Rotational Surfaces with Constant Mean Curvature in Pseudo-Euclidean 4-Space with Neutral Metric

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by  Yana Aleksieva

ROTATIONAL SURFACES WITH CONSTANT MEAN CURVATURE IN
PSEUDO-EUCLIDEAN 4-SPACE WITH NEUTRAL METRIC
YANA ALEKSIEVA, VELICHKA MILOUSHEVA
Abstract. We give the classification of constant mean curvature rotational surfaces of
6
elliptic, hyperbolic, and parabolic type in the four-dimensional pseudo-Euclidean space
1
0 with neutral metric.
2

n
a
J 1. Introduction

0
2 In the Minkowski 4-space E4 there exist three types of rotational surfaces with two-
1

dimensional axis – rotational surfaces of elliptic, hyperbolic or parabolic type, known also
]
G as surfaces invariant under spacelike rotations, hyperbolic rotations or screw rotations,
D respectively. A rotational surface of elliptic type is an orbit of a regular curve under the
. action of the orthogonal transformations of E4 which leave a timelike plane point-wise fixed.
h 1
t Similarly, a rotational surface of hyperbolic type is an orbit of a regular curve under the
a
m action of the orthogonal transformations of E41 which leave a spacelike plane point-wise
[ fixed. A rotational surface of parabolic type is an an orbit of a regular curve under the
action of the orthogonal transformations of E4 which leave a degenerate plane point-wise
1 1
v fixed.
8 ThemarginallytrappedsurfacesinMinkowski 4-spacewhichareinvariantunderspacelike
9
rotations (rotational surfaces of elliptic type) were classified by S. Haesen and M. Ortega in
1
5 [6]. The classification of marginally trapped surfaces in E4 which are invariant under boost
1
0
transformations (rotational surfaces of hyperbolic type) was obtained in [5] and the classifi-
.
1
cation of marginally trapped surfaces which are invariant under screw rotations (rotational
0
6 surfaces of parabolic type) is given in [7].
1 Motivated by the classification results of S. Haesen and M. Ortega about marginally
:
v trapped rotational surfaces in the Minkowski space, in [4] G. Ganchev and the second au-
i
X thor considered three types of rotational surfaces in the four-dimensional pseudo-Euclidean
r space E42, namely rotational surfaces of elliptic, hyperbolic, and parabolic type, which are
a analogous to the three types of rotational surfaces in E4. They classified all quasi-minimal
1
rotational surfaces of elliptic, hyperbolic, and parabolic type.
Constant mean curvature surfaces in arbitrary spacetime are important objects for the
special role they play in the theory of general relativity. The study of constant mean
curvature surfaces (CMC surfaces) involves not only geometric methods but also PDE and
complex analysis, that is why the theory of CMC surfaces is of great interest not only
for mathematicians but also for physicists and engineers. Surfaces with constant mean
curvature in Minkowski space have been studied intensively in the last years. See for
example [1], [2], [10], [11], [12].
Classification results for rotational surfaces in three-dimensional space forms satisfying
some classical extra conditions have also been obtained. For example, a classification of all
timelike andspacelike hyperbolic rotationalsurfaces withnon-zeroconstant meancurvature
2010 Mathematics Subject Classification. Primary 53B30, Secondary 53A35, 53B25.
Key words and phrases. Pseudo-Euclidean 4-space with neutral metric, CMC surfaces, rotational
surfaces.
1
2 YANAALEKSIEVA,VELICHKAMILOUSHEVA
in the three-dimensional de Sitter space S3 is given in [8] and a classification of the spacelike
1
and timelike Weingarten rotational surfaces of the three types in S3 is found in [9]. Chen
1
spacelike rotational surfaces of hyperbolic or elliptic type are described in [3].
In the present paper we study Lorentz rotational surfaces of elliptic, hyperbolic, and
parabolic type and give the classification of all such surfaces with non-zero constant mean
curvature in E4.
