A CLASS OF VARIATIONAL PROBLEMS FOR SUBMANIFOLDS IN A SPACE FORM

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1 Houston Journal of athematcs c 2009 Unversty of Houston Volume 35, No. 2, 2009 A CLASS OF VARIATIONAL PROBLES FOR SUBANIFOLDS IN A SPACE FOR LAN WU Communcated by n Ru Abstract. Let R n+p (c) be an (n + p)-dmensonal space form of constant sectonal curvature c, x : R n+p (c) an n-dmensonal submanfold n R n+p (c). For wth 1, x : R n+p (c) s called an extremal submanfold f t s a crtcal submanfold to the followng non-negatve functonal: F (x) := (S nh 2 ) dv, where S = (h )2 s the square of the length of the second fundamental,, form, and H s the mean curvature of. We note that when = n 2, the above functonal s Wllmore functonal. In ths paper, we prove an ntegral nequalty of Smons type for n-dmensonal compact extremal submanfolds n the (n + p)-dmensonal unt sphere S n+p and gve a characterzaton of Clfford torus and Veronese surface by use of our ntegral nequalty. Case = 1 and case = n were studed by Guo-L [4] and L [12] Introducton Let R n+p (c) be an (n+p)-dmensonal smply-connected Remannan manfold of constant sectonal curvature c, and we call R n+p (c) a space form. Let be an n-dmensonal submanfold of R n+p (c). When c = 0, R n+p (c) s an (n + p)- dmensonal Eucldean space; when c = 1, R n+p (c) s an (n + p)-dmensonal unt sphere S n+p ; when c = 1, R n+p (c) s an (n + p)-dmensonal hyperbolc space H n+p ( 1). If h denote the components of the second fundamental form of, S denotes the square of the length of the second fundamental form, H denotes 2000 athematcs Subect Classfcaton. Prmary 53C42, Secondary 53A10. Key words and phrases. Euler-Lagrangan equaton, ntegral nequalty, Clfford torus, Veronese surface. 435

2 436 LAN WU the mean curvature vector and H denotes the mean curvature of, then we have S = (h ) 2, H = H e, H = 1 h n, H = H,,, where e (n + 1 n + p) are orthonormal normal vector felds of n R n+p (c). We defne the followng non-negatve functon on ρ 2 = S nh 2, (1.1) whch vanshes exactly at the umblcal ponts of. For a fxed number wth 1, the followng functonal s a non-negatve functonal F (x) = ρ 2 dv = (S nh 2 ) dv, (1.2) whch vanshes f and only f s a totally umblcal submanfold, so the functonal F (x) measures how much x() devates from a totally umblcal submanfold. Remar 1.1. When n = 2, = 1, F (x) reduces to the well-nown Wllmore functonal W (x) and ts crtcal ponts are called Wllmore surfaces. Wllmore conecture can be stated that W (x) 4π 2 holds for all mmersed tor x : S 3, where S 3 s the 3-dmensonal unt sphere. The conecture was approached by Wllmore [26], L-Yau [15] and many others(see [25], [27] and references there). The Wllmore surfaces n a sphere were studed by Thomsen [22], Bryant [1], Pnall [18], Wener [24], ontel [16], L [8], L-Smon [13], L-Vrancen [14] and many others. When n 2, = n 2, F (x) s also called Wllmore functonal and ts crtcal ponts are called Wllmore submanfolds (see [5], [11], [12], [17], [23]). In ths paper, we frst calculate the Euler-Lagrangan equaton of F (x) gven by (1.2) Theorem 1.1. Let x : R n+p (c) be an n-dmensonal submanfold n an (n + p)-dmensonal space form R n+p (c). Then for 1, s an extremal submanfold of F (x) f and only f for n + 1 n + p the dentty (ρ 2 2 ) H + 2(n 1) (ρ 2 2 ) H, +ρ 2 2 [ h hβ l hβ l H β h β h n 2 ρ2 H ] β,,,l β,, +,(ρ 2 2 ), h + (n 1)ρ2 2 H = 0, (1.3) holds, where H = H, (see (2.14)-(2.16)).

