Selçuk J. Appl. Math. Selçuk Journal of Vol. 14. No. 1. pp. 71-81, 2013 Applied Mathematics
Self Similar Surfaces in Euclidean Spaces E. Etemo¼glu, K. Arslan, B. Bulca
Department of Mathematics, Uludag University, 16059, Bursa, Turkiye e-mail:e.etem oglu@ hotm ail.com , arslan@ uludag.edu.tr, bbulca@ uludag.edu.tr
Received Date: June 11, 2012 Accepted Date: February 26, 2013
Abstract. In the present study we consider self-similar surfaces imbedded in Euclidean space. We give necessary and su¢ cient conditions for the surface of revolution and surfaces with Monge patch in E3 to become self-similar. Fur-ther, we investigate self-similar surfaces in Euclidean 4-space E4:Additionally we give necessary and su¢ cient condition of spherical product surfaces and smooth surfaces given with the Monge patch in E4to become self-similar.
Key words: Monge patch; Surface of revolution surface; Self-similar surface. AMS Classi…cation: 53C40, 53C42.
1. Introduction
The mean curvature ‡ow (MCF) is the gradient ‡ow of the area functional on the space of n-dimensional submanifolds of a given Riemannian manifold. From the viewpoint of analysis, this ‡ow is governed by a non-linear parabolic equation. Self-similar ‡ows arise as special solutions of the MCF that preserve the shape of the evolving submanifold. The simplest and most important example of a self-similar ‡ow is when the evolution is a homothety. Such a self-similar submanifold M with mean curvature vector H satis…es the following non-linear, elliptic system:
H + X?= 0
where X? stands for the projection of the position vector X onto the normal
space. If is any strictly positive constant, the submanifold shrinks in …nite time to a single point under the action of the MCF, its shape remaining unchanged. If is strictly negative, the submanifold will expand, its shape compact. The case of vanishing is the well-known case of a minimal submanifold, which of course is stationary under the action of the ‡ow.
Before stating our own results, we mention some work that has been done on the subject: the planar curves which are self-shrinking where classi…ed in [2];
In particular, the only simple self-shrinking curves are the round circles. In [7], the existence of non-spherical self-similar hypersurfaces of revolution in En were shown; in [4], the author described rotational symmetric Lagrangian self-shrinkers and self-expanders in E2n. Very recently, wider classes of self-similar
Lagrangian submanifolds has been derived in [14]. Further, it was shown in [3] and [6] that the only Lagrangian self-similar submanifolds of E2n which
are foliated by (n-1)-dimensional spheres are the examples found in [4] ; in another direction spherical self-shrinkers have been characterized in [16]. In [5] the author give a characterization of the only self-similar surfaces of E3 known
until now: he …rst prove that the self-similar surfaces of revolution discovered by Angenent [7] are the only cyclic self-similar surfaces, and next that the cylinders over planar self-similar curves are the only ruled self-similar surfaces.
The paper is organized as follows: Section 2 provides some basic concepts of the surfaces in En. In Section 3 we give some results for the self-similar surfaces in E3: In Section 4 we investigate self-similar surfaces in Euclidean 4-space E4: Additionally we give necessary and su¢ cient condition of spherical product surfaces and surfaces with Monge patch in E4 to become self-similar.
2. Basic Concepts
Let M be a smooth surface in Engiven with the patch X(u; v) : (u; v) 2 D E2.
The tangent space to M at an arbitrary point p = X(u; v) of M span fXu; Xvg.
In the chart (u; v) the coe¢ cients of the …rst fundamental form of M are given by
(1) E = hXu; Xui ; F = hXu; Xvi ; G = hXv; Xvi ;
where h; i is the Euclidean inner product. We assume that g = EG F2 6= 0;
i.e. the surface patch X(u; v) is regular.
For each p 2 M, consider the decomposition TpEn = TpM Tp?M where
Tp?M is the orthogonal component of TpM in En. Let r be the Riemannian
connection of E4. Given any local vector …elds X1; X2 tangent to M . The
induced Riemannian connection on M is de…ned by rX1X2= ( erX1X2)
T;
where T meaning the tangent component.