2
2. Preliminaries
Let E4 be the pseudo-Euclidean 4-space endowed with the canonical pseudo-Euclidean
2
metric of index 2 given by g0 = dx21+dx22 dx23 dx24, where (x1,x2,x3,x4) is a rectangular
coordinatesystem ofE4. As usual, we den−oteby−, theindefinite inner scalar product with
2
h i
respect to g0. A non-zero vector v is called spacelike (respectively, timelike) if v,v > 0
h i
(respectively, v,v < 0). Avector v is called lightlike if it is nonzero andsatisfies v,v = 0.
A surface Mh 2 ini E4 is called Lorentz if the induced metric g on M2 is Lorenhtziani, i.e.
1 2 1
at each point p M2 we have the following decomposition E4 = T M2 N M2 with the
1 2 p 1 p 1
property that th∈e restriction of the metric onto the tangent space T M⊕2 is of signature
p 1
(1,1), and the restriction of the metric onto the normal space N M2 is of signature (1,1).
p 1
Denote by and ′ the Levi Civita connections of M2 and E4, respectively. Let x and
1 2
y denote vecto∇r fields∇tangent to M2 and ξ be a normal vector field. The formulas of Gauss
1
and Weingarten are given respectively by
′y = y +σ(x,y);
∇x ∇x
′
ξ = A x+D ξ,
∇x − ξ x
where σ is the second fundamental form, D is the normal connection, and A is the shape
ξ
operator with respect to ξ. In general, A is not diagonalizable.
ξ
The mean curvature vector field H of the surface M2 is defined as H = 1 trσ. A surface
1 2
M2 is called minimal if its mean curvature vector vanishes identically, i.e. H = 0. Asurface
1
M2 is quasi-minimal if its mean curvature vector is lightlike at each point, i.e. H = 0 and
1
H,H = 0. In this paper we consider Lorentz surfaces in E4 for which H,H = con6 st = 0.
2
h i h i 6
3. Lorentz rotational surfaces with constant mean curvature in E4
2
Let Oe1e2e3e4 be a fixed orthonormal coordinate system in the pseudo-Euclidean space
E42 such that e1,e1 = e2,e2 = 1, e3,e3 = e4,e4 = 1. Lorentz rotational surfaces of
h i h i h i h i −
elliptic, hyperbolic, and parabolic type are defined in [4]. Here we shall present shortly the
construction.
First we consider rotational surfaces of elliptic type. Let c : z = z(u), u J be a
smooth spacelike curve lying in the three-dimensional subspace E31 = span e1,e∈2,e3 and
parameterized by z(u) = (x1(u),x2(u),r(u),0); u J. Withouet losseof{generalit}y we
assume that c is parameterized by the arc-length, ∈i.e. (x′)2 + (x′)2 (r′)2 = 1, and
1 2
r(u) > 0, u J. e −
Let ′ be∈the surface in E4 defined by
2
M
′
(1) : z(u,v) = (x1(u),x2(u),r(u)cosv,r(u)sinv); u J, v [0;2π).
M ∈ ∈
The surface ′, defined by (1), is a Lorentz surface in E4, obtained by the rotation of the
2
M
spacelike curve c about the two-dimensional Euclidean plane Oe1e2. It is called a rotational
surface of elliptic type.
One can also obtain a rotational surface of elliptic type in E4 using rotation of a timelike
2
curve about the two-dimensional plane Oe3e4 (see [4]).
CONSTANT MEAN CURVATURE ROTATIONAL SURFACES 3
Next, we consider rotational surfaces of hyperbolic type. Let c : z = z(u), u J be
a smooth spacelike curve, lying in the three-dimensional subspace E31 = span e1,e2∈,e4 of
E42 and parameterized by z(u) = (r(u),x2(u),0,x4(u)); u J. Withoeut loess{of genera}lity
we assume that c is parameterized by the arc-length, i.e.∈(r′)2 + (x′)2 (x′)2 = 1, and
2 4
r(u) > 0, u J. e −
Let ′′ b∈e the surface in E4 defined by
2
M
(2) ′′ : z(u,v) = (r(u)coshv,x2(u),r(u)sinhv,x4(u)); u J, v R.
