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Conference onGeometric Structures in Mathematical Physics

Golden Sands, September 19 - 26, 2011

An Invariant Theory of

Marginally Trapped Surfaces in

Minkowski 4-space

G. Ganchev and V. Milousheva

Institute of Mathematics and Informatics,Bulgarian Academy of Sciences

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Aim: to develop the invariant theory of marginally trapped surfaces in

the four-dimensional Minkowski space R41

• The notion of ”marginally trapped surfaces”.

• Invariants of a marginally trapped surface .

• Fundamental theorem .

• Examples of marginally trapped surfaces .

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1. Introduction

The concept of trapped surfaces - introduced by Roger Penrose [Phys.

Rev. Lett., 1965].

A surface in a 4-dimensional spacetime is called marginally trapped if

it is closed, embedded, spacelike and its mean curvature vector is lightlike

at each point of the surface.

In the mathematical literature: a surface is called marginally trapped

it its mean curvature vector H is lightlike at each point.

Classification results:

Marginally trapped surfaces with positive relative nullity in Lorenz space

forms were classified by B.-Y. Chen and J. Van der Veken [Class.

Quantum Grav., 2007]. They also proved the non-existence of marginally

trapped surfaces in Robertson-Walker spaces with positive relative nullity

[J. Math. Phys., 2007] and classified marginally trapped surfaces with

parallel mean curvature vector in Lorenz space forms [Houston J. Math.,

2010].

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Marginally trapped surfaces in Minkowski 4-space which are invariant

under spacelike rotations were classified by S. Haesen and M. Ortega

[Gen. Relativ. Grav., 2009]. In [Class. Quantum Grav., 2007] they

classified boost invariant marginally trapped surfaces (rotational surfaces

of hyperbolic type). They also classified marginally trapped surfaces in

Minkowski 4-space which are invariant under a group of screw rotations,

i.e. a group of Lorenz rotations with an invariant lightlike direction [J.

Math. Anal. Appl., 2009].

We consider marginally trapped surfaces in the 4-dimensional Minkowski

space R41. Our approach to the study of these surfaces is based on the prin-

cipal lines generated by the second fundamental form. Using the principal

lines, we introduce a geometrically determined moving frame field at each

point of such a surface. The derivative formulas for this frame field imply

the existence of seven invariant functions. We prove a fundamental theo-

rem of Bonnet-type stating that these seven invariants under some natural

conditions determine the surface up to a motion in R41.

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2. Invariants of a marginally trapped surface

R41 - the Minkowski space endowed with the metric 〈, 〉 of signature (3, 1);

Oe1e2e3e4 - a fixed orthonormal coordinate system in R41, i.e.

e21 = e2

2 = e23 = 1, e2

4 = −1.

The standard flat metric: dx21 + dx2

2 + dx23 − dx2

4.

A surface M 2 in R41 is said to be spacelike if 〈, 〉 induces a Riemannian

metric g on M 2. Thus at each point p of a spacelike surface M 2 we have:

R41 = TpM

2 ⊕NpM2

with the property that the restriction of the metric 〈, 〉 onto TpM2 is of sig-

nature (2, 0), and the restriction of the metric 〈, 〉 onto NpM2 is of signature

(1, 1).

∇′ and ∇ - the Levi Civita connections on R41 and M 2, respectively;

σ - the second fundamental tensor;

D - the normal connection of M 2.

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The formulas of Gauss and Weingarten:

∇′xy = ∇xy + σ(x, y);

∇′xξ = −Aξx + Dxξ.

The mean curvature vector field:

H =1

2(σ(x, x) + σ(y, y)).

The Gauss curvature:

K = R(x, y, y, x),

where R(x, y) = ∇x∇y −∇y∇x −∇[x,y] is the curvature operator.

The curvature of the normal connection:

R⊥(x, y, ξ, η) = 〈R⊥(x, y)η, ξ〉,

where R⊥(x, y) = DxDy −DyDx−D[x,y] is the normal curvature operator.

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Let M 2 : z = z(u, v), (u, v) ∈ D (D ⊂ R2) - a spacelike surface.

Tangent space: TpM2 = span{zu, zv}, 〈zu, zu〉 > 0, 〈zv, zv〉 > 0.

E(u, v) = 〈zu, zu〉, F (u, v) = 〈zu, zv〉, G(u, v) = 〈zv, zv〉; W =√

EG− F 2.

Normal frame field - {n1, n2} such that 〈n1, n1〉 = 1, 〈n2, n2〉 = −1, and

the quadruple {zu, zv, n1, n2} is positively oriented in R41.