3 A CLASS OF VARIATIONAL PROBLES 437 We call x : R n+p (c) an extremal submanfold f t satsfes Euler- Lagrangan equaton (1.3). Remar 1.2. When = n 2, Theorem 1.1 was proved by Pedt-Wllmore [17], [11], L [12] and Wang [23] (also see Hu-L [6]); when = 1, Theorem 1.1 was proved by Guo-L [4]. In order to state our man result, we frst gve the followng mportant examples Example 1 (see [7] or [3]). Clfford tor ) C m,m = S m ( 1 2 S m ( 1 2 ), n = 2m (1.4) s an extremal hypersurface of F (x) n the (n + 1)-dmensonal unt sphere S n+1. In fact, the prncpal curvatures 1,, n of C m,m are We have from (1.5) 1 = = m = 1, m+1 = = n = 1, n = 2m. (1.5) H = 0, S = n, 3 = 0. (1.6) Thus we easly chec that (1.3) holds,.e., C m,m s an extremal hypersurface. In partcular, we note that ρ 2 of C m,m satsfes ρ 2 = n. (1.7) Example 2 (see [3] or [8]). Veronese surface. Let (x, y, z) be the natural coordnate system n R 3 and u = (u 1, u 2, u 3, u 4, u 5 ) be the natural coordnate system n R 5. We consder the mappng defned by u 1 = 1 yz, u 2 = 1 xz, u 3 = 1 xy, u 4 = (x2 y 2 ), u 5 = 1 6 (x2 + y 2 2z 2 ), where x 2 +y 2 +z 2 = 3. Ths defnes an sometrc mmerson of S 2 ( 3) nto S 4 (1). Two ponts (x, y, z) and ( x, y, z) of S 2 ( 3) are mapped nto the same pont of S 4. Ths real proectve plane mbedded n S 4 s called the Veronese surface. We now that Veronese surface s a mnmal surface wth ρ 2 = 4 3 n S4 (see [3] or [8], [13]), thus t satsfes (1.3),.e., t s an extremal surface. We also note that ρ 2 of the Veronese surface satsfes ρ 2 = 4 3. (1.8)

4 438 LAN WU Example 3. If x 0 : S n+p s an n-dmensonal (n 2) Ensten and mnmal submanfold, then t s an extremal submanfold of F (x). It can be checed drectly that n ths case (1.3) s satsfed by use of Gauss equaton (2.5) and mnmal condton H = 0. When = n 2, the functonal F (x) s Wllmore functonal. An extremal submanfold s called a Wllmore submanfold. In [12], the author proved the followng ntegral nequalty (also see [11], [8], [13]) Theorem 1.2 ([12]). Let be an n-dmensonal (n 2) compact Wllmore submanfold n the (n + p)-dmensonal unt sphere S n+p. Then we have ( ) n ρ n 2 1/p ρ2 dv 0. (1.9) In partcular, f 0 ρ 2 n 2 1/p, (1.10) then ether ρ 2 0 and s totally umblcal, or ρ 2 n 2 1/p. In the latter case, ether p = 1 and s a Wllmore torus W m,n m defned n [11], [5] or [12]; or n = 2, p = 2 and s the Veronese surface. When = 1, Guo-L studed the extremal submanfolds of the functonal F (x), and they proved the followng ntegral nequalty (see [4]) Theorem 1.3 ([4]). Let be an n-dmensonal (n 2) compact extremal submanfold of functonal F 1 (x) n the (n + p)-dmensonal unt sphere S n+p. Then we have ( ) n ρ 2 2 1/p ρ2 dv 0. (1.11) In partcular, f 0 ρ 2 n 2 1/p, (1.12) then ether ρ 2 0 and s totally umblcal, or ρ 2 n 2 1/p. In the latter case, ether p = 1, n = 2m and s a Clfford torus C m,m defned by (1.4); or n = 2, p = 2 and s the Veronese surface. In ths paper we prove the followng ntegral nequalty of Smons type for compact extremal submanfolds of functonal F (x) n S n+p. Theorem 1.4. Let 1 < n 2, be an n-dmensonal (n > 2) compact extremal submanfold of functonal F (x) n the (n+p)-dmensonal unt sphere S n+p. Then we have ( ) n ρ 2 2 1/p ρ2 dv 0. (1.13)