M ∈ ∈
The surface ′′, defined by (2), is a Lorentz surface in E4, obtained by hyperbolic rotation
2
of the spacelMike curve c about the two-dimensional Lorentz plane Oe2e4. ′′ is called a
M
rotational surface of hyperbolic type.
Similarly, one can obtaina rotational surface of hyperbolic type using hyperbolic rotation
of a timelike curve lying in span e2,e3,e4 about the two-dimensional Lorentz plane Oe2e4.
{ }
Rotational surfaces of hyperbolic type can also be obtained by hyperbolic rotations of
spacelike or timelike curves about the two-dimensional Lorentz planes Oe1e3, Oe1e4 and
Oe2e3.
Now, we shall consider rotational surfaces of parabolic type in E4. For convenience,
2
in the parabolic case we use the pseudo-orthonormal base e1,e4,ξ1,ξ2 of E42, such that
e2 +e3 e2 +e3 { }
ξ1 = , ξ2 = − . Note that ξ1,ξ1 = 0; ξ2,ξ2 = 0; ξ1,ξ2 = 1.
√2 √2 h i h i h i −
Let c be a spacelike curve lying in the subspace E31 = span e1,e2,e3 of E42 and parame-
{ }
terized by z(u) = x1(u)e1 +x2(u)e2 +x3(u)e3; u J, or equivalently,
∈
e x2(u)+x3(u) x2(u)+x3(u)
z(u) = x1(u)e1 + ξ1 + − ξ2; u J.
√2 √2 ∈
e
x2(u)+x3(u) x2(u)+x3(u)
Denote f(u) = , g(u) = − . Then z(u) = x1(u)e1 + f(u)ξ1 +
√2 √2
g(u)ξ2. Without loss of generality we assume that c is parameteerized by the arc-length, i.e.
(x′)2 +(x′)2 (x′)2 = 1, or equivalently (x′)2 2f′g′ = 1.
1 2 3 1
− −
A rotational surface of parabolic type is defined in the following way:
(3) ′′′ : z(u,v) = x1(u)e1 +f(u)ξ1+( v2f(u)+g(u))ξ2+√2vf(u)e4; u J, v R.
M − ∈ ∈
The rotational axis is the two-dimensional plane spanned by e1 (a spacelike vector field)
and ξ1 (a lightlike vector field).
Similarly, one can obtain a rotational surface of parabolic type using a timelike curve
lying in the subspace span e2,e3,e4 (see [4]).
{ }
In what follows, we find all CMC Lorentz rotational surfaces of elliptic, hyperbolic, and
parabolic type.
3.1. Constant mean curvature rotational surfaces of elliptic type. Let us consider
the surface ′ in E4, defined by (1). The tangent space of ′ is spanned by the vector
2
fields z = (Mx′,x′,r′cosv,r′sinv); z = (0,0, rsinv,rcosvM). Hence, the coefficients of
u 1 2 v
the first fundamental form of ′ are E = z ,−z = 1; F = z ,z = 0; G = z ,z =
u u u v v v
r2(u). Since the generating cMurve c is a sphacelikeicurve paramh eterizied by the ahrc-lengith,
−i.e. (x′)2 + (x′)2 (r′)2 = 1, then (x′)2 + (x′)2 = 1 + (r′)2 and x′x′′ + x′x′′ = r′r′′. We
1 2 1 2 1 1 2 2
− z
v
consider the orthonormal tangent frame field X = z ; Y = , and the normal frame field
u
r
4 YANAALEKSIEVA,VELICHKAMILOUSHEVA
n1,n2 , defined by
{ }
1
′ ′
n1 = ( x2,x1,0,0);
1+(r′)2 −
(4)
1
p ′ ′ ′ ′ ′ 2 ′ 2
n2 = r x1,r x2,(1+(r ) )cosv,(1+(r ) )sinv .
