Minimal Surfaces in Standard Three-Dimensional Geometry Sol³
We study minimal and totally geodesic surfaces in the standart three-dimensional geometry Sol³ with the left-invariant metric.
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Masaltsev, L.A. 2016-09-30T19:26:24Z 2016-09-30T19:26:24Z 2006 Minimal Surfaces in Standard Three-Dimensional Geometry Sol³ / L.A. Masaltsev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 1. — С. 104-110. — Бібліогр.: 5 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106585 We study minimal and totally geodesic surfaces in the standart three-dimensional geometry Sol³ with the left-invariant metric. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Minimal Surfaces in Standard Three-Dimensional Geometry Sol³ Article published earlier |
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Minimal Surfaces in Standard Three-Dimensional Geometry Sol³ |
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Minimal Surfaces in Standard Three-Dimensional Geometry Sol³ Masaltsev, L.A. |
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Minimal Surfaces in Standard Three-Dimensional Geometry Sol³ |
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Minimal Surfaces in Standard Three-Dimensional Geometry Sol³ |
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Minimal Surfaces in Standard Three-Dimensional Geometry Sol³ |
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Minimal Surfaces in Standard Three-Dimensional Geometry Sol³ |
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minimal surfaces in standard three-dimensional geometry sol³ |
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Masaltsev, L.A. |
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Masaltsev, L.A. |
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2006 |
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English |
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Журнал математической физики, анализа, геометрии |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Article |
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We study minimal and totally geodesic surfaces in the standart three-dimensional geometry Sol³ with the left-invariant metric.
|
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1812-9471 |
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https://nasplib.isofts.kiev.ua/handle/123456789/106585 |
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Minimal Surfaces in Standard Three-Dimensional Geometry Sol³ / L.A. Masaltsev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 1. — С. 104-110. — Бібліогр.: 5 назв. — англ. |
| work_keys_str_mv |
AT masaltsevla minimalsurfacesinstandardthreedimensionalgeometrysol3 |
| first_indexed |
2025-11-24T11:37:40Z |
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2025-11-24T11:37:40Z |
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1850461182999134208 |
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Journal of Mathematical Physics, Analysis, Geometry Short Notes
2006, vol. 2, No. 1, pp. 104�110
Minimal Surfaces in Standard Three-Dimensional
Geometry Sol3
L.A. Masaltsev
Department of Mechanics and Mathematics, V.N. Karazin Kharkov National University
4 Svobody Sq., Kharkov, 61077, Ukraine
E-mail:masaltsev@univer. kharkov.ua
Received, April 19, 2005
We study minimal and totally geodesic surfaces in the standart three-
dimensional geometry Sol3 with the left-invariant metric ds2 = e2zdx2 +
e�2zdy2 + dz2.
Key words: minimal surface, totally geodesic surface, Sol-manifold.
Mathematics Subject Classi�cation 2000: 53A10, 53C42.
A three-dimensional geometry Sol3 can be presented as matrix group
0
@ e�z 0 x
0 ez y
0 0 1
1
A ;
homeomorphic to R3 with the left-invariant metric ds2 = e2zdx2 + e�2zdy2 + dz2
[1, p. 127]. Its group of isometries is of dimension 3, consists from 8 components,
and the component of unit e = (0; 0; 0) coincides with Sol3, acting by left trans-
lations. The stabilizer of origin consists of 8 linear transformations of space R3
taking the form (x; y; z)! (�x;�y; z) and (x; y; z)! (�y;�x;�z). These eight
transformations are isomorphisms and isometries of group Sol3. In this note we
�nd some examples of ruled minimal surfaces and minimal surfaces, invariant
under the action of some 1-parameter group of isometris of Sol3. The techniques
of �nding ruled minimal surfaces is similar to those, wich we used in [2], but in
contrast to geometry Nil3, in geometry Sol3 there is a family of totally geodesic
surfaces.
