Table Of ContentZero f-mean curvature surfaces of revolution
in the Lorentzian product G2 × R
1
Doan The Hieua and Tran Le Namb
7 aCollege of Education, Hue University, Hue, Vietnam
1
0 bDong Thap University, Dong Thap, Vietnam
2
n
a January 10, 2017
J
8
]
G Abstract
D
We classify (spacelike or timelike) surfaces of revolution with zero f-mean cur-
h. vature in G2 ×R , the Lorentz-Minkowski 3-space R3 endowed with the Gaussian-
1 1
at Euclidean density e−f(x,y,z) = 1 e−x2+2y2. It is proved that an f-maximal surface of
m 2π
revolution is either a horizontal plane or a spacelike f-Catenoid. For the timelike
[
case, a timelike f-minimal surface is either a vertical plane containing z-axis, the
1 cylinder x2 +y2 = 1, or a timelike f-Catenoid. Spacelike and timelike f-Catenoids
v
are new examples of f-minimal surfaces in G2×R .
2 1
7
9 AMS Subject Classification (2000): Primary 53C25; Secondary 53A10; 49Q05
1
Keywords: Lorentzian product spaces, Gauss space, f-maximal surfaces, timelike f-minimal
0
. surfaces, spacelike f-Catenoid, timelike f-Catenoid
1
0
7
1
: 1 Introduction
v
i
X
In R3, together with the plane, Catenoid is the only minimal surface of revolution. If not
r
a counting the plane, it is the first minimal surface discovered by Leonhard Euler in 1744. The
counterpart of minimal surfaces in the Lorentz-Minkowski space R3, are (spacelike or timelike)
1
surfaces with zero mean curvature. Since the metric in R3 is not positive definite, there are three
1
types of vectors (spacelike, lightlike or timelike). Therefore, more complicated than rotations in
Euclidean space, in R3, there are three types of Lorentzian rotations depending on the causal of
1
the rotation axises. Maximal surfaces of revolution in R3 have been classified in [7]. Spacelike
1
and timelike surfaces of revolution with constant mean curvature in R3 have been studied in [8],
1
[9] and [10]. Recently, maximal surfaces in Lorentzian product spaces have been also studied by
some authors (see, for example, [1], [2], [3], [11] and [12]).
It is natural to study (spacelike or timelike) surfaces of revolution with zero weighted mean
curvature, also called f-mean curvature, in R3 endowed with a density, i.e., a positive function
1
defined on R3 used to weight the area (the length) of surfaces (curves).
1
1
In this paper, such a density that we considered is the Gaussian-Euclidean density, i.e., the
space is the Lorentzian product G2×R , where G2 is the Gauss plane. The space G2×R is a
1 1
special case of n-dimensional spacetime with a density that does not affect “time”. It should be
mentioned that the space we are living can be seen as a 4-dimensional spacetime with density,
the gravity, that affect “space” and does not affect “time”.
It is showed that the axis of an f-maximal surface of revolution in G2 × R must be the
1
z-axis. Then, solving the f-Maximal Surface Equation for surfaces of revolution we obtain new
non-trivial examples, called spacelike f-Catenoids. Beside horizontal planes, they are the only
f-maximal surfaces of revolution. This is the first result of the paper.
For the timelike case, by a similar proof, it is proved that the axis of a timelike f-minimal
surface of revolution must be the x-axis or the z-axis. If the rotation axis is the x-axis, the only
timelikef-minimalsurfacesofrevolutionareverticalplanescontainingthez-axis. Iftherotation
axis is the z-axis, there are a family of timelike f-minimal surfaces of revolution, called timelike
f-Catenoids, that convergences to another timelike f-minimal surface, the cylinder x2+y2 = 1.
The second main result of the paper is that a timelike f-minimal surface of revolution is either
the cylinder x2+y2 = 1, a vertical plane containing the z-axis or a timelike f-Catenoid.
