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

Ln+1 = (Rn+1, , ), , = −(dx0)2 +

n

• i=1

(dxi)2. Λn = {v ∈ Ln+1 : v, v = 0, v0 > 0} Assume ψ : Mn → Λn+1 is an inmersion such that the position vector field is not tangent to M. Then M inherits a Riemannian metric from Ln+2. What kind of n-dimensional Riemannian manifolds can be isometrically inmersed in Λn+1?

Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results

A Riemannian manifold Mn is said to be conformally flat if every point of the manifold lies in a local coordinate system (x1, ..., xn) with respect to which the Riemannian metric takes the form, ds2 = ω2

n

• i=1

(dxi)2 , for some non-vanishing function ω.

Brinkmann, 1923

Let Mn be a simply connected Riemannian manifold with n ≥ 3. Then Mn is conformally flat if and only if Mn can be isometrically inmersed into Λn+1 ⊂ Ln+2.a

• aA. Asperti and M. Dajczer, Conformally flat Riemannian manifolds as hypersurfaces
• f 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 : M2 → Λ3 ⊂ L4 be a spacelike surface (→ orientable). ψ, η ∈ X⊥(M2), η, η = 0, η, ψ = 1. Aψ = −Id ∇⊥ψ = ∇⊥η = 0 A spacelike surface which admits a lightlike normal vector field ξ with ∇⊥ξ = 0 and Aξ = −Id lies in a lightcone of L4.

Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results

Several formulaes ψ0 = x0 ◦ ψ Aη = −1 + ▽ ψ02 2ψ2 Id + 1 ψ0 ▽2 ψ0 K = −tr(Aη) = −△ log ψ0 + 1 ψ2 − → H = −1

2K · ψ − η

(→ − → H, − → H = K) If M2 is compact, then M2 must be a topological 2-sphere.

• M2−

→ H, − → H dA =

• M2

1 ψ2 dA = 4π.

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 L4 through Λ3 Euclidean plane φ : E2 → Λ3, φ(x, y) =

• x2+y2+1

2

, x2+y2−1

2

, x, y

• Francisco J. Palomo

Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results

Sphere S2(r) of constant Gauss curvature 1/r2 S2(u, r) = {v ∈ Λ3 : v, u = r} ∼ = S2(r) with u, u = −1, u0 < 0, r > 0. S2(u, r) are all the totally umbilical spheres of L4 through Λ3

Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results

Hyperbolic plane H2(1) σ(x, y) = − log y, y > 0 φσ(x, y) : H2(1) → Λ3, φσ(x, y) = 1

y

• x2+y2+1

2

, x2+y2−1

2

, x, y

• 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 E2, S2(1) or H2(1) Let (M2, g) be a 1-connected Riemannian 2-manifold. Then there is an isometric imbedding ψ : M2 → Λ3. a

• aH. Liu, M. Umehara and K. Yamada, The duality of conformally flat Riemannian

manifolds, arXiv:1001.4569v4.

Recall there is no isometric inmersion of S2(r) in S3

1 = {v ∈ L4 : v, v = 1} when r < 1.

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 : M2 → Λ3 be a spacelike surface and consider,

• ψ = −η : M2 → Λ3.
• ψ is a (spacelike) inmersion if and only if det(Aη) = d = 0 at every point.
• ψ is called the conjugate surface to ψ and η is called non-degenerate.
• ψ = ψ
• ψ∗X, Y = A2

η(X), Y := IIIη(X, Y )

K IIIη = K/d

Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results

Euclidean plane φ : E2 → Λ3, φ(x, y) =

• x2+y2+1

2

, x2+y2−1

2

, x, y

• ,

φ is constant. S2(u, r) = {v ∈ Λ3 : v, u = r} with u, u = −1, u0 < 0, r > 0.

• S2(u, r) = S2(u, 1/2r)

Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results

Hyperbolic plane φσ(x, y) : H2 → Λ3 with σ(x, y) = − log y, y > 0 The induced metric from φσ has constant Gauss curvature −4.

Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results

IIη(X, Y ) = −Aη(X), Y

Proposition

Suppose that ψ0 attains a local maximum value and η is non-degenerate (d = 0). Then IIη is a Riemannian metric on M2. Every compact spacelike surface through Λ3 is topological a 2-sphere.

• Theorem. Extrinsic characterization of S2(u, r)

Let ψ : M2 → Λ3 be a compact spacelike surface. Assume η is non-degenerate. Then the following assertions are equivalent: M2 is totally umbilical (= S2(u, r)). The Gauss-Kronecker curvature d = det(Aη) is constant. The Gauss curvature of IIη satisfies K η = 2 (⇔ area(M2, IIη) = 2π).

Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results

• Theorem. Intrinsic characterization of S2(u, r)

Let ψ : M2 → Λ3 be a complete spacelike surface. Assume: K is constant. ψ0 attains a local maximum value (⇒ K > 0). Then M2 is a totally umbilical sphere S2(u, r).

Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results

△f + λf = 0 0 = λ0 ≤ λ1 ≤ λ2 ≤ ....

• Theorem. Characterization of S2(u, r) in terms of λ1

Let ψ : M2 → Λ3 be a compact spacelike surface. For every u ∈ L4 with u, u = −1 and u0 < 0, λ1 ≤ 2 minψ, u. The equality holds for some u if and only if M2 = S2(u, r), r = ψ, u.

Francisco J. Palomo Spacelike surfaces through the lightcone

Introduction Examples and generalities Global results

λ1 ≤ 2 ψ(p0), u, λ1 ≤ 2 ψ(q0), w

Francisco J. Palomo Spacelike surfaces through the lightcone