Let (M ) and ?(M ) be the space of the smooth vector …elds tangent to M
and the space of the smooth vector …elds normal to M , respectively. Consider the second fundamental map
(2) h : (M ) (M ) !
?(M )
h(Xi; Xj) = erXiXj rXiXj 1 i; j 2:
For any arbitrary orthonormal frame …eld fN1; N2; :::; Nn 2g of M, recall the shape operator A : ?(M ) (M ) ! (M) ANiX = ( erXiNi) T; X i2 (M):
This operator is bilinear, self-adjoint and for any tangent vector …elds X1; X22
TpM satis…es the following equation:
(3) hANkXj; Xii = hh(Xi; Xj); Nki = c
k ij,
where 1 i; j 2; 1 k n 2:
The equation (2) is called Gaussian formula, where
rXiXj = 2 X k=1 k ijXk ; 1 i; j 2 and (4) h(Xi; Xj) = n 2X k=1 ckijNk ; 1 i; j 2: Here k
ij are called Christo¤ el symbols and ckij are the coe¢ cients of the second
fundamental form [13].
Further, the mean curvature vector of a regular patch X(u; v) is de…ned by
(5) !H = 1 2W2 n 2 X k=1 (ck11G + ck22E 2ck12F )Nk:
Recall that a surface in En is said to be minimal if its mean curvature vanishes identically [10].
De…nition 1.A surface is self-similar if the mean curvature vector H satis…es the following non-linear, elliptic system:
!H + X? = 0
where X? stands for the projection of the position vector X onto the normal
space [6]
3. Self-Similar Surfaces in Euclidean 3-Spaces
Let X : U ! E3 a local parametrization of a surface M . We denote by the
by (1). Let N be the unit normal vector …eld of M then the coe¢ cients of the second fundamental form are de…ned to be:
e = hXuu; N i ; f = hXuv; N i ; g = hXvv; N i :
In order to simplify further calculations, we introduce the following coe¢ cients, which are proportional to the previous ones:
(6) ee = hXuu; Xu Xvi ; ef = hXuv; Xu Xvi ; eg = hXvv; Xu Xvi :
Rather than the classical formula for the mean curvature, 2H =eG + gE 2f F
EG F2 ;
it will be more convenient to use the following one (see, [5]) :
(7) 2H = eeG + egE 2 ef F
(EG F2)3=2 :
In codimension one, the self-similar equation!H + X? = 0 becomes scalar,
namely: H + < X; N >= 0: Moreover, in E3 we have:
(8) hX; Ni = 1
EG F2hX; Xu Xvi =
1
EG F2det(X; Xu; Xv):
Finally, from Equations (7) and (8) we deduce:
Lemma 1. [5] A surface M of E3 is self-similar if and only if, for any local
parametrization X : U ! E3 of M , the following formula holds:
(9) eeG + egE 2 ef F + 2 (EG F2)det(X; Xu; Xv) = 0.
Corollary 1.Let M be a smooth surface of E3 with constant mean curvature.
Then M is self-similar if and only if (EG F2)det(X; X
u; Xv) is a nonzero
constant.
Remark 1. The equality eeG + egE 2 ef F = 0 implies that M is minimal. So, the case of vanishing is the well-known case of a minimal surface.
We give the following results.
Proposition 1. Let M be a surface of revolution given with the regular patch (10) X(u; v) = (f (u); g(u) cos v; g(u) sin v)):
Then M is a self similar surface if and only if the following equality holds: (11) g (gufuu fuguu) + fu fu2+ g2u + 2 g fu2+ g2u (f gu gfu) = 0:
Proof. Di¤erentiating (10) and using (6) and (9) we get the result. As a consequence of Proposition 1 we obtain the following result.
Corollary 2.Let M be a surface of revolution given with the regular patch (10). Then M is a self similar surface if and only if the following formulas hold;
g (gufuu fuguu) + fu fu2+ gu2 = c1;
g fu2+ gu2 (f gu gfu) = c2;
where c1 and c2 are nonzero constant.