2 Preliminaries
Forsimplicity,allconceptsaswellasresultsinthissectionareintroducedin3-dimensionalspace.
For more details about Lorentz-Minkowski spaces, manifolds with density or the Gauss space we
refer the reader to [13], [15], [16], [18], [19] and references therein.
Let R3 be the Lorentz-Minkowski 3-space endowed with the Lorentzian scalar product
1
(cid:104) , (cid:105) = dx2+dy2−dz2.
A nonzero vector x ∈ R3 is called spacelike, lightlike or timelike if (cid:104)x,x(cid:105) > 0, (cid:104)x,x(cid:105) = 0, or
1
(cid:104)x,x(cid:105) < 0, respectively.
(cid:112)
Thenormofthevectorxisthendefinedby(cid:107)x(cid:107) = |(cid:104)x,x(cid:105)|.Twovectorsx = (x ,y ,z ), x =
1 1 1 1 2
(x ,y ,z ) ∈ R3 are said to be orthogonal if (cid:104)x ,x (cid:105) = 0, i.e., x x + y y − z z = 0. The
2 2 2 1 1 2 1 2 1 2 1 2
Lorentzian vector product of x and x , denoted by x ∧x , is defined by
1 2 1 2
(cid:12) (cid:12)
(cid:12)e1 e2 −e3(cid:12)
(cid:12) (cid:12)
x1∧x2 = (cid:12)x1 y1 z1 (cid:12),
(cid:12) (cid:12)
(cid:12)x2 y2 z2 (cid:12)
where {e ,e ,e } is the canonical basis of R3. For every x ∈ R3,
1 2 3 1 1
(cid:104)x,x ∧x (cid:105) = det(x,x ,x ).
1 2 1 2
It follows that x ∧x is orthogonal to both x and x .
1 2 1 2
A surface in R3 is called spacelike (timelike) if its induced metric from R3 is Riemannian
1 1
(Lorentzian) or equivalently, every normal vector of the surface is timelike (spacelike).
Forexample,letαbeaplanewhosegeneralequationisAx+By+Cz+D = 0, A2+B2−C2 (cid:54)=
0.Itiseasytoseethat,thevectorn = (A,B,−C)isanormalvectorofα.Theplaneαisspacelike
or timelike if and only if n is timelike or spacelike, respectively.
The following formula for computing the mean curvature of a (spacelike or timelike) surface
in R3 is well-known (see [14], for instance)
1
Eg−2Ff +Ge
H = (cid:15) , (1)
2(EG−F2)
2
where(cid:15) = −1,ifthesurfaceisspacelike;(cid:15) = 1,ifthesurfaceistimelike;E,F,Garethecoefficients
of the first fundamental form and e,f,g are the coefficients of the second fundamental form.
A spacelike (timelike) surface is called maximal (timelike minimal) if its mean curvature H
is zero everywhere.
There are three kinds of rotations in R3: rotations about a spacelike axis, rotations about a
1
timelike axis and rotations about a lightlike axis (see [8], for instance). Below are the matrices
of some typical kinds of rotations that will be used in the proof of Lemma 3 and Lemma 7.
1. The matrix corresponding to a rotation about the y-axis is
coshθ 0 sinhθ
0 1 0 .
sinhθ 0 coshθ
2. The matrix corresponding to a rotation about the lightlike axis x = z, y = 0 is
v2 v2
1− −v
2 2
v 1 −v .
−v2 v2
−v 1+
2 2
3. The matrix corresponding to a rotation about the z-axis is
cosθ −sinθ 0
sinθ cosθ 0.
0 0 1
A surface of revolution is a surface in R3 obtained by rotating a curve γ, the generatrix,
1
around an axis of rotation l, assuming that γ and l are in a plane.