Work is partialy supported with the grant DFFD of Ministery of education and science
of Ukraine 01.07/00132.
c
L.A. Masaltsev, 2006
Minimal Surfaces in Standard Three-Dimensional Geometry Sol3
1. Minimal ruled surfaces in geometry Sol
3
Acting as in [2], we at �rst write down the system of ordinary di�erential
equations for geodesics of Sol3:
x00(t) + 2x0(t)z0(t)� 0;
y00(t)� 2y0(t)z0(t)� 0;
z00(t)� e2z(x0)2 + e�2z(y0)2 = 0:
The obvious solutions to the system are 1) �vertical� geodesics (x = x0; y =
y0; z = t), 2) �horizontal� geodesics (x = � 1p
2
e�z0t + x0; y = � 1p
2
ez0t + y0; z =
z0)). Find at �rst all ruled minimal surfaces, composed of �vertical� geodesics:
r(s; t) = (x(s); y(s); t). Compute the �rst and the second fundamental forms of
this surface.
Proposition 1. 1) The �rst fundamental form of the surface r(s; t) = (x(s);
y(s); t) in the geometry Sol3 is
I = (e2t(x0s)
2 + e�2t(y0s)
2)ds2 + dt2;
2) the second fundamental form of the surface is
II =
(x00y0 � x0y00)ds2 + 4x0y0dsdt
(e2t(x0)2 + e�2t(y0)2)1=2
:
P r o o f. The nonzero Cristo�el symbols of Sol3 metric are �1
13 = 1, �2
23 = �1,
�3
11 = �e2z, �3
22 = e�2z . The tangent vectors to the surface are rs = (x0; y0; 0),
rt = (0; 0; 1), and from here we easily obtain the �rst fundamental form. The
normal vector is n =
(e�2t
y
0
;�e2tx0;0)
(e2tx02+e�2ty02)1=2
. The coe�cients of the second fundamental
form can be computed using formulas (43.4), (43.5) from [3, p. 180], which for
given surface in Sol3 take the form (latin indices vary from 1 to 2, and greek
indices from 1 to 3):
bij = e2tn1(r1ij + �1
��r
�
;i
r�;j) + e�2tn2(r2ij +�2
��r
�
;i
r�;j) + n3(r3ij + �3
��r
�
;i
r�;j):
Corollary 1. The ruled minimal surfaces, composed from 'vertical' geodesics
in Sol3, are the surfaces of the form r(s; t) = (s; as+b; t) or r(s; t) = (as+b; s; t),
where a; b � arbitrary constants.
P r o o f. For the surface, composed from 'vertical' geodesics, minimality
condition 2H = b11g22 � 2b12g12 + b22g11 = 0 takes the form x00y0 � x0y00 = 0,
whence the statement follows.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 105
L.A. Masaltsev
Corollary 2. The totally geodesic surfaces in Sol3, composed from �vertical�
geodesics are the surfaces of the form r(s; t) = (s; b; t) and r(s; t) = (a; s; t).
P r o o f. It must be ful�lled the condition b11 = b12 = 0, whence the
statement follows.
Remark, that isometries of Sol3 of the form (x; y; z) ! (y; x;�z) transform
�vertical� totally geodesic surfaces, �parallel� to xOz to the �vertical� totally
geodesic surfaces, �parallel� to yOz.
Proposition 2. Arbitrary minimal surface, composed from �vertical� geode-
sics, is stable.