Derivative formulas:

∇′zu

zu = zuu = Γ111 zu + Γ2

11 zv + c111 n1 − c2

11 n2;

∇′zu

zv = zuv = Γ112 zu + Γ2

12 zv + c112 n1 − c2

12 n2;

∇′zv

zv = zvv = Γ122 zu + Γ2

22 zv + c122 n1 − c2

22 n2,

Γkij are the Christoffel’s symbols, ck

ij, i, j, k = 1, 2 are given by

c111 = 〈zuu, n1〉; c1

12 = 〈zuv, n1〉; c122 = 〈zvv, n1〉;

c211 = 〈zuu, n2〉; c2

12 = 〈zuv, n2〉; c222 = 〈zvv, n2〉.

Obviously, the surface M 2 lies in a 2-plane if and only if M 2 is totally

geodesic, i.e. ckij = 0, i, j, k = 1, 2. So, we assume that at least one of the

coefficients ckij is not zero.

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The second fundamental form II of the surface M 2 at a point p ∈ M 2

is introduced by the following functions

L =2

W

∣∣∣∣∣∣c111 c1

12

c211 c2

12

∣∣∣∣∣∣ ; M =1

W

∣∣∣∣∣∣c111 c1

22

c211 c2

22

∣∣∣∣∣∣ ; N =2

W

∣∣∣∣∣∣c112 c1

22

c212 c2

22

∣∣∣∣∣∣ .

Let X = λzu + µzv, (λ, µ) 6= (0, 0) be a tangent vector at a point p ∈ M 2.

Then

II(λ, µ) = Lλ2 + 2Mλµ + Nµ2, λ, µ ∈ R.

The condition L = M = N = 0 characterizes points at which the space

span{zuu, zuv, zvv} is one-dimensional. We call such points flat points of

the surface. A surface consists of flat points if and only if it is developable

or lies in a 3-dimensional space.

Further we consider surfaces free of flat points, i.e. (L, M, N) 6= (0, 0, 0).

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The second fundamental form II determines conjugate, asymptotic, and

principal tangents at a point p of M 2 in the standard way.

The equation for the asymptotic tangents at a point p ∈ M 2 is

Lλ2 + 2Mλµ + Nµ2 = 0.

The equation for the principal tangents at a point p ∈ M 2 is∣∣∣∣∣∣E F

L M

∣∣∣∣∣∣ λ2 +

∣∣∣∣∣∣E G

L N

∣∣∣∣∣∣ λµ +

∣∣∣∣∣∣F G

M N

∣∣∣∣∣∣ µ2 = 0.

A line c : u = u(q), v = v(q); q ∈ J ⊂ R on M 2 is said to be an

asymptotic line, respectively a principal line, if its tangent at any

point is asymptotic, respectively principal.

The second fundamental form II generates two invariant functions:

k =LN −M 2

EG− F 2 , κ =EN + GL− 2FM

2(EG− F 2).

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It is interesting to note that the ”umbilical” points, i.e. points at which

the coefficients of the first and the second fundamental forms are propor-

tional (L = ρE, M = ρF, N = ρG, ρ 6= 0), are exactly the points at which

the mean curvature vector H is zero.

So, the spacelike surfaces consisting of ”umbilical” points in R41 are ex-

actly the minimal surfaces. Minimal spacelike surfaces are characterized in

terms of the invariants k and κ by the equality κ2 − k = 0.

For surfaces free of minimal points κ2 − k > 0 and at each point of the

surface there exist exactly two principal tangents.

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Let M 2 : z = z(u, v), (u, v) ∈ D be a marginally trapped surface.

The mean curvature vector is lightlike at each point of the surface, i.e.

〈H, H〉 = 0.

Then there exists a pseudo-orthonormal normal frame field {n1, n2}, such

that n1 = H and

〈n1, n1〉 = 0; 〈n2, n2〉 = 0; 〈n1, n2〉 = −1.

We assume that M 2 is free of flat points. Then at each point of the

surface there exist exactly two principal lines and we assume that M 2 is

parameterized by principal lines. Let us denote x =zu√E

, y =zv√G

.

Thus at each point p ∈ M 2 we obtain a special frame field {x, y, n1, n2},which we call a geometric frame field of M 2.

