Simons type formulas for submanifolds with parallel mean curvature in product spaces and applications D OREL F ETCU XIV TH I NTERNATIONAL C ONFERENCE ON G EOMETRY , I NTEGRABILITY AND Q UANTIZATION June 8–13, 2012 Varna, Bulgaria
References D. Fetcu and H. Rosenberg, Surfaces with parallel mean curvature in S 3 × R and H 3 × R , Michigan Math. J., to appear, arXiv:math.DG/1103.6254v1 . D. Fetcu, C. Oniciuc, and H. Rosenberg, Biharmonic submanifolds with parallel mean curvature in S n × R , J. Geom. Anal., to appear, arXiv:math.DG/1109.6138v1 . D. Fetcu and H. Rosenberg, On complete submanifolds with parallel mean curvature in product spaces , Rev. Mat. Iberoam., to appear, arXiv:math.DG/1112.3452v1 .
Using Simons inequalities to study minimal, cmc and pmc submanifolds ◮ 1968 - J. Simons - a formula for the Laplacian of the second fundamental form of a submanifold in a Riemannian manifold
Using Simons inequalities to study minimal, cmc and pmc submanifolds ◮ 1968 - J. Simons - a formula for the Laplacian of the second fundamental form of a submanifold in a Riemannian manifold - for a minimal hypersurface Σ m in S m + 1 this formula is 1 2 ∆ | A | 2 = | ∇ A | 2 + | A | 2 ( m −| A | 2 ) ≥ | A | 2 ( m −| A | 2 ) where ∇ and A are defined by ¯ ∇ X V = − A V X + ∇ ⊥ ¯ ∇ X Y = ∇ X Y + σ ( X , Y ) and X V
- for a minimal submanifold with arbitrary codimension in S n : Theorem (Simons - 1968) Let Σ m be a closed minimal submanifold in S n . Then | A | 2 − m ( n − m ) � � � | A | 2 ≥ 0 . 2 n − 2 m − 1 Σ m
- for a minimal submanifold with arbitrary codimension in S n : Theorem (Simons - 1968) Let Σ m be a closed minimal submanifold in S n . Then | A | 2 − m ( n − m ) � � � | A | 2 ≥ 0 . 2 n − 2 m − 1 Σ m Corollary Let Σ m be a closed minimal submanifold in S n with m ( n − m ) | A | 2 ≤ 2 n − 2 m − 1 . Then, either Σ m is totally geodesic or | A | 2 = m ( n − m ) 2 n − 2 m − 1 .
Definition If the mean curvature vector field H = 1 m trace σ of a submanifold Σ m in a Riemannian manifold is parallel in the normal bundle, i.e. ∇ ⊥ H = 0 , then Σ m is called a pmc submanifold. If | H | = constant , then Σ m is a cmc submanifold.
Definition If the mean curvature vector field H = 1 m trace σ of a submanifold Σ m in a Riemannian manifold is parallel in the normal bundle, i.e. ∇ ⊥ H = 0 , then Σ m is called a pmc submanifold. If | H | = constant , then Σ m is a cmc submanifold. ◮ 1969 - K. Nomizu, B. Smyth; 1973 - B. Smyth - Simons type formula for cmc hypersurfaces and, in general, pmc submanifolds in a space form ◮ 1971 - J. Erbacher - Simons type formula for pmc submanifolds in a space form: | ∇ ∗ A | 2 + cm {| A | 2 − m | H | 2 } 1 2 ∆ | A | 2 = + ∑ n + 1 α , β = m + 1 { ( trace A β )( trace ( A 2 α A β )) + trace [ A α , A β ] 2 − ( trace ( A α A β )) 2 } ,
◮ 1977 - S.-Y. Cheng, S.-T. Yau - a general Simons type equation for operators S , acting on a submanifold of a Riemannian manifold and satisfying ( ∇ X S ) Y = ( ∇ Y S ) X ◮ 1970 - S.-S. Chern, M. do Carmo, S. Kobayashi; 1994 - H. Alencar, M. do Carmo - gap theorems for minimal hypersurfaces and cmc hypersurfaces, respectively, in S n ( c ) ◮ 1994 - W. Santos - a gap theorem for pmc submanifolds in S n ( c ) ◮ other studies on pmc submanifolds in space forms: - 1984, 1993, 2005, 2010, 2011 - H.-W. Xu et al. - 2001 - Q. M. Cheng, K. Nonaka - 2009 - K. Araújo, K. Tenenblat ◮ 2010 - M. Batista - Simons type formulas for cmc surfaces in M 2 ( c ) × R
A Simons type formula for submanifolds in M n ( c ) × R Theorem (F., Oniciuc, Rosenberg - 2011) Let Σ m be a submanifold of M n ( c ) × R , with mean curvature vector field H and shape operator A . If V is a normal vector field, parallel in the normal bundle, with trace A V = constant , then | ∇ A V | 2 + c { ( m −| T | 2 ) | A V | 2 − 2 m | A V T | 2 1 2 ∆ | A V | 2 = + 3 ( trace A V ) � A V T , T �− m ( trace A V ) � H , N �� V , N � + m ( trace ( A N A V )) � V , N �− ( trace A V ) 2 } + ∑ n + 1 α = m + 1 { ( trace A α )( trace ( A 2 V A α )) − ( trace ( A V A α )) 2 } , where { E α } n + 1 α = m + 1 is a local orthonormal frame field in the normal bundle, and T and N are the tangent and normal part, respectively, of the unit vector ξ tangent to R .