P r o o f. For the surface in Sol3 with parametrization r(s; t) = (s; as+ b; t)
the coe�cients of the �rst fundamental form are g11 = e2t + a2e�2t, g12 = 0,
g22 = 1. The coe�cients of the second fundamental form are b11 = b22 = 0,
b12 = 2a
(e2t+a2e�2t)1=2
. Nonzero components of Rimann tensor of geometry Sol3
are R1212 = 1, R1313 = �e2z, R2323 = �e�2z. The unique nonzero component of
Ricci tensor of geometry Sol3 is R33 = �2. Since the normal to studed surface
is of the form n =
(ae�2t
;�e2t;0)
(e2t+a2e�2t)1=2
, the Ricci curvature in the normal direction is
Ric(n; n) = R��n
�n� = 0. The norm of squared second fundamental jjbjj2 (the
sum of squared principal curvatures) is jjbjj2 = 8a2
(e2t+a2e�2t)2
. For the Laplace�
Beltrami �M operator of the surface we obtain the following expression:
�M =
1
e2t + a2e�2t
�
@2
@s2
+ (e2t � a2e�2t)
@
@t
�
+
@2
@t2
:
Hence, for the Jacobi operator L = �M +Ric(n; n)+ jjbjj2 we �nd the expres-
sion
L =
1
e2t + a2e�2t
�
@2
@s2
+ (e2t � a2e�2t)
@
@t
�
+
@2
@t2
+
8a2
(e2t + a2e�2t)2
:
It is directly checked that the following positive function f(t) = (e2t+a2e�2t)�1=2
solves the equation Lf = 0. Then according to theorem of Fisher�Colbrie�Schoen
[4, Th. 1] the studed minimal surface is stable.
To solve the problem of classi�cation of all totally geodesic surfaces in the
geometry Sol3 we need the expressions for the coe�cients of the �rst and second
fundamental forms of the surface r(x; y) = (x; y; z(x; y)), which has nondegenerate
projection on the plane xOy. In this case the tangent vectors and normal to the
surface are rx = (1; 0; zx), ry = (0; 1; zy), n =
(�zxe�2z
;�zye2z ;1)
(z2xe
�2z+z2ye
2z+1)1=2
. The coe�cients
of the �rst and second fundamental forms are
g11 = e2z + z2x; g12 = zxzy; g22 = e�2z + z2y ;
106 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1
Minimal Surfaces in Standard Three-Dimensional Geometry Sol3
b11 =
zxx � 2z2x � e2z
(z2xe
�2z + z2ye
2z + 1)1=2
; b12 =
zxy
(z2xe
�2z + z2ye
2z + 1)1=2
;
b22 =
zyy + 2z2y + e�2z
(z2xe
�2z + z2ye
2z + 1)1=2
:
Proposition 3. There is no totally geodesic surface in the geometry Sol3 with
nondegenerate projection to the xOy.
P r o o f. Suppose that there exists the totally geodesic surface in the form
(x; y; z(x; y)). Then the condition b12 = 0 implies z(x; y) = �(x) + (y): The
conditions b11 = b22 = 0 yield the following system for �(x) and (y):
�xx � 2�2x � e2(�+ ) = 0; yy + 2 2
y + e�2(�+ ) = 0:
It can be rewritten in the form
(e�2�(x))00xx = �2e (y); (e2 (y))00yy = �2e�2�(x):
Since the left hand side of the second equation does not depend of x, we can
di�erentiate it two times by x, getting (e�2�)00xx = 0, but then the �rst equation
takes the form �2e2 (y) = 0, that is impossible.
We will �nd now all ruled minimal surfaces composed of "horizontal" geode-
sics. This surface admits the parametrization
x(s; t) =
1p
2
e�z(s)t+ a(s); y(s; t) =
1p
2
ez(s)t+ b(s); z(s; t) = z(s):
The problem consists in �nding of triple of unknown functions (a(s); b(s); z(s)),
which yield minimal surface in the geometry Sol3. Note, that by virtue of men-
tioned dihedral isometries it is su�cient to restrict search to the case of the pointed
out surfaces. The tangent vectors and normal to the surface are
r0s = (� 1p
2
e�zz0t+ a0;
1p
2
ezz0t+ b0; z0); r0t = (
1p
2
e�z;
1p
2
ez ; 0);
n =
(z0;�e2zz0;
p
2ezz0t� a0e2z + b0)
A
;
where A =
�
2e2zz02 + (
p
2ezz0t� a0e2z + b0)2
�1=2
.