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With respect to the geometric frame field we have the following derivative

formulas of M 2:

∇′xx = γ1 y + (1 + ν) n1; ∇′

xn1 = µ y + β1 n1;

∇′xy = −γ1 x + λ n1 + µ n2; ∇′

yn1 = µ x + β2 n1;

∇′yx = −γ2 y + λ n1 + µ n2; ∇′

xn2 = (1 + ν) x + λ y − β1 n2;

∇′yy = γ2 x + (1− ν) n1; ∇′

yn2 = λ x + (1− ν) y − β2 n2.

The functions ν, λ, µ, γ1, γ2, β1, β2 are invariants of the surface deter-

mined by the principal directions and the mean curvature vector field:

ν = −〈σ(x, x)− σ(y, y)

2, n2〉, λ = −〈σ(x, y), n2〉, µ = −〈σ(x, y), n1〉,

β1 = −〈∇′xn1, n2〉, β2 = −〈∇′

yn1, n2〉, γ1 = −y(ln√

E), γ2 = −x(ln√

G).

The invariants k, κ and the Gauss curvature K of M 2 are expressed by

the functions ν, λ, and µ as follows:

k = 4µ2(ν2 − 1); κ = −2µν; K = 2λµ. (1)

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In terms of these new invariants we can characterize some basic classes

of marginally trapped surfaces.

Proposition 2.1. Let M 2 be a marginally trapped surface free of flat points.

Then M 2 is a flat surface if and only if λ = 0.


Then M 2 is a surface with flat normal connection if and only if

ν = 0.


Then M 2 is a surface with parallel mean curvature vector field if

and only if β1 = β2 = 0.

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The allied vector field of a normal vector field ξ of an n-dimensional

submanifold Mn of (n + m)-dimensional Riemannian manifold Mn+m is

defined by B.-Y. Chen by the formula

a(ξ) =‖ξ‖n

m∑k=2

{tr(A1 ◦ Ak)}ξk,

where {ξ1 =ξ

‖ξ‖, ξ2, . . . , ξm} is a local orthonormal frame of the normal

bundle of Mn, and Ai = Aξiis the shape operator with respect to ξi.

In particular, the allied vector field a(H) of the mean curvature vector

field H is called the allied mean curvature vector field of Mn in

Mn+m.

B.-Y. Chen defined the A-submanifolds to be those submanifolds of

Mn+m for which a(H) vanishes identically. The A-submanifolds are also

called Chen submanifolds. It is easy to see that minimal submanifolds,

pseudo-umbilical submanifolds and hypersurfaces are Chen submanifolds.

These Chen submanifolds are said to be trivial A-submanifolds.

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S. Haesen and M. Ortega [J. Math. Anal. Appl., 2009] extended the

notion of allied mean curvature vector field to the case when the normal

space is a two-dimensional Lorentz space and the mean curvature vector

field is lightlike as follows.

Denote by {H, H⊥} a pseudo-orthonormal basis of the normal space such

that 〈H, H〉 = 0; 〈H⊥, H⊥〉 = 0; 〈H, H⊥〉 = −1. The null allied mean

curvature vector field is defined as

a(H) =1

2tr(AH ◦ AH⊥) H⊥. (2)

For a marginally trapped surface M 2 we found that the null allied mean

curvature vector field of M 2 is expressed as follows:

a(H) = λµ n2 =K

2n2.


Then M 2 is a flat surface if and only if M 2 has vanishing null allied

mean curvature vector field.

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3. Fundamental theorem

In the local theory of surfaces in Euclidean space a statement of significant

importance is a theorem of Bonnet-type giving the natural conditions under

which the surface is determined up to a motion. A theorem of this type was

proved for surfaces with flat normal connection by B.-Y. Chen (Geometry

of submanifolds, 1973).

In [Cent. Eur. J. Math., 2010] G. Ganchev and V. Milousheva proved

a fundamental theorem of Bonnet-type for surfaces in R4 free of minimal

points. In [Mediterr. J. Math., 2012] we considered spacelike surfaces in

R41 whose mean curvature vector at any point is a non-zero spacelike vector

or timelike vector. Using the geometric moving frame field on such a sur-

face and the corresponding derivative formulas, we proved a fundamental

theorem of Bonnet-type for this class of surfaces, stating that any such

a surface is determined up to a motion in R41 by eight invariant

functions satisfying some natural conditions . In the same way

we prove a fundamental theorem of Bonnet-type for marginally trapped

surfaces.