5 A CLASS OF VARIATIONAL PROBLES 439 In partcular, f 0 ρ 2 n 2 1/p, (1.14) then ether ρ 2 0 and s totally umblcal, or ρ 2 n 2 1/p. In the latter case, ether p = 1, n = 2m and s a Clfford torus C m,m defned by (1.4); or n = 2, p = 2 and s the Veronese surface. Remar 1.3. When = 1, Theorem 1.4 reduces to Theorem Prelmnares Let x : R n+p (c) be an n-dmensonal submanfold n an (n + p)-dmensonal space form R n+p (c). Let {e 1,, e n } be a local orthonormal bass of wth respect to the nduced metrc, and {θ 1,, θ n } be ther dual forms. Let e n+1,, e n+p be the local unt orthonormal normal vector felds. In ths paper we mae the followng conventon on the range of ndces: 1,, n; n + 1, β, γ n + p; 1 A, B, C n + p. Then we have the structure equatons dx = θ e, (2.1) de = θ e +, h θ e θ x, (2.2) The Gauss equatons are de =, h θ e + β θ β e β. (2.3) R l = (δ δ l δ l δ )c + (h h l h lh ), (2.4) R = (n 1)cδ + n H h, h h, (2.5) n(n 1)R = n(n 1)c + n 2 H 2 S, (2.6) where R s the normalzed scalar curvature of and S =,, (h )2 s the norm square of the second fundamental form, H = H e = 1 n ( h )e s the mean curvature vector and H = H s the mean curvature of. The Codazz equatons are (c.f. [9], [10]) h = h, (2.7)

6 440 LAN WU where the covarant dervatve of h s defned by h θ = dh + h θ + h θ + β h β θ β. (2.8) The second covarant dervatve of h s defned by h lθ l = dh + h lθ l + h lθ l + h lθ l + l l l l β h β θ β. (2.9) By exteror dfferentaton of (2.8), we have the followng Rcc denttes h l h l = m h mr ml + m h mr ml + β h β R βl. (2.10) The Rcc equatons are R β = (h h β hβ h ). (2.11) We defne the followng non-negatve functon on ρ 2 = S nh 2, (2.12) whch vanshes exactly at the umblcal ponts of. For 1, the followng non-negatve functonal F (x) = ρ 2 dv = (S nh 2 ) dv, (2.13) vanshes f and only f s a totally umblcal submanfold, so the functonal F (x) measures how much x() devates from totally a umblcal submanfold. We defne the frst, second covarant dervatves and the Laplacan of the mean curvature vector feld H = H e n the normal bundle N() as follows H, θ = dh + β H,θ = dh, + H,θ + β H β θ β, (2.14) H β, θ β, (2.15) H = H,, H = 1 h n. (2.16)

7 A CLASS OF VARIATIONAL PROBLES Proof of Theorem 1.1 Let x 0 : R n+p (c) be an n-dmensonal compact submanfold of an (n+p)- dmensonal space form R n+p (c). Now we calculate the frst varaton of the functonal F (x) Let x : R R n+p (c) be a smooth varaton of x 0 such that x(, 0) = x 0. Along x : R R n+p (c), we choose a local orthonormal bass {e A } for T R n+p (c) wth dual bass {ω A }, such that {e (, t)} forms a local orthonormal bass for x t : {t} R n+p (c). Snce T ( R) = T T R, the pullbac of {ω A } and {ω AB } on R n+p (c) through x : R R n+p (c) have the decomposton x ω = a dt, x ω = θ + a dt, (3.1) x ω = θ + a dt, x ω = θ + a dt, x ω β = θ β + a β dt, (3.2) where {a, a, a, a, a β } are local functons on R wth a = a, a β = a β and V = d x t = a dx 0 (e ) + a e, (3.3) dt t=0 s the varaton vector feld of x t : R n+p (c). We note that the one forms {θ, θ, θ, θ β } are defned on {t}, and for t = 0, they reduce to the forms wth the same notaton on. We denote by d the dfferental operator on T, then we have d = d +dt t on T ( R). Lemma 3.1. Under the above notatons, we have θ t = (a, + a )θ, h a θ, (3.4) a = a, + h a, (3.5) θ t = ( a, + a h β a β h β + ca δ )θ, (3.6) where h and the covarant dervatves a,, a, and a, are defned on {t} by θ = h θ, (3.7) a, θ = d a + a θ, (3.8)