The calculations yield the following values for the coe�cients of the �rst and
second fundamental forms:
g12 =
1p
2
(eza0 + e�zb0); g22 = 1;
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 107
L.A. Masaltsev
b11 =
ez
A
(z00(b0e�z � a0ez) + 2z02(a0ez + b0e�z)
+(a00ez � b00e�z)z0 + (eza0 + e�zb0)(
p
2z0t� a0ez + b0e�z)2);
b12 =
p
2ez
A
(z0t� eza0p
2
+
e�zb0p
2
)2; b22 = 0:
The minimality condition 2H = b11g22 � 2b12g12 + b22g11 = 0 leads to the
equation
z00(b0e�z � a0ez) + (a00ez � b00e�z)z0+(a0ez + b0e�z)(2z02 +(
p
2z0t� a0ez + b0e�z)2)
= 2(a0ez + b0e�z)(z0t+ b0e�z � a0ez)2
After conversion we get linear by variable t equation:
z00(b0e�z � a0ez) + (a00ez � b00e�z)z0
+(eza0 + e�zb0)(2z02 + 2(
p
2� 2)(b0e�z � a0ez)z0t� (b0e�z � a0ez)2) = 0:
From here follows, that it must be ful�lled the system of two equations, getting
by setting equal to zero the coe�cient by t and constant term:
z0(a0ez + b0e�z)(a0ez � b0e�z) = 0;
z00(b0e�z � a0ez) + (a00ez � b00e�z)z0 + (a0ez + b0e�z)(2z02 � (b0e�z � a0ez)2) = 0:
The analysis of the system gives that 1) if z0 = 0, the solution is complete
minimal surface z = z0 (analog of the plane), 2) if a
0ez�b0e�z=0, then di�erentiate
this relation we get a00ez � b00e�z = �z0(a0ez + b0e�z).
Then the second equation of the system takes the form z02(a0ez + b0e�z) = 0,
if z0 6= 0, we may assume that z(s) = s, and then we �nd that a = a0; b = b0 �
arbitrary constants. The solution obtained we can write in the form
x(s; t) =
1p
2
e�st+ a0; y(s; t) =
1p
2
est+ b0; z(s; t) = s: (1)
Finally, let us consider the last possibility 3) a0ez + b0e�z = 0. Di�erentiating
this relation, we get a00ez = �b00e�z + 2b0z0e�z. Substituting it to the second
equation of the system, we get the following equation z00b0 � z0b00 + z02b0 = 0.
Integrating it we �nd z0 = b
0
b+c
. Whence, integrating it once more, we get b =
c1e
z + c2. Then from the relation a0 = �e�2zb0 we �nd a = c1e
�z + c3. Hence the
solution we get in the form x(s; t) = 1p
2
e�z(s)t+ c1e
�z + c3, y(s; t) =
1p
2
ez(s)t+
c1e
z + c2, z(s; t) = z(s).
108 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1
Minimal Surfaces in Standard Three-Dimensional Geometry Sol3
It is evident that if we introduce new variables �s = z(s), �t = t+
p
2c1, then we
get the parametrization (1), that we �nd earlier in the case 2). Hence, we have
proved the following statement.
Proposition 4. Arbitrary complete minimal ruled surface composed from
�horizontal� geodesics is either analog of the �plane� z = z0, or the analog of the
�helicoid� with the parametrization (1) (as well the surfaces obtained from they by
dihedral isometries in Sol3) .