1+(r′)2
(cid:0) (cid:1)
p
Note that X,X = 1; X,Y = 0; Y,Y = 1; n1,n1 = 1; n1,n2 = 0; n2,n2 = 1.
h i h i h i − h i h i h i −
Calculating the second partial derivatives of z(u,v) and the components of the second
fundamental form, we obtain the formulas:
x′x′′ x′′x′ r′′
1 2 1 2
σ(X,X) = − n1 + n2,
1+(r′)2 1+(r′)2
σ(X,Y) = 0p, p
1+(r′)2
σ(Y,Y) = n2,
−p r
which imply that the normal mean curvature vector field H of ′ is expressed as follows
M
1
′ ′′ ′′ ′ ′′ ′ 2
(5) H = r(x1x2 x1x2)n1 +(rr +(r ) +1)n2 .
2r 1+(r′)2 −
(cid:0) (cid:1)
rp2(x′x′′ x′′x′)2 (rr′′ +(r′)2 +1)2
1 2 1 2
Hence, H,H = − − . In the present paper we are
h i 4r2(1+(r′)2)
interested in rotational surfaces with non-zero constant mean curvature. So, we assume
that r2(x′x′′ x′′x′)2 (rr′′ +(r′)2 +1)2 = 0.
1 2 1 2
− − 6
It follows from (4) that
x′x′′ x′′x′ r′
∇′Xn1 = − 1 12+−(r1′)22 X + 1+(r′)2(x′1x′2′ −x′1′x′2)n2,
′ p
∇Yn1 = 0,
(6) r′′ r′
∇′Xn2 = 1+(r′)2 X + 1+(r′)2(x′1x′2′ −x′1′x′2)n1,
p
1+(r′)2
∇′Yn2 = p r Y.
whIifchx′1ixm′2′p−lyxth′1′xa′2t t=he0n,orrmr′′a+l v(erc′t)o2r+fie1ld6=n10,istchoennstfarnomt. H(6e)ncwee, tgheets∇ur′Xfanc1e= ∇′ l′Yiens1in=th0e,
hyperplane E32 = span X,Y,n2 . M
So, further we consi{der rotat}ional surfaces of elliptic type satisfying x′x′′ x′′x′ = 0 in
1 2 1 2
− 6
an open interval I J.
⊂
In the next theorem we give a local description of all constant mean curvature rotational
surfaces of elliptic type.
Theorem 3.1. Given a smooth positive function r(u) : I R R, define the functions
⊂ →
(rr′′ +(r′)2 +1)2 4C2r2(1+(r′)2)
ϕ(u) = η ± du, η = 1, C = const = 0
Z p r(1+(r′)2) ± 6
and
x1(u) = 1+(r′)2 cosϕ(u)du,
R p
x2(u) = 1+(r′)2 sinϕ(u)du.
R p
CONSTANT MEAN CURVATURE ROTATIONAL SURFACES 5
Then the spacelike curve c : z(u) = (x1(u),x2(u),r(u),0) is a generating curve of a constant
mean curvature rotational surface of elliptic type.
Conversely, any constantemean curvature rotational surface of elliptic type is locally con-
structed as above.
Proof: Let ′ be a general rotational surface of elliptic type generated by a spacelike
M
curve c : z(u) = (x1(u),x2(u),r(u),0); u J. We assume that c is parameterized by the
′ ′′ ′′ ′ ∈ ′
arc-length and x x x x = 0 for u I J. Using (5) we get that is of constant
1 2 1 2
mean curveature if an−d only i6f ∈ ⊂ M
r2(x′x′′ x′′x′)2 (rr′′ +(r′)2 +1)2
1 2 1 2 2
(7) − − = εC , ε = sign H,H , C = const = 0.
4r2(1+(r′)2) h i 6
Since the curve c is parameterized by the arc-length, we have (x′)2 + (x′)2 = 1 + (r′)2,
1 2
which implies that there exists a smooth function ϕ = ϕ(u) such that
x′(u) = 1+(r′)2 cosϕ(u),
1
(8) x′(u) = p1+(r′)2 sinϕ(u).
2
p
It follows from (8) that x′x′′ x′′x′ = (1+(r′)2)ϕ′. Hence, condition (7) is written in terms
1 2 1 2
−
of r(u) and ϕ(u) as follows:
(rr′′ +(r′)2 +1)2 4C2r2(1+(r′)2)
′
(9) ϕ(u) = η ± , η = 1.
p r(1+(r′)2) ±
Formula (9) allows us to recover the function ϕ(u) from r(u), up to integration constant.