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Theorem 3.1. Let γ1, γ2, ν, λ, µ, β1, β2 be smooth functions, defined in a

domain D, D ⊂ R2, and satisfying the conditions

µu

µ(2 γ2 + β1)> 0;

µv

µ(2 γ1 + β2)> 0;

−γ1√

E√

G = (√

E)v; −γ2√

E√

G = (√

G)u;

2λ µ =1√E

(γ2)u +1√G

(γ1)v −((γ1)

2 + (γ2)2);

2λ γ2 − 2ν γ1 − λ β1 + (1 + ν) β2 =1√E

λu −1√G

νv;

2λ γ1 + 2ν γ2 + (1− ν) β1 − λ β2 =1√E

νu +1√G

λv;

γ1 β1 − γ2 β2 + 2ν µ = − 1√E

(β2)u +1√G

(β1)v,

where√

E =µu

µ(2 γ2 + β1),√

G =µv

µ(2 γ1 + β2).

Let {x0, y0, (n1)0, (n2)0} be vectors at a point p0 ∈ R41, such that x0, y0

are unit spacelike vectors, 〈x0, y0〉 = 0, (n1)0, (n2)0 are lightlike vectors, and

〈(n1)0, (n2)0〉 = −1. Then there exist a subdomain D0 ⊂ D and a unique

marginally trapped surface M 2 : z = z(u, v), (u, v) ∈ D0 free of flat points,

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such that M 2 passes through p0, the functions γ1, γ2, ν, λ, µ, β1, β2 are the

geometric functions of M 2 and {x0, y0, (n1)0, (n2)0} is the geometric frame

of M 2 at the point p0.

Any marginally trapped surface free of flat points is deter-

mined up to a motion in R41 by these seven invariants satisfying

some natural conditions.

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4. Meridian surfaces in R41

{e1, e2, e3, e4} - orthonormal frame in R41, i.e. e2

1 = e22 = e2

3 = 1, e24 = −1.

S2(1) - the 2-dimensional sphere in the Euclidean space R3 = span{e1, e2, e3}.f = f(u), g = g(u) - smooth functions, defined in an interval I ⊂ R, such

that f 2(u)− g2(u) > 0, f(u) > 0, where f(u) =df(u)

du, g(u) =

dg(u)

du.

The standard rotational hypersurface M′ in R41, obtained by the rota-

tion of the meridian curve m : u → (f(u), g(u)) about the Oe4-axis, is

parameterized as follows:

M′ : Z(u, w1, w2) = f(u) l(w1, w2) + g(u) e4,

where l(w1, w2) is the unit position vector of S2(1) in R3. The hypersurface

M′ is a rotational hypersurface with timelike axis.

Let c : l = l(v) = l(w1(v), w2(v)), v ∈ J, J ⊂ R - a smooth curve on

S2(1), parameterized by the arc-length, i.e. 〈l′, l′〉 = 1.

We construct a surface M′m in R4

1 in the following way:

M′m : z(u, v) = f(u) l(v) + g(u) e4, u ∈ I, v ∈ J. (3)

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Since M′m is a one-parameter system of meridians of M′, we call M′

m a

meridian surface on M′.

Invariants of M′m:

k = −κ2m(u) κ2(v)

f 2(u); κ = 0; K = − κm(u) g(u)

f(u)√

f 2(u)− g2(u),

κ(v) is the spherical curvature of c and κm(u) is the curvature of the merid-

ian curve m.

The mean curvature vector field H is given by

H =κ

2fn1 +

κmf

√f 2 − g2 + g

2f

√f 2 − g2

n2. (4)

Equality (4) implies that M′m is a minimal surface (H = 0) if and only if

κ = 0; κmf

√f 2 − g2 + g = 0.

We exclude this case and consider the case when M′m is marginally

trapped, i.e. H 6= 0 and 〈H, H〉 = 0.

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Proposition 4.1. The meridian surface M′m is marginally trapped if

and only if

κ2(v) =

(κm(u)f(u)

√f 2(u)− g2(u) + g(u)

)2

f 2(u)− g2(u), κ 6= 0.

The equality above implies

κ(v) = a = const, a 6= 0;

κm(u)f(u)√

f 2(u)− g2(u) + g(u)√f 2(u)− g2(u)

= ±a.(5)

We assume that the meridian curve m is given by

f = u; g = g(u).

Then equation (5) takes the form:

ug + g − (g)3 = ±a(1− g2)32 . (6)

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All solutions of differential equation (6) are given by the formula

g(u) =±a

a2 + 1

√(±au + c)2 + u2

+c

(a2 + 1)32

ln

(√a2 + 1 u± ac√

a2 + 1+

√(±au + c)2 + u2

)+ b,

(7)

where b and c are constants, c 6= 0.