2. Minimal surfaces in Sol3, invariant under action
of 1-parameter subgroup
It is known that if the metric on the Lie group is biinvariant then every 1-
parameter subgroup is geodesic with respect to the Levi�Civita connection [5,
p. 184]. In the case of Sol3 considered left-invariant metric is not biinvariant, so
in general not every 1-parameter subgroup is geodesic. The law of multiplication
in the Sol3 can be written in the form
(x; y; z)(x0; y0; z0) = (x+ e�zx0; y + ezy0; z + z0):
The basis of the Lie algebra sol3 consists of the vectors e1 =
0
@ 0 0 1
0 0 0
0 0 0
1
A ;
e2 =
0
@ 0 0 0
0 0 1
0 0 0
1
A ; e3 =
0
@ �1 0 0
0 1 0
0 0 0
1
A ; with brackets [e1; e2] = 0, [e1; e3] = e1,
[e2; e3] = �e2.
Denote by Ga;b(t) the 1-parameter subgroup exp(ae1+be2)t= E+(ae1+be2)t.
Consider the surface in Sol3 , generated with the aid of curve r(s) = (s; 0; z(s))
in the following way:
R(s; t) = Ga;b(t)r(s) = (at; bt; 0)(s; 0; z(s)) = (at+ s; bt; z(s)):
It is evident, that the surface R(s; t) is invariant under the action of the group
Ga;b, that is Ga;b(�t)R(s; t) = Ga;b(�t+ t)R(s; 0).
Note that 1-parameter subgroup Ga;b(t), in general, is not a geodesic of Sol3,
excepting the case jaj = jbj = 1p
2
, when we get �horizontal� geodesic. We will �nd
minimal surfaces R(s; t) in Sol3, invariant under the action of subgroup Ga;b(t).
The tangent vectors and normal to the surface R(s; t) are
Rs = (1; 0; z0); Rt = (a; b; 0); n =
(bz0e�2z;�az0e2z;�b)
B
;
where B = (a2z02e2z + b2z02e�2z + b2)1=2.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 109
L.A. Masaltsev
Calculation of the �rst and second fundamental forms yields
g11 = e2z + z02; g12 = ae2z ; g22 = a2e2z + b2e�2z ;
b11 =
b
B
(2z02 � z00 + e2z); b12 =
ab
B
(2z02 + e2z); b22 = � b
B
(�a2e2z + b2e�2z):
Minimality condition 2H = g11b22�2g12b12+g22b11 = 0 leads to the equation
z00(a2e2z + b2e�2z) + z02(a2e2z � b2e�2z) = 0:
Integrating it, we get z02(a2e2z+b2e�2z) = c, and further on
R p
a2e2z + b2e�2zdz
= cs.
So the following statement is valid.
Proposition 5. Minimal surfaces in Sol3, invariant under the action of the
subgroup exp(ae1 + be2)t, admit the parametrization
R(s; t) = (at+ s; bt; z(s));
where the function z(s) can be found from the equation
R p
a2e2z + b2e�2zdz = cs:
R e m a r k. The minimal surface equation in Sol3, which admits nondegen-
erate projection on xOy, takes the form
(e�2z + z2y)zxx � 2zxzyzxy + (e2z + z2x)zyy � e�2zz2x + e2zz2y = 0:
The analog of helicoid, founded in Sect. 1, admits in the domain (x > 0; y > 0)
the parametrization (x; y; 1
2
ln( y
x
)), and among the surfaces discussed in 2, there
are (x; y;�ln(c� x)), which obtained, taking a = 0, b = 1.
References
[1] P. Scott, The geometries of 3-manifolds. � Bull. London Math. Soc. 15 (1983),
No. 56, 401�487.
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35�45. (Russian)
[3] L.P. Eisenhart, Riemannian geometry. Princeton univ., Princeton, 1926.
[4] D. Fisher-Colbrie and R. Schoen, The structure of complete stable minimal surface
in 3-manifold of nonnegative scalar curvature. � Comm. Pure Appl. Math. 33
(1980), 199�211.
[5] V.V. Tro�mov, Introduction to the geometry of manifolds with symmetries. Publ.
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110 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1
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