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Centroaffine Surfaces of Cohomogeneity One

  • Atsushi Fujioka
  • Hitoshi FuruhataEmail author
Article
  • 102 Downloads

Abstract

We characterize centroaffine surfaces of cohomogeneity one which have vanishing Tchebychev vector fields. Moreover, we classify centroaffine minimal surfaces of cohomogeneity one which have centroaffine metrics of constant curvature.

Keywords

Centroaffine minimal surfaces Proper affine spheres Rotation surfaces Cohomogeneity one 

Mathematics Subject Classification

53A15 53A05 

1 Introduction

The catenoid and plane are the only minimal surfaces of revolution, which is an elementary and beautiful theorem in Euclidean geometry. It is natural to hope to obtain the corresponding results in centroaffine geometry. The first problem is to establish what surfaces of revolution in centroaffine geometry mean. There might be some options. In fact, Yang et al. (2014) studied surfaces of equiaffine revolution, that is, surfaces in the forms:
$$\begin{aligned} f(x,y)= \left\{ \begin{array}{lcc} (\psi (x)\cos y, \psi (x)\sin y, x), &{}&{} \mathrm {(a)} \\ (\psi (x)\cosh y, \psi (x)\sinh y, x), &{}&{} \mathrm {(b)} \\ (x, x y, x y^2/2+\psi (x)), &{}&{} \mathrm {(c)}\\ (x, y, y^2/2+\psi (x)) &{}&{} \mathrm {(d)} \end{array} \right. \end{aligned}$$
as problems to determine the function \(\psi \) in one variable. These expressions come from the definition due to Blaschke as surfaces of which equiaffine normals meet a fixed line, and from succeeding works in equiaffine geometry (e.g. Lee 1995; Manhart 2004).
In centroaffine geometry, it seems natural to consider a surface of cohomogeneity one, that is, a centroaffine surface of the form
$$\begin{aligned} f(x,y)=\gamma (x)e^{yA}, \end{aligned}$$
(1.1)
where \(\gamma \) is a curve in \({\mathbf {R}}^3\) and \(\left\{ e^{yA} \right\} _{y \in {\mathbf {R}}}\) is a one-parameter subgroup of the centroaffine transformation group \(GL(3, {\mathbf {R}})\) for \(A\in \mathfrak {gl}(3,{\mathbf {R}})\). We can call such a surface a centroaffine rotation surface if the one-parameter transformation group preserves a fixed line through the origin and if the curve \(\gamma \) lies on a plane containing the fixed line. In fact, Ohdera (2014) and Watanabe (2015) calculated some centroaffine invariants for such surfaces. However, in this paper, we will study surfaces of cohomogeneity one more generally.

As mentioned above, the condition that the mean curvature vanishes gives a strong restriction for rotation surfaces in Euclidean geometry. On the other hand, in centroaffine geometry it does not so, because the set of centroaffine minimal surfaces is enormous. In fact, it includes the set of centroaffine surfaces with vanishing Tchebychev vector field, which can be considered as proper affine spheres centered at the origin.

First, we check that such a surface is of cohomogeneity one if the centroaffine metric is flat (Example 3.2). Moreover, we can give a characterization of a surface of cohomogeneity one which has vanishing Tchebychev vector field and non-flat centroaffine metric (Theorems 5.1, 5.4). In order to obtain this characterization, we find a standard parametrization of such a surface in terms of asymptotic line coordinates or a holomorphic coordinate (Lemma 3.3).

Second, we classify centroaffine minimal surfaces of cohomogeneity one which have centroaffine metrics of constant curvature as follows. Using Lemma 3.3 again, we classify such surfaces into three categories, by considering if the curvature of the centroaffine metric equals zero or not, and if the affine support function depends only on one variable or not (Lemma 6.1). For each case, we present the explicit expressions of \(\gamma \) and A (Theorems 6.26.4, 6.7). Similarly, we classify centroaffine minimal surfaces of cohomogeneity one which have definite centroaffine metrics of constant curvature (Sect. 7).

2 Preliminaries

A centroaffine surface in the three-dimensional vector space \({\mathbf {R}}^3\) is given by an immersion \(f:M\rightarrow {\mathbf {R}}^3\) from a two-dimensional manifold M into \({\mathbf {R}}^3\) with a transversal vector field \(\xi \) defined as the restriction of the radial vector field to f:
$$\begin{aligned} \xi =-\left. \left( \sum _{i=1}^{3}x_i\frac{\partial }{\partial x_i}\right) \right| _{f}, \end{aligned}$$
where \((x_1, x_2, x_{3})\) are affine coordinates on \({\mathbf {R}}^{3}\). Then the Gauss-Weingarten formulas for f are given by
$$\begin{aligned} D_Xf_{*}Y=f_{*}\nabla _XY+h(X,Y)\xi ,\ D_X\xi =-f_{*}X \ \ (X,Y\in {\mathfrak {X}}(M)), \end{aligned}$$
where D is the standard flat connection on \({\mathbf {R}}^{3}\) and \({\mathfrak {X}}(M)\) is the set of vector fields on M. Note that \(\nabla \) defines a torsion-free affine connection on M, called the induced connection. On the other hand, h defines a symmetric (0, 2)-tensor field on M. We call f to be nondegenerate, definite or indefinite if h is nondegenerate, definite or indefinite, respectively.
If f is nondegenerate, we call h the centroaffine metric, and denote the Levi-Civita connection for h by \(\hat{\nabla }\). Then we have a (1, 2)-tensor field C, a vector field T and a (1, 1)-tensor field \({\mathcal {T}}\) on M defined by
$$\begin{aligned} C=\nabla -\hat{\nabla },\ T=\frac{1}{2}\text{ tr }_hC,\ {\mathcal {T}}=\hat{\nabla }T, \end{aligned}$$
called the difference tensor field, the Tchebychev vector field and the Tchebychev operator, respectively. It is known that \(T=0\) if and only if f is a proper affine sphere centered at the origin, i.e., the Blaschke normals of f meet at the origin regarding f as a Blaschke surface (Wang 1994, Example 1). In this paper, we call such a surface a proper affine sphere for short. Moreover, f is called to be centroaffine minimal if it is an extremal for the volume integral of h, which is equivalent to the condition that \(\text{ tr }\,\,{\mathcal {T}}=0\) (Wang 1994, Theorem 2). In particular, proper affine spheres are centroaffine minimal. We also note that the Pick function J and the square norm of the Tchebychev vector field \(\Vert T\Vert _h^2\) are given by
$$\begin{aligned} J=\frac{1}{2}h(C,C),\ \Vert T\Vert _h^2=h(T,T). \end{aligned}$$