A density on R3 is a positive function, denoted by e−f, used to weight area (length) of
1
surfaces (curves). The weighted mean curvature or the f-mean curvature of a (spacelike or
timelike) surface, denoted by H , is defined by
f
1
H = H + (cid:104)∇f,N(cid:105).
f
2
A spacelike (timelike) surface Σ is called f-maximal (timelike f-minimal) if H = 0 everywhere,
f
i.e., H = −1(cid:104)∇f,N(cid:105).
2
Gauss space G2, is just R2 with the Gaussian probability density
e−f(x,y) = 1 e−x2+2y2,
2π
where (x,y) ∈ G2.
Therefore, the Lorentzian product G2 ×R can be seen as R3 = R2 ×R endowed with the
1 1 1
Gaussian-Euclidean density
e−f(x,y,z) = 1 e−x2+2y2,
2π
where (x,y,z) ∈ G2 ×R . It should be noted that the last coordinate is not dependent on the
1
density.
Let Σ be an oriented (spacelike or timelike) surface in G2×R , N be a unit normal vector
1
field on Σ and ρ be the projection onto the z-axis. Then at any point p ∈ Σ, we have the
following.
3
Lemma 1. (Geometric meaning of the quantity (cid:104)∇f,N(cid:105))
(cid:0) (cid:1)
|(cid:104)∇f,N(cid:105)(p)| = d ρ(p),T Σ .|N| ,
E p E
where d and | | denote the Euclidean distance the Euclidean length, respectively.
E E
Proof. Suppose that p = (x ,y ,z ) and N(p) = (a,b,c), a2+b2−c2 = ±1. An equation of T Σ
0 0 0 p
is of the form ax+by−cz+d = 0. We have ∇f(p) = (x ,y ,0) and ρ(p) = (0,0,z ). Therefore
0 0 0
(cid:0) (cid:1)
|(cid:104)∇f,N(cid:105)(p)| = |ax +by | = |cz −d| = d ρ(p),T Σ .|N| .
0 0 0 E p E
By Lemma 1, it is not hard to prove the followings.
Corollary 2. In G2×R ,
1
1. horizontal planes are f-maximal surfaces;
2. vertical planes have constant f-mean curvature, such a plane containing the z-axis is time-
like f-minimal;
3. circular cylinders about the z-axis have constant f-mean curvature, such a cylinder is
timelike f-minimal if and only if the radius is 1.
3 Spacelike f-maximal surfaces of revolution in G2 × R
1
3.1 Spacelike f-Catenoids in G2 × R
1
In the xz-plane, consider the curve γ that is the graph of the function (see Figure 1).
S
(cid:90) u(cid:114) 1
h(u) = dτ, u ∈ R.
1+τ2eτ2+C
0
Rotating γ the about the z-axis, we obtain a surface of revolution (see Figure 2), denoted
S
by Σ , that can be parametrized as follows.
S
(cid:32) (cid:90) u(cid:114) 1 (cid:33)
X(u,v) = ucosv,usinv, dτ ,
1+τ2eτ2+C
0
where C is a constant. It is easy to verify that the curve is spacelike and therefore the surface Σ
S
is spacelike. The surface Σ has a singular point, that is the origin. By a direct computation, it
S
follows that the f-mean curvature of Σ is zero, i.e., Σ is f-maximal. We call Σ a spacelike
S S S
f-Catenoid.
4
1.5
1
0.5
O
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 3.5
-0.5
-1
-1.5
Figure 1. The generatrix γ
Figure 2. Spacelike f-Catenoid.
3.2 Classification of f-maximal surfaces of revolution in G2 ×R
1
In the Lorentz-Minkowski space R3, because the mean curvature of a (spacelike or timelike)
1
surfaces is invariant under Lorentzian transformation, when studying surfaces of revolution of
constant mean curvature, if the rotation axis is timelike, spacelike or lightlike we can suppose it
isthez-axis, thex-axis, orthelightlikeaxisx = z,y = 0,respectively. InthespaceG2×R ,with
1
the appearance of the density, we can not do this because the f-mean curvature is not invariant
under some Lorentzian transformations. Since the density is dependent on the distance from
points to the z-axis and not dependent on the last coordinate, the f-mean curvature of a surface
does not change under rotations about as well as translations along the z-axis (see Lemma (1).