Using formulas (8), we can recover x1(u) and x2(u) from the functions ϕ(u) and r(u), up
to integration constants. Consequently, the constant mean curvature rotational surface of
′
elliptic type is constructed as described in the theorem.
M
Conversely, given a smooth function r(u) > 0, we can define the function
(rr′′ +(r′)2 +1)2 4C2r2(1+(r′)2)
ϕ(u) = η ± du,
Z p r(1+(r′)2)
where η = 1, and consider the functions
±
x1(u) = 1+(r′)2 cosϕ(u)du,
R p
x2(u) = 1+(r′)2 sinϕ(u)du.
R p
A straightforward computation shows that the curve c : z(u) = (x1(u),x2(u),r(u),0) is
a spacelike curve generating a constant mean curvature rotational surface of elliptic type
according to formula (1). e
(cid:3)
Remark: In the special case when rr′′ + (r′)2 +1 = 0, i.e r(u) = √ u2 +2au+b, a =
± −
const = 0, b = const, the function ϕ(u) is expressed as
6
2C u a a2 +b u a
ϕ(u) = − √ u2 +2au+b+ arcsin − +d , d = const.
√a2 +b (cid:18) 2 − 2 √a2 +b (cid:19)
6 YANAALEKSIEVA,VELICHKAMILOUSHEVA
3.2. Constant mean curvature rotational surfaces of hyperbolic type. Now, we
′′
shall consider the rotational surface of hyperbolic type , defined by (2). The tan-
gent space of ′′ is spanned by the vector fields z = M(r′coshv,x′,r′sinhv,x′); z =
u 2 4 v
(rsinhv,0,rcoMshv,0), and the coefficients of the first fundamental form of ′′ are E =
1; F = 0; G = r2(u). M
The generatin−g curve c is a spacelike curve parameterized by the arc-length, i.e. (r′)2 +
(x′)2 (x′)2 = 1, and hence (x′)2 (x′)2 = (r′)2 1. We assume that (r′)2 = 1, otherwise
2 4 4 2
the su−rface lies in a 2-dimensional−plane. Denote−by ε the sign of (r′)2 1. 6
z −
v
We use the orthonormal tangent frame field X = z ; Y = and the normal frame field
u
r
n1,n2 , defined by
{ }
1
′ ′
n1 = (0,x4,0,x2);
ε((r′)2 1)
−
(10) p 1
′ 2 ′ ′ ′ 2 ′ ′
n2 = (1 (r ) )coshv, r x2,(1 (r ) )sinhv, r x4 .
ε((r′)2 1) − − − −
− (cid:0) (cid:1)
p
The frame field X,Y,n1,n2 satisfies X,X = 1; X,Y = 0; Y,Y = 1; n1,n1 =
{ } h i h i h i − h i
ε; n1,n2 = 0; n2,n2 = ε.
Th he meian curhvatureivec−tor field H of ′′ is expressed by the following formula
M
ε
′ ′′ ′′ ′ ′′ ′ 2
H = r(x4x2 x4x2)n1 (rr +(r ) 1)n2 .
2r ε((r′)2 1) − − −
− (cid:0) (cid:1)
If x′2x′4′ x′2′x′4 = 0, rpr′′ +(r′)2 1 = 0, ′′ lies in the hyperplane span X,Y,n2 . So, we
assume−that x′x′′ x′′x′ = 0 in−an 6openMinterval I J. { }
2 4 2 4
− 6 ⊂
The local classification of constant mean curvature rotational surfaces of hyperbolic type
is given by the following theorem:
Theorem 3.2. Case (A). Given a smooth positive function r(u) : I R R, such that
(r′)2 > 1, define the functions ⊂ →
(rr′′+(r′)2 1)2 4C2r2((r′)2 1)
ϕ(u) = η − ± − du, η = 1, C = const = 0,
Z p r((r′)2 1) ± 6
−
and
x2(u) = (r′)2 1 sinhϕ(u)du,
−
x4(u) = R p(r′)2 1 coshϕ(u)du.