Sketch of the proof. 2 ∆ | A V | 2 = | ∇ A V | 2 + � trace ∇ 2 A V , A V � ◮ Weitzenböck formula: 1
Sketch of the proof. 2 ∆ | A V | 2 = | ∇ A V | 2 + � trace ∇ 2 A V , A V � ◮ Weitzenböck formula: 1 ◮ C ( X , Y ) = ( ∇ 2 A V )( X , Y ) = ∇ X ( ∇ Y A V ) − ∇ ∇ X Y A V ◮ consider an orthonormal basis { e i } m i = 1 in T p Σ m , p ∈ Σ m , extend e i to vector fields E i in a neighborhood of p such that { E i } is a geodesic frame field around p , and denote X = E k m ( trace ∇ 2 A V ) X = ∑ C ( E i , E i ) X . i = 1
◮ Codazzi equation of Σ m : ( ∇ X A V ) Y = ( ∇ Y A V ) X + c � V , N � ( � Y , T � X −� X , T � Y )
◮ Codazzi equation of Σ m : ( ∇ X A V ) Y = ( ∇ Y A V ) X + c � V , N � ( � Y , T � X −� X , T � Y ) ◮ Ricci commutation formula: C ( X , Y ) = C ( Y , X )+[ R ( X , Y ) , A V ]
◮ Codazzi equation of Σ m : ( ∇ X A V ) Y = ( ∇ Y A V ) X + c � V , N � ( � Y , T � X −� X , T � Y ) ◮ Ricci commutation formula: C ( X , Y ) = C ( Y , X )+[ R ( X , Y ) , A V ] ◮ Codazzi equation + Ricci formula ⇒ C ( E i , E i ) X = ∇ X (( ∇ E i A V ) E i )+[ R ( E i , X ) , A V ] E i + c � A V E i , T � ( � E i , T � X −� X , T � E i ) − c � V , N � ( � A N E i , E i � X −� A N X , E i � E i )
◮ Codazzi equation of Σ m : ( ∇ X A V ) Y = ( ∇ Y A V ) X + c � V , N � ( � Y , T � X −� X , T � Y ) ◮ Ricci commutation formula: C ( X , Y ) = C ( Y , X )+[ R ( X , Y ) , A V ] ◮ Codazzi equation + Ricci formula ⇒ C ( E i , E i ) X = ∇ X (( ∇ E i A V ) E i )+[ R ( E i , X ) , A V ] E i + c � A V E i , T � ( � E i , T � X −� X , T � E i ) − c � V , N � ( � A N E i , E i � X −� A N X , E i � E i ) ◮ ∇ E i A V is symmetric + Codazzi eq. + trace A V = constant ⇒ ∑ m i = 1 ( ∇ E i A V ) E i = c ( m − 1 ) � V , N � T ◮ R ( X , Y ) Z = c {� Y , Z � X −� X , Z � Y −� Y , T �� Z , T � X + � X , T �� Z , T � Y + � X , Z �� Y , T � T −� Y , Z �� X , T � T } + ∑ n + 1 α = m + 1 {� A α Y , Z � A α X −� A α X , Z � A α Y } ,
◮ Codazzi equation of Σ m : ( ∇ X A V ) Y = ( ∇ Y A V ) X + c � V , N � ( � Y , T � X −� X , T � Y ) ◮ Ricci commutation formula: C ( X , Y ) = C ( Y , X )+[ R ( X , Y ) , A V ] ◮ Codazzi equation + Ricci formula ⇒ C ( E i , E i ) X = ∇ X (( ∇ E i A V ) E i )+[ R ( E i , X ) , A V ] E i + c � A V E i , T � ( � E i , T � X −� X , T � E i ) − c � V , N � ( � A N