This observation is useful for the rest of the paper to simplify some calculations.
Lemma 3. A spacelike surface of revolution Σ in G2×R can be parametrized as follows.
1
1. If the rotation axis is spacelike
X(u,v) = (ucoshθ+g(u)sinhθcoshv,usinhv+a,usinhθ+g(u)coshθcoshv). (2)
2. If the rotation axis is lightlike
(cid:18) v2 v2(cid:19)
X(u,v) = u−[u−g(u)] +a,v[u−g(u)],g(u)−[u−g(u)] . (3)
2 2
5
3. If the rotation axis is timelike
X(u,v) = (ucosvcoshθ+g(u)sinhθ,usinv+a,ucosvsinhθ+g(u)coshθ). (4)
Proof. 1. The case l is spacelike. Under a suitable rotation about the z-axis, we can
assume that the plane containing the generatrix γ and the rotation axis l are parallel to
or coincident with the xz-plane. If l and the z-axis are not intersect, we assume that
the common perpendicular line of l and the z-axis is the y-axis. If l and the z-axis are
intersect, we assume that the intersection point is the origin O. Let {H} = (0,a,0) be the
intersection point of l and xy-plane and let θ be the angle between l and Ox.
There exist a Lorentz transformation that maps Σ to a surface of revolution Σ obtained
1
by rotating a spacelike γ , that lies in the xz-plane, about the x-axis. This transformation
1
is a composition of a translation along y-axis by a vector v = (0,a,0) and a rotation
about y-axis of angle θ. Because the curve γ is spacelike, it can be parametrized as
1
γ (u) = (u,0,g(u)), u ∈ I ⊂ R, g (cid:54)= 0, 1 − g(cid:48)2 > 0. Then, a parametrization of Σ is
1 1
X(u,v) = (u,g(u)sinh,g(u)coshv) and therefore a parametrization of Σ is
coshθ 0 sinhθ u 0
X(u,v) = 0 1 0 g(u)sinhv+a
sinhθ 0 coshθ g(u)coshv 0
= (ucoshθ+g(u)sinhθcoshv,g(u)sinhv+a,g(u)sinhθ+g(u)coshθcoshv)
2. The case l is lightlike.
By a suitable rotation about the z-axis, we can assume that the plane containing the
generatrix γ and the rotation axis l is the xz-plane and l is parallel to e +e . Let {H} =
1 3
(a,0,0)betheintersectionpointoflandthexy-planeandsupposethatγ(u) = (u,0,g(u)).
Then, a parametrization of Σ is
v2 v2
1− −v
u a
2 2
X(u,v) = v 1 −v 0 +0
−v2 v2 g(u) 0
−v 1+
2 2
(cid:18) v2 v2(cid:19)
= u−[u−g(u)] +a,v[u−g(u)],g(u)−[u−g(u)] .
2 2
3. The case l is timelike.
By the same arguments as in the case l is spacelike, but in this case, θ is the angle between
l and the z-axis and Σ is the surface of revolution obtained by rotating γ about the
1 1
z-axis.
A parametrization of Σ is (ucosv,usinv,g(u)). Therefore, a parametrization of Σ is
1
coshθ 0 sinhθ ucosv 0
X(u,v) = 0 1 0 usinv+a
sinhθ 0 coshθ g(u) 0
= (ucosvcoshθ+g(u)sinhθ,usinv+a,ucosvsinhθ+g(u)coshθ).
6
Theorem 4. An f-maximal surface of revolution Σ in G2 ×R is either a horizontal plane or
1
a spacelike f-Catenoid.
