Affine translation surfaces in Euclidean 3-space - Project Euclid

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in E3. Here using the standard coordinate system of Euclidean 3-space E3, a surface rрu; vЮ in E3 will be written as rрu; vЮјрxрu; vЮ;yрu; vЮ;zрu; vЮЮ. The.
No. 9]

Proc. Japan Acad., 89, Ser. A (2013)

111

Affine translation surfaces in Euclidean 3-space By Huili L IU and Yanhua YU Department of Mathematics, Northeastern University, Shenyang 110004, P. R. China (Communicated by Kenji F UKAYA,

M.J.A.,

Oct. 15, 2013)

Abstract: In this paper we define affine translation surface and classify minimal affine translation surfaces in three dimensional Euclidean space. Key words:

Translation surface; mean curvature; minimal surface.

1. Introduction. In the classical theories of minimal surfaces in three dimensional Euclidean space E3 (simply, Euclidean 3-space E3 ), it is well known that the Scherk surface ð1:1Þ

1 cos cx zðx; yÞ ¼ log c cos cy

zðx; yÞ ¼ fðxÞ þ gðyÞ

in E3 . Here using the standard coordinate system of Euclidean 3-space E3 , a surface rðu; vÞ in E3 will be written as rðu; vÞ ¼ ðxðu; vÞ; yðu; vÞ; zðu; vÞÞ. The surface which can be written as (1.2) is usually called translation surface in Euclidean 3-space E3 ([5], [6]). In this note we define affine translation surfaces zðx; yÞ ¼ fðxÞ þ gðy þ axÞ in Euclidean 3-space E3 and get the following main result. Theorem A (minimal affine translation surfaces). Let rðx; yÞ ¼ ðx; y; zðx; yÞÞ be a minimal affine translation surface. Then, either zðx; yÞ is linear, or can be written as pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 cosðc 1 þ a2 xÞ zðx; yÞ ¼ log ð1:3Þ ; cos½cðy þ axÞ c where a and c are constants and ac 6¼ 0. 2. Affine translation surfaces. Let rðu; vÞ be a regular surface with arbitrary parameter ðu; vÞ in Euclidean 3-space E3 . Using the standard coordinate system of E3 we denote the parametric representation of the surface rðu; vÞ by ð2:1Þ

rðu; vÞ ¼ ðx; y; zÞ ¼ ðxðu; vÞ; yðu; vÞ; zðu; vÞÞ:

Definition 2.1.

An affine translation sur-

2000 Mathematics Subject Classification. 53A10, 53C42.

doi: 10.3792/pjaa.89.111 #2013 The Japan Academy

ð2:2Þ

rðu; vÞ ¼ ðxðu; vÞ; yðu; vÞ; zðu; vÞÞ ¼ ðu; v; fðuÞ þ gðv þ auÞÞ ¼ ðx; y; fðxÞ þ gðy þ axÞÞ

is the minimal surface of the type ð1:2Þ

face in Euclidean 3-space E3 is defined as a parameter surface rðu; vÞ in E3 which can be written as

Primary 53A05,

for some non zero constant a and functions fðxÞ and gðy þ axÞ ([1–3], [4], [7]). By a direct calculation, the first fundamental form of rðx; yÞ can be written as 8 I ¼ Edx2 þ 2F dxdy þ Gdy2 ; > > > < E ¼ 1 þ ðf 0 þ ag0 Þ2 ; ð2:3Þ > F ¼ g0 ðf 0 þ ag0 Þ; > > : G ¼ 1 þ g02 ; where 8 dfðxÞ > 0 > ; > : g0 ¼ ¼ ; dv dðy þ axÞ where v ¼ y þ ax. The second fundamental form of rðx; yÞ can be written as 8 II ¼ Ldx2 þ 2Mdxdy þ Ndy2 ; > > > < L ¼ ðf 00 þ a2 g00 ÞD1 ; ð2:5Þ > M ¼ ag00 D1 ; > > : N ¼ g00 D1 ; where ð2:6Þ

D2 ¼ EG  F 2 ¼ 1 þ ðf 0 þ ag0 Þ2 þ g02 :

The Gauss curvature of rðx; yÞ can be written as ð2:7Þ

K ¼ f 00 g00 D4 ¼

f 00 g00 ½1 þ ðf 0 þ ag0 Þ2 þ g02 2

:

The mean curvature of rðx; yÞ can be written as 1 ð2:8Þ H ¼ ½f 00 ð1 þ g02 Þ þ g00 ð1 þ a2 þ f 02 ÞD3 2

112

H. LIU and Y. YU

¼

f 00 ð1 þ g02 Þ þ g00 ð1 þ a2 þ f 02 Þ 3 02 2

2½1 þ ðf 0 þ ag0 Þ2 þ g 

:

ð3:8Þ

From (2.6) we have ð2:9Þ DDy ¼ g0 g00 þ ag00 ðf 0 þ ag0 Þ

þ 2 tan½cðy þ axÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi  ½ 1 þ a2 tan½c 1 þ a2 x þ a tan½cðy þ axÞdxdy

¼ f 0 f 00 þ af 00 g0 þ a2 f 0 g00 þ ag0 g00 þ a3 g0 g00 :

ð3:1Þ

f 00 ð1 þ g02 Þ þ g00 ð1 þ a2 þ f 02 Þ  0:

I ¼ Edx2 þ 2F dxdy þ Gdy2 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ f1 þ ½ 1 þ a2 tanðc 1 þ a2 xÞ þ a tan½cðy þ axÞ2 gdx2

¼ af 0 g00 þ g0 g00 þ a2 g0 g00 ; ð2:10Þ DDx ¼ aDDy þ f 0 f 00 þ af 00 g0 3. Minimal affine translation surfaces. In this section we consider minimal affine translation surfaces in Euclidean 3-space E3 . If the mean curvature H of the affine translation surface rðx; yÞ in E3 vanishes identically, from (2.8) we have

[Vol. 89(A),

þ sec2 ½cðy þ axÞdy2 : The Gauss curvature of the affine translation surface given by (3.7) is ð3:9Þ

K¼

c2 A2 sec2 ðcAxÞ sec2 ½cY  sec2 ½cY  þ ½A tanðcAxÞ þ a tan½cY 2

Where

pffiffiffiffiffiffiffiffiffiffiffiffiffi A ¼ 1 þ a2 Y ¼ y þ ax:

Then ð3:2Þ

f 00 g00 þ ¼ 0: 2 02 1þa þf 1 þ g02

Differentiating (3.2) with respect to y we get  00  d g ð3:3Þ ¼ 0: dðy þ axÞ 1 þ g02 Differentiating (3.2) with respect to x we get   d f 00 ð3:4Þ dx 1 þ a2 þ f 02  00  d g þa ¼ 0: dðy þ axÞ 1 þ g02 Therefore we have ð3:5Þ

f 00 g00 ¼  ¼ c; 1 þ a2 þ f 02 1 þ g02

where c is a constant. The constant c ¼ 0 means that f 00 ¼ g00  0 and the affine translation surface rðu; vÞ is a plane. Let c 6¼ 0 and solving (3.5) we get 8 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 > > < fðxÞ ¼ log cosðc 1 þ a2 xÞ; c ð3:6Þ 1 > > : gðy þ axÞ ¼  log cos½cðy þ axÞ: c Theorem 3.1. Let rðx; yÞ ¼ ðx; y; zðx; yÞÞ be a minimal affine translation surface. Then either zðx; yÞ is linear or can be written as pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 cosðc 1 þ a2 xÞ zðx; yÞ ¼ log ð3:7Þ : c cos½cðy þ axÞ The metric of the affine translation surface given by (3.7) is

:

Definition 3.1. The minimal affine translation surface (3.7) is called generalized Scherk surface or affine Scherk surface in Euclidean 3-space. Remark 3.1. If a ¼ 0, the minimal affine translation surface or generalized Scherk surface rðx; yÞ given by (3.7) is the classical Scherk surface ð3:10Þ

zðx; yÞ ¼

1 cos cx log : c sin cy

In this case, the surface is translated along two orthonormal directions. Therefore the classical Scherk surface may be called minimal orthonormal translation surface in Euclidean 3-space E3 . It is easy to see that the metrics of (3.7) and (3.10) are different (they are homothetic). Remark 3.2. The Gauss curvature K of the generalized Scherk surface (3.7) can be written as K¼

c2 ð1 þ a2 Þ ; Aðx; yÞ

where

pffiffiffiffiffiffiffiffiffiffiffiffiffi Aðx; yÞ ¼ cos2 ðc 1 þ a2 xÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi þ ½ 1 þ a2 sinðc 1 þ a2 xÞ cos½cðy þ axÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi þ a sin½cðy þ axÞ cosðc 1 þ a2 xÞ2 :

Therefore, when 8 > > > > > :y ¼

  1  pffiffiffiffiffiffiffiffiffiffiffiffiffi n  ; 2 c 1 þ a2   1  m   ax; c 2

No. 9]

Affine translation surfaces in Euclidean 3-space

where m; n ¼ 1; 2; . . . ; the Gauss curvature of the generalized Scherk surface tends to the infinity. Acknowledgements. This work was supported by NSFC (No. 11071032 and 11371080); Joint Research of NSFC and NRF (No. 11111140377). The first author was partially supported by Chern Institute of Mathematics; Beijing International Center for Mathematical Research; Northeastern University. References [ 1 ] W. Blaschke, Vorlesungen u ¨ber Differentialgeometrie I, Springer, Berlin, 1921. [ 2 ] W. Blaschke, Vorlesungen u ¨ber Differentialgeometrie II, Springer, Berlin, 1923.

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[ 3 ] W. Blaschke, Vorlesungen u ¨ber Differentialgeometrie III, Springer, Berlin, Heidelberg, New York, 1929. [ 4 ] A. M. Li, U. Simon and G. S. Zhao, Global affine differential geometry of hypersurfaces, de Gruyter Expositions in Mathematics, 11, de Gruyter, Berlin, 1993. [ 5 ] K. Kenmotsu, Surfaces with constant mean curvature, translated from the 2000 Japanese original by Katsuhiro Moriya and revised by the author, Translations of Mathematical Monographs, 221, Amer. Math. Soc., Providence, RI, 2003. [ 6 ] K. Kenmotsu, Surfaces of revolution with periodic mean curvature, Osaka J. Math. 40 (2003), no. 3, 687–696. [ 7 ] U. Simon, A. Schwenk-Schellschmidt and H. Viesel, Introduction to the affine differential geometry of hypersurfaces, Lecture Notes of the Science University of Tokyo, Sci. Univ. Tokyo, Tokyo, 1991.