8 442 LAN WU a, θ = d a + β a β θ β, (3.9) a, θ = d a + a θ + β a β θ β. (3.10) Proof. These can be checed by drect calculatons. In fact, substtutng (3.1) and (3.2) nto the followng equatons, respectvely d(x ω ) = x (dω ) = x ( ω ω + ω ω ), d(x ω ) = x (dω ) = x ( ω ω + β d(x ω ) = x (dω ) = x ( ω ω + β ω β ω β ), ω β ω β cω ω ), and comparng the terms n T dt for each equaton on the both sdes, we can get (3.4), (3.5) and (3.6), respectvely. Lemma 3.2. h t = a, + ( ) a h + a h + h a + β a βh β + cδ a +,β h hβ a β. (3.11) Remar 3.1. Lemma 3.2 was proved by Cao-L [2] (see Lemma 5.2 of [2]). We note that Lemma 3.2 was proved by Hu-L n [6] for any n-dmensonal compact submanfolds n an (n + p)-dmensonal Remannan manfold N n+p. Some nterestng results about related varatonal problems for hypersurfaces n space forms can be found n Relly [19], Rosenberg [20]. Proof. Dfferentatng (3.7) wth respect to t and usng (3.4), (3.6) we get h t = a, + a h a β h β + ca δ β ) (h a, + h a + h h β a β.,β

9 A CLASS OF VARIATIONAL PROBLES 443 Covarant dfferentatng (3.5) over {t} and usng the Codazz equaton (2.7) for x t : R n+p (c), we get a, = a, + ( ) a, h + a h ( ) a, h + a h. = a, + Combnng the above two equatons, we prove Lemma 3.2. Settng = n (3.11) and mang summaton over wth usng, a h = 0, we get H t = 1 n a a + H,a + β a β H β + 1 h n h β a β + ca. (3.12),,β From (3.11) and the followng facts S = (h ) 2, a β h β h = 0,,β,,, a h h = 0, we obtan 1 S 2 t = h a, S, a + nch a + From (3.12) and,β a βh H β = 0, we obtan,,,,β h h h β a β. (3.13) n H 2 = H a + n (H 2 ), a + H h 2 t 2 h β a β + nc H a.,β (3.14) Let 1, for x t : R n+p (c), we consder the functonal F (x t ) = ρ 2 dv = (S nh 2 ) θ 1 θ n. (3.15) From (3.4), we have t (θ 1 θ n ) = θ 1 θ t θ n = ( a, + a h a ) θ1 θ n = ( a, n ) H a θ1 θ n. (3.16)