Proof. SinceLorentztransformationspreservethemeancurvatureofsurfaces,alongacoordinate
curve u = u , the mean curvature H of Σ is constant. Therefore if the f-mean curvature H of
0 f
Σ is zero, along any coordinate curve u = u , (cid:104)∇f,N(cid:105) must be constant. This fact will be used
0
to eliminate the case that the rotation axis l is spacelike or lightlike.
The followings are obtained by straightforward computations.
• If l is spacelike and (2) is a parametrization of Σ, then
1
(cid:104)∇f,N(cid:105) = [usinhθcoshθcoshv+g(u)(sinh2θcosh2v+sinh2v
(cid:112)
1−[g(cid:48)(u)]2
(cid:112)
+g(cid:48)(u)sinhθcoshvcoshθ)+ug(cid:48)(u)cosh2θ+asinhv 1−[g(cid:48)(u)]2].
The condition “(cid:104)∇f,N(cid:105) is not constant” is equvalent to that “for any u
∂ (cid:112)
Q := 1−[g(cid:48)(u)]2(cid:104)∇f,N(cid:105),
∂v
must be zero for every v.” By a straightforward computation, we obtain
(cid:18) (cid:19)
usinh2θ sinh2θ
Q = sinhv+g(u) sinh2θsinh2v+sinh2v+g(cid:48)(u) sinhv +acoshv
2 2
sinh2θ
= g(u)sinh2vcosh2θ+[u+g(u)g(cid:48)(u)] sinhv+acoshv.
2
It is not hard to see that if for any u ∈ I, Q vanishes for every v then g(u) = 0. This is
impossible because g (cid:54)= 0.
• If l is lightlike and (3) is a parametrization of Σ, then
(cid:104)∇f,N(cid:105) = 1 (cid:20)u−[u−g(u)] v2(cid:21)(cid:20)g(cid:48)(u)+ v2 (cid:2)1−g(cid:48)(u)(cid:3)(cid:21)+v2[u−g(u)][g(cid:48)(u)−1]
g(cid:48)2(u)−1 2 2
= 1 (cid:2)(g(u)−u)(1−g(cid:48)(u))v4+2(u+g(u)g(cid:48)(u)−2g(u))v2(cid:3)+ug(cid:48)(u).
4(g(cid:48)2(u)−1)
We can verify that (cid:104)∇f,N(cid:105) is not constant.
• The case l is timelike and (4) is a parametrization of Σ, A direct computation shows that
(cid:20) (cid:21)
1 sinh2θ
(cid:104)∇f,N(cid:105) = ug(cid:48)(u)(1+cos2vsinh2θ)+[u+g(u)g(cid:48)(u)] cosv+g(u)sinh2θ
(cid:112)
1−[g(cid:48)(u)]2 2
ag(cid:48)(u)sinv
+ . (5)
(cid:112)
1−[g(cid:48)(u)]2
We can see that, for any u, (cid:104)∇f,N(cid:105) is constant if and only if θ = a = 0, i.e., l must be the z-axis.
The parametrization of Σ is now become
X(u,v) = (ucosv,usinv,g(u)). (6)
7
A direct computation shows that
−1(1−g(cid:48)2)g(cid:48)+ug(cid:48)(cid:48)
H = ,
2 u(1−g(cid:48)2)3/2
−g(cid:48)u
(cid:104)∇f,N(cid:105) = .
(cid:112)
1−g(cid:48)2
Therefore, Σ is f-maximal if and only if g satisfies the following equation
(1−g(cid:48)2)g(cid:48)+ug(cid:48)(cid:48)+u2g(cid:48)(1−g(cid:48)2) = 0. (7)
We solve equation (7).
• It is clear that g(u) = a, where a is constant, is a solution of (7), i.e., Σ is a vertical plane.
• Now locally we can suppose that g(cid:48)(u) (cid:54)= 0 for every u ∈ J, where J ⊂ I. Multiply both
sides of (7) by g(cid:48) and set h = g(cid:48)2, we get
dh u2+1
− = 2 h(1−h).
du u
Solving this equation, we obtain
1−h
ln = u2+lnu2+C, C ∈ R,
h
or
1
h = .
