensonal complex proectve space wth constant holomorphc sectonal curvature 4. Then the n + 1-dmensonal complex quadrc {z 0, z 1,, z n+1 C n+2 z z z 2 n+1 = 0} defnes a compact complex hypersurface Q n n CP n+1. It s well nown that Q n s Kähler-Ensten and thus has constant scalar curvature. From Theorem 5.1, Q n s a Wllmore submanfold n CP n Wllmore Lagrangan submanfolds n a complex space form In ths secton, we wll consder the case that x : M n M m 4c s a totally real submanfold and also the Lagrangan submanfold see secton 5 for ther defnton. We wll use the followng conventon on the range of ndces n ths secton: A, B, C, D = 1,, m, 1,, m ;,,, l = 1, 2,, n. From the assumpton, we can choose {e A } such that, restrcted to M n, {e 1,, e n } s a local orthonormal bass for T M n. The Gauss and Codazz equaton now read R l = c δ δ l δ l δ + h α h α l h α l h α, 6.1 α h α = h α,,,, α. 6.2 We also have the relatons cf. [ 48 ] h = h = h,,,. 6.3 From 5.1 and 6.3, a drect calculaton gves: - for α { 1,, n } R βα h β = 4 n c Hα, H β Rβα = n + 3 c H α. 6.4,,β,β - for α { n + 1,, m; n + 1,, m } R βα h β = H β Rβα = n c H α. 6.5,,β,β

21 295 Applyng 6.1, 6.4 and 6.5 n Theorem 3.1, we get Theorem 6.1. A totally real submanfold x : M n M m 4c s Wllmore f and only f 1 For α {1,, n }, there hold 0 = ρ n 2[,,,β h β hβ hα,,β H β h β hα ρ 2 H α n 1 c H α] H α ρ n 2 + n 1 ρ n 2 H α + 2 n 1 ρ n 2 Hα, + ρ n 2, hα., 2 For α {n + 1,, m, n + 1,, m }, there hold 0 = ρ n 2[ h β hβ hα H β h β hα ρ 2 H α] 6.7,,,β,,β H α ρ n 2 + n 1 ρ n 2 H α + 2 n 1 ρ n 2 Hα, + ρ n 2, hα., We wll emphasze the mportant specal case, n = m, of Lagrangan submanfolds. We note that n recent years, due to ther bacgrounds n mathematcal physcs, specal Lagrangan submanfolds have been extensvely studed see [ 14 ], [ 32 ] and [ 40 ]. Theorem 6.2. A Lagrangan submanfold x : M n M n 4c s Wllmore f and only f 0 = ρ n 2[,,,β h β hβ hα,,β H β h β hα ρ 2 H α n 1 c H α] H α ρ n 2 + n 1 ρ n 2 H α + 2 n 1 ρ n 2 Hα, + ρ n 2, hα, for α., The followng result s an mmedately consequence of Theorem 6.1, whch can be proved by a smlar argument as n provng Corollary 3.1.

22 296 Proposton 6.1. A totally real surface x : M 2 M m 4c s Wllmore f and only f H α 2 H 2 H α + 3 c H α +,,β H β h β hα = 0, α = 1, 2 ; 6.9 H α 2 H 2 H α +,,β H β h β hα = 0, α {3,, m, 3,, m } Corollary 6.1. [ 29 ] Every totally real mnmal surface n a complex space form s Wllmore. Remar 6.1. Proposton 6.1 can be compared wth Proposton 5 n [ 29 ]. Counterpart of Proposton 4.1 holds for totally real submanfolds n complex space forms. Proposton 6.2. Every mnmal and Ensten totally real submanfold x : M n M m 4c s Wllmore. Proof. From 6.6 and 6.7, we only need to prove that ρ n 2 ρ n 2, hα = ,,,β h β hβ hα +, Snce M s mnmal and Ensten, the Gauss equaton 6.1 gves h β hβ = n 1 c δ R = n 1 c δ R n δ, 6.12,β ρ 2 = S = nn 1 c R = const, 6.13 whch mply that 6.11 holds. Thus we prove Proposton 6.2. Example 6.1. Let RP n 1 be the n-dmensonal real proectve space wth constant sectonal curvature 1. RP n 1 can be sometrcally mmersed nto CP n 4 as a totally geodescally Lagrangan submanfold. From Proposton 6.2 we now that RP n 1 s a compact Wllmore submanfold of CP n 4. Example 6.2. [ 32 ] The Clfford torus T n CP n 4. Consder the sometrc embeddng of n + 1-torus T n+1 : S 1 1 n + 1 S 1 1 n + 1 S 2n+1 1 C n+1, ths embeddng s Lagrangan n C n+1 and t s mnmal n S 2n+1 1. Snce the standard acton by S 1 on C n+1 restrcts to both the above torus T n+1

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