Theorem 4.2. The meridian surface M′m is marginally trapped if and

only if the curve c on S2(1) has constant spherical curvature κ = a, a 6= 0,

and the meridian m is defined by (7).

The invariants ν, λ, µ, γ1, γ2, β1, β2 of M′m are:

ν = 0; λ =−au∓ c

au; µ = ∓ ac

2u3 ;

γ1 = γ2 = −√

(±au + c)2 + u2√

2u2;

β1 = β2 =

√(±au + c)2 + u2

√2u4

(c(±au + c)− u2).

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Meridian surfaces lying on the standard rotational hypersurface in R41

with spacelike axis :

S21(1) - the timelike sphere in the Minkowski space R3

1 = span{e2, e3, e4},i.e. S2

1(1) = {V ∈ R31 : 〈V, V 〉 = 1}.

S21(1) is a timelike surface in R3

1 known as the de Sitter space.

f = f(u), g = g(u) - smooth functions, defined in an interval I ⊂ R, such

that f 2(u) + g2(u) > 0, f(u) > 0.

l(w1, w2) - the unit position vector of S21(1) in R3

1.

The rotational hypersurface M′′ in R41, obtained by the rotation of the

meridian curve m : u → (f(u), g(u)) about the Oe1-axis is parameterized

as follows:

M′′ : Z(u, w1, w2) = f(u) l(w1, w2) + g(u) e1.

M′′ is a rotational hypersurface with spacelike axis.

We consider a smooth spacelike curve c : l = l(v) = l(w1(v), w2(v)),

v ∈ J, J ⊂ R on S21(1), parameterized by the arc-length, i.e. 〈l′, l′〉 = 1.

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We construct a surface M′′m in R4

1 in the following way:

M′′m : z(u, v) = f(u) l(v) + g(u) e1, u ∈ I, v ∈ J. (8)

The surface M′′m, defined by (8), lies on the rotational hypersurface M′′.

Since M′′m is a one-parameter system of meridians of M′′, we call M′′

m a

meridian surface on M′′.

Invariants of M′′m:

k = −κ2m(u) κ2(v)

f 2(u); κ = 0; K =

κm(u) g(u)

f(u)√

f 2(u) + g2(u).

The mean curvature vector field H is given by

H = −κmf

√f 2 + g2 + g

2f

√f 2 + g2

n1 −κ

2fn2.

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Theorem 4.3. The meridian surface M′′m is marginally trapped if and

only if the curve c on S21(1) has constant spherical curvature κ = a, a 6= 0,

and the meridian m is defined by

g(u) =±a

1− a2

√u2 − (±au + c)2

+c

(1− a2)32

ln

(√1− a2 u∓ ac√

1− a2+

√u2 − (±au + c)2

)+ b,

where b and c are constants, c 6= 0.

The invariants of the marginally trapped meridian surface M′′m are ex-

pressed in a similar way as the invariants of the marginally trapped merid-

ian surface M′m. The marginally trapped meridian surface M′′

m has non-

parallel mean curvature vector field.

So, we constructed new examples of marginally trapped surfaces, which

are surfaces with non-parallel mean curvature vector field. Moreover, they

are non-flat (K 6= 0) and with non-constant Gauss curvature (K 6= const).

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References

[1] Chen, B.-Y., ”Geometry of submanifolds,” Marcel Dekker, Inc., New

York, 1973.

[2] Chen, B.-Y., Van der Veken, J., ”Marginally trapped surfaces in Loren-

zian space with positive relative nullity,” Class. Quantum Grav. 24,

551–563 (2007).

[3] Chen, B.-Y., Van der Veken, J., ”Spacial and Lorenzian surfaces in

Robertson-Walker space-times,” J. Math. Phys. 48, 073509, 12 pp,

(2007).

[4] Chen, B.-Y., Van der Veken, J., ”Classification of marginally trapped

surfaces with parallel mean curvature vector in Lorenzian space forms,”

Houston J. Math. 36, 421–449 (2010).

[5] Ganchev, G., Milousheva, V., ”Invariants and Bonnet-type theorem

for surfaces in R4,” Cent. Eur. J. Math. 8 (6), 993–1008 (2010).

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[6] Ganchev, G., Milousheva, V., ”An invariant theory of spacelike sur-

faces in the four-dimensional Minkowski space,” Mediterr. J. Math.,

DOI: 10.1007/s00009-010-0108-2.

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