1+u2eu2+C
Therefore,
(cid:114)
1
g(cid:48)(u) = ± ,
1+u2eu2+C
and
(cid:90) u(cid:114) 1
g(u) = ± dτ, u ∈ J.
1+τ2eτ2+C 0
u0
The function g is defined over R, therefore we can assume that I = R, u = 0, i.e., γ ≡ γ
0 s
or γ ≡ γ , where γ is the graph of the function
s s
(cid:90) u(cid:114) 1
h(u) = − dτ, u ∈ R.
1+τ2eτ2+C
0
It is clear that γ and γ generate the same surface of revolution Σ .
s s S
8
4 Timelike f-minimal surfaces of revolution in G2 × R
1
4.1 Timelike f-Catenoids in G2 × R
1
In the xz-plane consider the curve γ that is the graph of the function
T
(cid:115)
(cid:90) u eτ2
h(u) = dτ,
eτ2 −Cτ2
u0
where C is a positive constant and u belongs to the domain D of the function. The domain D
0
is determined by the following lemma.
Lemma 5. Consider the function h : R −→ R defined by h(u) = eu2 −Cu2, where C > 0. Then
1. If 0 < C < e, then h(u) > 0, ∀u ∈ R.
2. If C = e, then h(u) > 0, ∀u (cid:54)= −1,1.
3. If C > e, then there exist 0 < u < 1 < u , such that h(u) > 0, ∀u ∈ (−∞,−u ) ∪
1 2 2
(−u ,u )∪(u ,+∞).
1 1 2
Proof. Because the function h is even, we just consider the case u ≥ 0. Taking the derivative of
the function, we obtain
(cid:16) (cid:17)
h(cid:48)(u) = 2u eu2 −C .
1. If C ≤ 1, then h(cid:48)(u) > 0,∀u > 0. The function h is monotonically increasing and therefore
h(u) > 0,∀u ≥ 0. Note that h(0) = 1.
√ √
2. If C > 1, the function has the only minimum point at u = lnC and h( lnC) = C −
ClnC. We consider the following subcases.
√
• The case 1 < C < e. Because h( lnC) = C −lnC > 0, h(u) > 0,∀u ≥ 0.
• The case C = e. We can see that h(u) > 0,∀u (cid:54)= 1 and h(1) = 0.
√
• The case C > e. Because h( lnC) = C−ClnC < 0, there exist two values 0 < u <
1
1 < u such that h(u ) = h(u ) = 0, h(u) > 0, ∀u ∈/ [u ,u ] and h(u) ≤ 0, ∀u ∈
2 1 2 1 2
[u ,u ].
1 2
√
x 0 u1 lnc u2 +∞
h(cid:48)(x) − − 0 + +
h(x) 1 +∞
0 0
C(1−lnC)
By Lemma 5, the domain D and u are chosen as follows.
0
1. If 0 < C < e, then D = R and u = 0.
0
9
2. IfC = e,thenD = (−∞,−1), u = −1orD = (−1,1), u = 0ororD = (1,+∞),u = 1.
0 0 0
3. If C > e, then D = (−∞,−u ), u = −u or D = (−u ,u ), u = 0; or D =
2 0 2 1 1 0
(u ,+∞),u = u .
2 0 2
Rotate the curve about the z-axis, we obtain a surface of revolution, denoted by Σ , that can
T
be parametrized as follows.
(cid:115)
(cid:90) u eτ2
X(u,v) = ucosv,usinv, dτ.
eτ2 −Cτ2
u0
By a direct computation, it follows that the curve is timelike, Σ is timelike. Moreover Σ is
T T
timelike f-minimal. We call Σ a timelike f-Catenoid.
T
4.5
4
3.5
3
2.5
2
1.5
1
0.5
O
-3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
Figure 3. Generatrices corresponding to C = 2,1.5,1,0.5 and the line x = z, respectively
10