−
R p
Then the spacelike curve c : z(u) = (r(u),x2(u),0,x4(u)) is a generating curve of a constant
mean curvature rotational surface of hyperbolic type.
Case (B). Given a smoothepositive function r(u) : I R R, such that (r′)2 < 1, define
⊂ →
the functions
(rr′′+(r′)2 1)2 4C2r2((r′)2 1)
ϕ(u) = η − ± − du, η = 1, C = const = 0,
Z p r((r′)2 1) ± 6
−
and
x2(u) = 1 (r′)2 coshϕ(u)du,
−
x4(u) = R p1 (r′)2 sinhϕ(u)du.
−
R p
Then the spacelike curve c : z(u) = (r(u),x2(u),0,x4(u)) is a generating curve of a constant
mean curvature rotational surface of hyperbolic type.
e
CONSTANT MEAN CURVATURE ROTATIONAL SURFACES 7
Conversely, any constant mean curvature rotational surface of hyperbolic type is locally
described by one of the cases given above.
The proof of the theorem is similar to the proof of Theorem 3.1.
In the case rr′′ +(r′)2 1 = 0, i.e. r(u) = √u2 +2au+b, a = const = 0, b = const,
− ± 6
the function ϕ(u) is expressed by the formula
2ηC u+a ε(a2 b)
ϕ(u) = √u2 +2au+b − ln u+a+√u2 +2au+b +d .
ε(a2 b) (cid:18) 2 − 2 (cid:12) (cid:12) (cid:19)
− (cid:12) (cid:12)
p (cid:12) (cid:12)
3.3. Constant mean curvature rotational surfaces of parabolic type. Now we shall
′′′
consider the rotational surface of parabolic type , defined by formula (3). The length
′′′ M
of the mean curvature vector field of is given by
M
1
2 ′′ ′ ′ ′′ 2 ′′ ′ 2 2
H,H = f (x f x f ) (ff +(f ) ) .
h i 4f2f′2 1 − 1 −
(cid:0) (cid:1)
In the case x′′f′ x′f′′ = 0, ff′′+(f′)2 = 0 the surface ′′′ lies in a hyperplane of E4.
1 1 2
So, we assume that−x′′f′ x′f′′ = 0 in an o6pen interval I MJ.
1 1
− 6 ⊂
Inthefollowing theoremwe givealocal description ofconstant meancurvaturerotational
surfaces of parabolic type.
Theorem 3.3. Given a smooth function f(u) : I R R, define the functions
⊂ →
1
ϕ(u) = f′ A ((ln ff′ )′)2 4C2 du , C = const = 0, A = const,
(cid:18) ±Z f′q | | ± (cid:19) 6
and
ϕ2(u) 1
x1(u) = Z ϕ(u)du; g(u) = Z 2f′(u−) du.
Then the curve c : z(u) = x1(u)e1 + f(u)ξ1 + g(u)ξ2 is a spacelike curve generating a
constant mean curvature rotational surface of parabolic type.
Conversely, any coenstant mean curvature rotational surface of parabolic type is locally
constructed as described above.
In the special case ff′′+(f′)2 = 0, i.e. f(u) = √2au+b, a = const = 0,b = const, we
± 6
obtain the following expression for the function ϕ(u):
1 2CB 3
ϕ(u) = A √2au+b , A = const,B = const = 0.
√2au+b (cid:18) ± 3a (cid:16) (cid:17) (cid:19) 6
Acknowledgments: The second author is partially supported by the National Science
Fund, Ministry of Education and Science of Bulgaria under contract DFNI-I 02/14.
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Faculty of Mathematics and Informatics, Sofia University, 5 James Bourchier blvd.,
1164 Sofia, Bulgaria
E-mail address: yana a [email protected]
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G.
Bonchev Str. bl. 8, 1113, Sofia, Bulgaria; ”L. Karavelov” Civil Engineering Higher
School, 175 Suhodolska Str., 1373 Sofia, Bulgaria
E-mail address: [email protected]