Example 1. The surface of revolution given with the parametrization f (u) = u and g(u) = a cos h(ua + b) is a minimal surface (i.e. catenoid) and for = 0 it is self-similar.
Example 2. The surface of revolution given with the parametrization f (u) = u and g(u) = const: is a ‡at surface (i.e. cylinder) and for g = p1
2 it is
self-similar.
Proposition 2. Let M be smooth surface given with the Monge patch
(12) X(u; v) = (u; v; f (u; v)):
Then M is a self similar surface if and only if
fuu 1 + fv2 + fvv 1 + fu2 2fufvfuv+ 2 1 + fu2+ fv2 (f ufu vfv) = 0
holds.
Proof. Di¤erentiating (12) and using (6) and (9) we get the result.
Corollary 3. Let M be a smooth surface given with the Monge patch (12). Then M is a self similar surface if and only if the following formulas hold;
(13) fuu 1 + f
2
v + fvv 1 + fu2 2fufvfuv = c1;
1 + f2
u+ fv2 (f ufu vfv) = c2;
where c1 and c2 are nonzero constants.
De…nition 2. The surface M de…ned as the sum of two space curves (u) = (u; 0; f (u)) and (v) = (0; v; g(v)) is called a translation surface in E3: So, the
translation surfaces are de…ned with the Monge patch (14) X(u; v) = (u; v; f (u) + g(v)):
The following results are well-known;
Example 3. The surface of revolution given with the parametrization f (u) =
1
alog jcos(au)j and g(v) = 1
alog jcos(av)j ( a is a nonzero constant) is a
min-imal surface (i.e. surface of Scherk) and for = 0 it is self-similar. For more detail see [15].
Theorem 1. [15] Let M be a translation surface with constant mean curvature H 6= 0 in 3-dimensional Euclidean space E3: Then M is congruent to a surface
given with the parametrization
f (u) = p 1 a2 2H p 1 4H2u2 (15) g(v) = av: where a < 1 is non-zero positive constant.
By the use of (13)-(15) we get the following result.
Corollary 4. Translation surface with constant mean curvature H 6= 0 in 3-dimensional Euclidean space E3 can not be self-similar.
4. Self-Similar Surfaces in Euclidean 4-Spaces
2-dimensional surfaces in E4 are interesting object for investigation of
geome-ters. Here we have some di¢ cult problems which wait its solutions. Hence the investigation of various classes of surfaces in E4 with point of view of in‡uence of the principal invariant -the vector of mean curvature H on the behavior of surfaces is an actual problem.
De…nition 3.Let X : U ! E4 a local parametrization of a surface M . For a
self similar surface M in E4 the following equality holds:
hH; Nii + hX; Nii = 0; i = 1; 2:
That is equivalent to
(16) hXuu; Nii G + hXvv; Nii E 2 hXuv; Nii F = 2 (EG F2) hX; Nii :
We deduce that the surface given with a regular patch X(u; v) in E4 is self similar if and only if the following equalities hold:
(17) c
1
11G + c122E 2c112F = 2 (EG F2) hX; N1i ;
c2
De…nition 4. Let : R ! E3 be an Euclidean space curve and : R ! E2 Euclidean plane curve. Put (u) = (f1(u); f2(u); f3(u)) and (v) =
(g1(v); g2(v)): Then their spherical product patch is given by
X = : E2 ! E4; X(u; v) = (f1(u); f2(u); f3(u)g1(v); f3(u)g2(v));
u 2 I = (u0; u1); v 2 J = (v0; v1), which is a surface in E4:
For the case f1(u) = 0 or f2(u) = 0; the patch X = : E2 ! E3 becomes
a spherical product of two 2D curves. Recently, G. Ganchev and V. Milousheva considered the general product of the space curve (u) = (f1(u); f2(u); f3(u))
with the circle (v) = (cos v; sin v) such that
(18) X(u; v) = (u) (v) = (f1(u); f2(u); f3(u) cos v; f3(u) sin v);
u 2 I; 0 v < 2 ; where (u) is parametrized with respect to the arc-length, i.e. (f10)2+ (f20)2+ (f30)2= 1 and f3(u) > 0; [12]:
We obtain the following result.
