Spacelike surfaces through the lightcone of 4-dimensional - PowerPoint PPT Presentation

Introduction Examples and generalities Global results Spacelike surfaces through the lightcone of 4-dimensional Lorentz-Minkowski spacetime Francisco J. Palomo Departamento de Matem´ atica Aplicada PADGE 2012 Leuven (Belgium), August 27-30 Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results F.J. Palomo and A. Romero, On spacelike surfaces in 4-dimensional Lorentz-Minkowski spacetime through a lightcone, P. Roy. Soc. Edinb. A Mat. (to appear). F.J. Palomo, F. Rodr´ ıguez and A. Romero, Compact spacelike surfaces in the lightcone with non-degenerate lightlike Gauss map, (submitted) Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results n � L n +1 = ( R n +1 , � , � ) , � , � = − ( dx 0 ) 2 + ( dx i ) 2 . i =1 Λ n = { v ∈ L n +1 : � v , v � = 0 , v 0 > 0 } Assume ψ : M n → Λ n +1 is an inmersion such that the position vector field is not tangent to M . Then M inherits a Riemannian metric from L n +2 . What kind of n -dimensional Riemannian manifolds can be isometrically inmersed in Λ n +1 ? Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results A Riemannian manifold M n is said to be conformally flat if every point of the manifold lies in a local coordinate system ( x 1 , ..., x n ) with respect to which the Riemannian metric takes the form, ds 2 = ω 2 � ( dx i ) 2 � � n , i =1 for some non-vanishing function ω . Brinkmann, 1923 Let M n be a simply connected Riemannian manifold with n ≥ 3. Then M n is conformally flat if and only if M n can be isometrically inmersed into Λ n +1 ⊂ L n +2 . a a A. Asperti and M. Dajczer, Conformally flat Riemannian manifolds as hypersurfaces of the light cone, Can. Math. Bull. , 32 (1989), 281–285 What is the situation for n = 2? Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results Let ψ : M 2 → Λ 3 ⊂ L 4 be a spacelike surface ( → orientable). ψ, η ∈ X ⊥ ( M 2 ) , � η, η � = 0 , � η, ψ � = 1 . A ψ = − Id ∇ ⊥ ψ = ∇ ⊥ η = 0 A spacelike surface which admits a lightlike normal vector field ξ with ∇ ⊥ ξ = 0 and A ξ = − Id lies in a lightcone of L 4 . Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results Several formulaes ψ 0 = x 0 ◦ ψ A η = − 1 + � ▽ ψ 0 � 2 Id + 1 ▽ 2 ψ 0 2 ψ 2 ψ 0 0 K = − tr ( A η ) = −△ log ψ 0 + 1 ψ 2 0 − → ( → �− → H , − → H = − 1 2 K · ψ − η H � = K ) If M 2 is compact, then M 2 must be a topological 2-sphere. � � M 2 �− → H , − → 1 H � dA = dA = 4 π. ψ 2 M 2 0 Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results Every simply conected Riemannian 2 -manifold admits an isometric imbedding in L 4 through Λ 3 Euclidean plane � � φ : E 2 → Λ 3 , x 2 + y 2 +1 , x 2 + y 2 − 1 φ ( x , y ) = , x , y 2 2 Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results Sphere S 2 ( r ) of constant Gauss curvature 1 / r 2 S 2 ( u , r ) = { v ∈ Λ 3 : � v , u � = r } ∼ = S 2 ( r ) with � u , u � = − 1 , u 0 < 0 , r > 0. S 2 ( u , r ) are all the totally umbilical spheres of L 4 through Λ 3 Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results Hyperbolic plane H 2 (1) σ ( x , y ) = − log y , y > 0 � � x 2 + y 2 +1 , x 2 + y 2 − 1 φ σ ( x , y ) : H 2 (1) → Λ 3 , φ σ ( x , y ) = 1 , x , y y 2 2 Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results Uniformization theorem Every 1-connected Riemannian 2-manifold is conformally equivalent to E 2 , S 2 (1) or H 2 (1) Let ( M 2 , g ) be a 1-connected Riemannian 2-manifold. Then there is an isometric imbedding ψ : M 2 → Λ 3 . a a H. Liu, M. Umehara and K. Yamada, The duality of conformally flat Riemannian manifolds, arXiv:1001.4569v4. Recall there is no isometric inmersion of S 2 ( r ) in 1 = { v ∈ L 4 : � v , v � = 1 } when r < 1. S 3 Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results A general method to construct spacelike surfaces (conjugate surface) Let ψ : M 2 → Λ 3 be a spacelike surface and consider, ψ = − η : M 2 → Λ 3 . � � ψ is a (spacelike) inmersion if and only if det ( A η ) = d � = 0 at every point. � ψ is called the conjugate surface to ψ and η is called non-degenerate. � � ψ = ψ � ψ ∗ � X , Y � = � A 2 η ( X ) , Y � := III η ( X , Y ) K III η = K / d Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results Euclidean plane � � φ : E 2 → Λ 3 , φ ( x , y ) = x 2 + y 2 +1 , x 2 + y 2 − 1 , � , x , y φ is constant. 2 2 S 2 ( u , r ) = { v ∈ Λ 3 : � v , u � = r } with � u , u � = − 1 , u 0 < 0 , r > 0. � S 2 ( u , r ) = S 2 ( u , 1 / 2 r ) Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results Hyperbolic plane φ σ ( x , y ) : H 2 → Λ 3 with σ ( x , y ) = − log y , y > 0 The induced metric from � φ σ has constant Gauss curvature − 4. Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results II η ( X , Y ) = −� A η ( X ) , Y � Proposition Suppose that ψ 0 attains a local maximum value and η is non-degenerate ( d � = 0). Then II η is a Riemannian metric on M 2 . Every compact spacelike surface through Λ 3 is topological a 2-sphere. Theorem. Extrinsic characterization of S 2 ( u , r ) Let ψ : M 2 → Λ 3 be a compact spacelike surface. Assume η is non-degenerate. Then the following assertions are equivalent: M 2 is totally umbilical (= S 2 ( u , r )). The Gauss-Kronecker curvature d = det ( A η ) is constant. The Gauss curvature of II η satisfies K η = 2 ( ⇔ area ( M 2 , II η ) = 2 π ). Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results Theorem. Intrinsic characterization of S 2 ( u , r ) Let ψ : M 2 → Λ 3 be a complete spacelike surface. Assume: K is constant. ψ 0 attains a local maximum value ( ⇒ K > 0). Then M 2 is a totally umbilical sphere S 2 ( u , r ). Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results △ f + λ f = 0 0 = λ 0 ≤ λ 1 ≤ λ 2 ≤ .... Theorem. Characterization of S 2 ( u , r ) in terms of λ 1 Let ψ : M 2 → Λ 3 be a compact spacelike surface. For every u ∈ L 4 with � u , u � = − 1 and u 0 < 0, 2 λ 1 ≤ min � ψ, u � . The equality holds for some u if and only if M 2 = S 2 ( u , r ), r = � ψ, u � . Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results 2 2 λ 1 ≤ � ψ ( p 0 ) , u � , λ 1 ≤ � ψ ( q 0 ) , w � Francisco J. Palomo Spacelike surfaces through the lightcone