10 444 LAN WU Dfferentatng (3.15) wth respect to t, we get by use of (3.13), (3.14) and (3.16) F (x t) t = ρ2 ρ2 2 t θ 1 θ n + ρn t (θ 1 θ n ) = { [ ρ2 2 2 h a, 2 H a + (ρ2 ), a + 1 ] ρ2 a, +2 ( h β hβ l h l H β h β h n 2 ρ2 H )a }dv.,,l,β,,β Notng that ρ 2 2 (ρ 2 ), a + ρ 2 a, = ( ρ 2 a ) (3.17),, (3.18) and s compact (wthout boundary), and h = nh,, h = n H, (3.19), t follows from (3.17), (3.18) and Green s formula that F (x t) t {ρ2 2[,,l,β hβ hβ l h l,,β Hβ h β h (ρ 2 2 ), h + 2(n 1) (ρ 2 2 ), H, = 2 n 2 ρ2 H ] +, (ρ 2 2 ) H + (n 1)ρ 2 2 H }a dv (3.20) From (3.3) and (3.20) wth restrcton to t = 0, we have proved Theorem The Lemmas For a submanfold x : R n+p (c), we defne the followng two operators : C () Γ(T ) and : Γ(T ) C () f = (nh δ h )f, e, (4.1) ξ = (nh δ h )ξ,, (4.2) where f, are the components of the second dervatve of f, ξ = Γ(T ) and ξ, are the components of the second dervatve of ξ. ξ e

11 A CLASS OF VARIATIONAL PROBLES 445 Lemma 4.1. Let be compact, then operators and are adont,.e. f ξdv = where <, > s the nner product n Γ(T ). In partcular, we have < ξ, f > dv, (4.3) whch s equvalent to ρ 2 2 Hdv = < H, (ρ 2 2 ) > dv, (4.4) n (ρ 2 2 ), h H dv = n 2 H2 (ρ 2 2 )dv + n 2 ρ2 2 H H dv n (4.5) ρ2 2 H, h dv. Proof. Let ξ = ξ e, we have the followng calculaton: f ξdv = f (nh δ h )ξ, = [f(nh δ h )ξ, ],dv f (nh δ h )ξ, dv = [f (nh δ h )ξ ], dv + f, (nh δ h )ξ dv = < ξ, f > dv, where we used Green s formula and f, = f,. In partcular, we get (4.4) by choosng f = ρ 2 2 and ξ = H = H e n (4.3). Lemma 4.2 (see Lemma 4.2 of [12]). Let be an n-dmensonal (n 2) submanfold n R n+p (c), then we have h 2 3n2 n + 2 H 2, (4.6) where h 2 =,,,(h )2, H 2 = (H, )2, H, s defned by (2.14).,

12 446 LAN WU Lemma 4.3. Let x : R n+p (c) be an n-dmensonal submanfold n R n+p (c). Then for (ρ2 ) ρ 2 2[ n(n 1) H 2 + +n,β,,,m,,, (h h ) H β h β m h h m +ncρ2 + nh 2 ρ 2 (2 1 p )ρ4 1 2 (nh2 ) ]. Proof. It can be checed drectly that (4.7) 1 2 ρ2 = 2( 1)ρ 2 2 ρ ρ2 2 ρ ρ2 2 ρ 2. (4.8) Defne tensors h = h H δ, (4.9) σ β =, h h β, σ β =, h h β. (4.10) From Lemma 4.5 of [12], we have (ust note that n [12] t s case c = 1) 1 2 ρ2 h 2 n 2 H 2 +,,, (h h ) + n +ncρ 2 + nh 2 ρ 2 (2 1 p )ρ4 1 2 (nh2 ).,β,,,m Puttng (4.11) nto (4.8), we get (4.7) by use of Lemma 4.2. From (4.9) and (4.10), we have by a drect calculaton h = 0, σ β = σ β nh H β, ρ 2 = H β h β m h h m (4.11) σ = S nh 2, (4.12) β,,,l h h β l hβ l = β,,,l h h β l h β l + 2 β,, H β hβ h + H ρ 2 + nh H 2. (4.13) Puttng (4.12) and (4.13) nto (1.3), we can get