Proposition 3. Let M be a spherical product surface given with the regular patch (18). Then M is a self similar surface if and only if
2f
3 f300+ 2 f3(f1f100+ f2f200+ f3f300) = 0
and
2 f3ff1(f20f300 f30f200) + f2(f10f300 f30f100) + f3(f10f200 f20f100)g 1= 0
hold. Where is the Frenet curvature of the curve and 1 = f10f200(u)
f100f20(u) is the curvature of the projection of on the Oe1e2- plane.
Proof. The tangent space of M is spanned by the vector …elds @X
@u = (f
0
1(u); f20(u); f30(u) cos v; f30(u) sin v);
@X
@v = (0; 0; f3(u) sin v; f3(u) cos v):
Hence, the coe¢ cients of the …rst fundamental form of the surface are E = hXu(u; v); Xu(u; v)i = 1;
F = hXu(u; v); Xv(u; v)i = 0;
G = hXv(u; v); Xv(u; v)i = f3(u)2;
where h; i is the standard scalar product in R4:The second partial derivatives of
X(u; v) are expressed as follows
(19)
Xuu(u; v) = (f100(u); f200(u); f300(u) cos v; f300(u) sin v);
Xuv(u; v) = (0; 0; f30(u) sin v; f30(u) cos v);
Further, the normal space of M is spanned by the vector …elds (20) N1= 1 (f100; f200; f300cos v; f300sin v) N2= 1 (f 0 2f300 f30f200; f30f100 f10f300; (f 0 1f200 f20f100) cos v; (f10f200 f20f100) sin v):
Using (3), (19) and (20) we can calculate the coe¢ cients of the second funda-mental form ck ij are as follows [8] : (21) c1 11=< Xuu(u; v); N1>= ; c112=< Xuv(u; v); N1>= 0; c1 22=< Xvv(u; v); N1>= f3f300; c2 11=< Xuu(u; v); N2>= 0; c212=< Xuv(u; v); N2>= 0; c2 22=< Xvv(u; v); N2>= 1 f3 :
Substituting (21) in to (17), after some calculation we get the result.
Corollary 5. Let M be a spherical product surface given with the regular patch (18). If 2f3 f300= 0 and 1= 0 then M is a minimal surface and for = 0
it is self-similar. For more detail see [9].
De…nition 5. In the considering work we use the representation of surfaces in the explicit form
(22) X(u; v) = (u; v; f (u; v); g(u; v));
where f and g are some smooth functions. The parametrization (22) is called Monge patch in E4 (see, [1]):
First we obtain the following result.
Proposition 4. Let M be a smooth surface given with the Monge patch (22). Then M is a self similar surface if and only if
fuuG + fvvE 2fuvF + 2 EG F2 (f ufu vfv) = 0
(Aguu Bfuu) G + (Agvv Bfvv) E 2 (Aguv Bfuv) F
+2 EG F2 fAg Bf + u (Bf
hold, where E = 1 + (fu)2+ (gu)2; F = fufv+ gugv; G = 1 + (fv)2+ (gv)2; and A = 1 + (fu)2+ (fv)2; B = fugu+ fvgv ; C = 1 + (gu)2+ (gv)2:
Proof. The tangent space of M is spanned by the vector …elds @X
@u = (1; 0; fu; gu); @X
@v = (0; 1; fv; gv):
Hence, the coe¢ cients of the …rst fundamental form of the surface are
(23)
E = < Xu(u; v); Xu(u; v) > = 1 + (fu)2+ (gu)2;
F = < Xu(u; v); Xv(u; v) > = fufv+ gugv;
G = < Xv(u; v); Xv(u; v) > = 1 + (fv)2+ (gv)2;
where h; i is the standard scalar product in E4:The second partial derivatives of
X(u; v) are expressed as follows
(24)
Xuu(u; v) = (0; 0; fuu; guu);
Xuv(u; v) = (0; 0; fuv; guv);
Xvv(u; v) = (0; 0; fvv; gvv):
Further, the normal space of M is spanned by the vector …elds
(25) N1= 1 p A( fu; fv; 1; 0) N2= 1 WpA(Bfu Agu; Bfv Agv; B; A):
funda-mental form h are as follows [9] : (26) c1 11=< Xuu(u; v); N1>= fuu p A; c1 12=< Xuv(u; v); N1>= fuv p A; c1 22=< Xvv(u; v); N1>= fvv p A; c2 11=< Xuu(u; v); N2>= Bfuu+ Aguu WpA ; c212=< Xuv(u; v); N2>= Bfuv+ Aguv WpA ; c222=< Xvv(u; v); N2>= Bfvv+ Agvv WpA : So, substituting (26) into (17) we get the result.