E i , E i � X −� A N X , E i � E i ) ◮ ∇ E i A V is symmetric + Codazzi eq. + trace A V = constant ⇒ ∑ m i = 1 ( ∇ E i A V ) E i = c ( m − 1 ) � V , N � T ◮ R ( X , Y ) Z = c {� Y , Z � X −� X , Z � Y −� Y , T �� Z , T � X + � X , T �� Z , T � Y + � X , Z �� Y , T � T −� Y , Z �� X , T � T } + ∑ n + 1 α = m + 1 {� A α Y , Z � A α X −� A α X , Z � A α Y } , ◮ Ricci eq. � R ⊥ ( X , Y ) V , U � = � [ A V , A U ] X , Y � + � ¯ R ( X , Y ) V , U � ⇒ [ A V , A U ] = 0 , ∀ U ∈ N Σ m
pmc surfaces in M 3 ( c ) × R • Let Σ 2 be a non-minimal pmc surface in M 3 ( c ) × R • Consider the orthonormal frame field { E 3 = H | H | , E 4 } in the normal bundle ⇒ E 4 = parallel • φ 3 = A 3 −| H | I and φ 4 = A 4 • φ ( X , Y ) = σ ( X , Y ) −� X , Y � H = � φ 3 X , Y � E 3 + � φ 4 X , Y � E 4 • | φ | 2 = | φ 3 | 2 + | φ 4 | 2 = | σ | 2 − 2 | H | 2
pmc surfaces in M 3 ( c ) × R • Let Σ 2 be a non-minimal pmc surface in M 3 ( c ) × R • Consider the orthonormal frame field { E 3 = H | H | , E 4 } in the normal bundle ⇒ E 4 = parallel • φ 3 = A 3 −| H | I and φ 4 = A 4 • φ ( X , Y ) = σ ( X , Y ) −� X , Y � H = � φ 3 X , Y � E 3 + � φ 4 X , Y � E 4 • | φ | 2 = | φ 3 | 2 + | φ 4 | 2 = | σ | 2 − 2 | H | 2 Proposition (F., Rosenberg - 2011) If Σ 2 is an immersed pmc surface in M n ( c ) × R , then 1 2 ∆ | T | 2 = | A N | 2 − 1 2 | T | 2 | φ | 2 − 2 � φ ( T , T ) , H � + c | T | 2 ( 1 −| T | 2 ) −| T | 2 | H | 2 .
Theorem (F., Rosenberg - 2011) Let Σ 2 be an immersed pmc 2 -sphere in M n ( c ) × R , such that 1. | T | 2 = 0 or | T | 2 ≥ 2 3 and | σ | 2 ≤ c ( 2 − 3 | T | 2 ) , if c < 0 ; 2. | T | 2 ≤ 2 3 and | σ | 2 ≤ c ( 2 − 3 | T | 2 ) , if c > 0 . Then, Σ 2 is either a minimal surface in a totally umbilical hypersurface of M n ( c ) or a standard sphere in M 3 ( c ) .
Theorem (F., Rosenberg - 2011) Let Σ 2 be an immersed pmc 2 -sphere in M n ( c ) × R , such that 1. | T | 2 = 0 or | T | 2 ≥ 2 3 and | σ | 2 ≤ c ( 2 − 3 | T | 2 ) , if c < 0 ; 2. | T | 2 ≤ 2 3 and | σ | 2 ≤ c ( 2 − 3 | T | 2 ) , if c > 0 . Then, Σ 2 is either a minimal surface in a totally umbilical hypersurface of M n ( c ) or a standard sphere in M 3 ( c ) . Proof. ◮ Q ( X , Y ) = 2 � σ ( X , Y ) , H �− c � X , ξ �� Y , ξ � ⇒ Q ( 2 , 0 ) = holomorphic
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