13 A CLASS OF VARIATIONAL PROBLES 447 Lemma 4.4. Let x : R n+p (c) be an n-dmensonal extremal submanfold of functonal F (x) n R n+p (c). Then we have nρ 2 2 H h β h β l h l,,l,,β = nρ 2 2 σ β H H β + ( n2 2 n)h2 ρ 2,β n(n 1) (ρ 2 2 ) (H 2 ) n (ρ 2 2 ), h H n(n 1)ρ 2 2 H H + n (ρ 2 2 )H 2. (4.14) 5. The Proof of Theorem 1.4 Integratng (4.7) over and puttng (4.14) nto t, we have 0 { n(n 1)ρ2 2 H 2 + ρ 2 2 (h h ),,, +ρ 2 [nc (2 1 p )ρ2 ] + [ n2 2 H2 ρ 2 nρ 2 2,β H H β σ β ] n(n 1)ρ 2 2 H H + n (ρ 2 2 )H 2 n(n 1) (ρ 2 2 ) (H 2 ) n (ρ 2 2 ), h H n 2 ρ2 2 (H 2 )}dv. (5.1) Snce the (p p)-matrx ( σ β ) s symmetrc, t can be assumed to be dagonalzed for a sutable choce of e n+1,, e n+p. We set then we have,β H H β σ β = σ β = σ δ β, (5.2) (H ) 2 σ Puttng (4.5) nto (5.1), we get by use of (5.3) 0 { n(n 1)ρ2 2 H 2 + ρ 2 2 (H ) 2 σ β = H 2 ρ 2. (5.3),,, β (h h ) +ρ 2 [nc (2 1 p )ρ2 ] + ( n2 2 n)h2 ρ 2 n(n 1) (ρ 2 2 ) (H 2 ) +nρ 2 2 H H n 2 ρ2 2 (H 2 ) n(n 1) (ρ 2 2 )H 2 nρ 2 2 H, h }dv. (5.4) By use of Green s formula and the assumpton 1 < n 2, we can get from (5.4)

14 448 LAN WU 0 ρ2 [nc (2 1 p )ρ2 + ( n2 2 n)h2 ]dv ρ2 [(nc (2 1 p )ρ2 ]dv. (5.5) Choosng c = 1 n (5.5), we have obtaned the ntegral nequalty (1.13) n Theorem 1.4. If (1.14) holds, then we conclude from (5.5) that ether ρ 2 0, or ρ 2 n/(2 1 p ). In the former case, we now that S nh2,.e. s totally umblcal; n the latter case,.e., ρ 2 =,,( h ) 2 n/(2 1 ), (5.6) p (5.5) becomes an equalty, and we conclude that ether H = 0 f n 3, or n = 2. If n 3 and H = 0, from the an Theorem of Chern-Do Carmo-Kobayash [3] we get p = 1 and ( ) ( ) m n m = S m S n m. (5.7) n n Combnng (5.7) wth (1.3), we conclude that n = 2m and = C m,m, thus we prove Theorem 1.4. If n = 2, our Theorem 1.3 comes from Theorem 3 of [8] (also see [13]). We complete the proof of Theorem 1.4. Remar 5.1. From the proof of Theorem 1.4, n fact we prove the followng Theorem 5.1 Theorem 5.1. Let 1, and be an n-dmensonal (n 2) compact extremal submanfold of functonal F (x) n an (n + p)-dmensonal space form R n+p (c). Then we have In partcular, f 0 0 ρ 2 ρ 2 [nc (2 1 p )ρ2 + ( n2 2 n)h2 ]dv. (5.8) 1 2 1/p (nc + n( n 2 1)H2 ), (5.9) then ether ρ 2 0 and s totally umblcal, or ρ /p (nc + n( n 2 1)H2 ). In the latter case, we have c > 0, ether p = 1, s S m (a) S n m (b); or n = 2, p = 2 and s the Veronese surface. Remar 5.2. Theorem 1.2-Theorem 1.4 follow from the above Theorem 5.1 n case c = 1.

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16 450 LAN WU [26] Wllmore T.J., Notes on embedded surfaces, Ann. Stnt. Unv. Al. I. Cuza, Ias, Sect. I a at. (N.S.) 11B (1965), [27] Wllmore T.J., Remannan Geometry, Oxford Scence Publcatons, Clarendon Press, Oxford, Receved September 20, 2007 Department of athematcs, School of Informaton, Renmn Unversty of Chna, , Beng, People s Republc of Chna E-mal address: wulan@ruc.edu.cn