De…nition 6. The surface given with the parametrization (22) by the parame-trization
(27) f (u; v) = f3(u) + g3(v); g (u; v) = f4(u) + g4(v)
is called translation surface in Euclidean 4-space E4[11]: The following results are well-known;
Corollary 6. Let M be a translation surface of E4 given with (27). If M is
given with the parametrization fk(u) =
ck
c2 3+ c24
log cos(pau) + cu + eku;
gk(v) =
ck
c2 3+ c24
log cos(pbv) + dv + pkv; k = 3; 4;
then M is a minimal surface and for = 0 it is is self-similar. For more detail see [11].
References
1. Aminov, Yu. (1994): Surfaces inE4with a Gaussian curvature coinciding with a
Gaussian torsion up to the sign. Mathematical Notes, 56, 1211-1215.
2. Abresch, U., Langer, J. (1986): The normalized curve shortening ‡ow and homo-thetic solutions, J. of Di¤. Geom., 23, 175–196.
3. Anciaux, H., Castro, I., Romon, P. (2006): Lagrangian submanifolds ofR2nwhich
4. Anciaux, H. (2006): Construction of equivariant self-similar solutions to the mean curvature ‡ow in Cn, Geom. Dedicata, 120, no.1, 37–48.
5. Anciaux, H. (2009): Two non existence results for the self-similar equation in Euclidean 3-space, arXiv:0904.4269v1.
6. Anciaux, H., Romon, P. (2008): Cyclic and ruled Lagrangian surfaces in complex Euclidean space, arXiv:math/0703645v2
7. Angenent, S. (1992): Shrinking donuts, in Nonlinear di¤usion reaction equations & their equilibrium, States 3, editor N.G. Lloyd, Birkhauser, Boston.
8. Bulca, B., Arslan, K., Bayram, B.K. and Öztürk, G. (2012): Spherical product sur-faces inE4;Analele Stiinti…ce ale Universitatii Ovidius Constanta, Seria Matematica,
20,41-54.
9. Bulca, B., Arslan, K. (2013): Surfaces Given with The Monge Patch inE4; J.
Math. Physics, Anal., Geom. (Accepted).
10. Chen, B.Y. (1973): Geometry of Submanifols,. Dekker, New York.
11. Dillen, F.,Verstraelen, L., Vrancken, L. and Za…ndratafa, G. (1995): Classi…cation of Polynomial Translation Hypersurfaces of Finite Type. Results in Math, 27, 244-249. 12. Ganchev, G. and Milousheva, V. (2008): On the Theory of Surfaces in the Four-dimensional Euclidean Space. Kodai Math. J., 31, 183-198.
13. Gray, A (1993): Modern di¤erential geometry of curves and surfaces. CRC Press, Boca Raton Ann Arbor London Tokyo.
14. Joyce, D., Lee, Y.I., Tsui, M.P.(2010): Self-similar solutions and translating solitons for Lagrangian mean curvature ‡ow, J. Di¤er. Geom., 84, 127-161.
15. Liu, H. (1999): Translation surfaces with constant mean curvature in 3-dimensional space. J. Geom., 64, 141-149.
16. Smoczyk, K. (2005): Self-shrinkers of the mean curvature ‡ow in arbitrary codi-mension, IMRN 48, 2983–3004.