3 Examples and the Normal Form

We shall begin with giving examples of centroaffine surfaces of cohomogeneity one.

Example 3.1

Surfaces of type (a), (b), (c) in Sect. 1 are written in the form (1.1) as follows:
$$\begin{aligned} (\text{ a })\ \gamma (x)&=(\psi (x), 0, x),\ A=\left( \begin{array}{ccc} 0 &{} 1 &{} 0 \\ -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{array}\right) , \\ (\text{ b })\ \gamma (x)&=(\psi (x), 0, x),\ A=\left( \begin{array}{ccc} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{array}\right) , \\ (\text{ c })\ \gamma (x)&=(x, 0, \psi (x)),\ A=\left( \begin{array}{ccc} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \end{array} \right) . \end{aligned}$$
We remark that \(A \in \mathfrak {sl}(3, \mathbf {R})\). On the other hand, a surface of type (d) is considered as
$$\begin{aligned}&\mathrm {pr}\left( (x, 0, \psi (x), 1) \left( \begin{array}{cccc} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} y &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} y &{} \frac{y^2}{2} &{} 1 \end{array} \right) \right) \\&\quad = \mathrm {pr}\left( (x, 0, \psi (x), 1) \exp \left( y \left( \begin{array}{cccc} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \end{array} \right) \right) \right) , \end{aligned}$$
where \(\mathrm {pr} : \mathbf {R}^4 \ni (x_1,x_2,x_3, x_4) \mapsto (x_1, x_2, x_3) \in \mathbf {R}^3\) is the standard projection. Such a surface is not of cohomogeneity one in centroaffine geometry.

Example 3.2

As is shown in (Liu and Wang 1995, Theorem 4.2), a nondegenerate flat centroaffine surface \(f=(X,Y,Z):M\rightarrow {\mathbf {R}}^3\) with vanishing Tchebychev operator is a piece of the following surfaces up to centroaffine congruence:
$$\begin{aligned}&(1)\ X^pY^qZ^r=1,\ pqr(p+q+r)\not =0,\\&(2)\ \left\{ \exp \left( -p\tan ^{-1}\frac{X}{Y}\right) \right\} (X^2+Y^2)^qZ^r=1,\ r(2q+r)(p^2+q^2)\not =0,\\&(3)\ Z=-X(p\log X+q\log Y),\ q(p+q)\not =0,\\&(4)\ Z=\pm X\log X+\frac{Y^2}{X},\ (5)\ f=(e^x,\psi _1(x)e^y,\psi _2(x)e^y), \end{aligned}$$
where \(p,q,r\in {\mathbf {R}}\) and \(\psi _1\) and \(\psi _2\) are linearly independent solutions to the differential equation: \(\psi ''-\psi '-\theta \psi =0\) for an arbitrary function \(\theta =\theta (x)\). See also (Furuhata and Vrancken 2006, Example 2.9).
All the above surfaces are of cohomogeneity one. Indeed, depending on the cases (1)–(5), we can choose \(\gamma \) and A in (1.1) up to centroaffine congruence as follows:
$$\begin{aligned} (1)&\ \gamma (x)=(e^x,1,e^{-\frac{p}{r}x}),\ A=\left( \begin{array}{ccc}0&{}0&{}0\\ 0&{}1&{}0\\ 0&{}0&{}-\frac{q}{r}\end{array}\right) ,\\ (2)&\ \gamma (x)=(0,x,x^{-\frac{2q}{r}}),\ A=\left( \begin{array}{ccc}0&{}-1&{}0\\ 1&{}0&{}0\\ 0&{}0&{}\frac{p}{r}\end{array}\right) ,\\ (3)&\ \gamma (x)=\left( e^x,\frac{q}{p+q}x,1\right) ,\ A=\left( \begin{array}{ccc}1&{}0&{}0\\ 0&{}1&{}0\\ 0&{}1&{}1\end{array}\right) ,\\ (4)&\ \gamma (x)=(x,\pm x^2 ,1),\ A=\left( \begin{array}{ccc}1&{}0&{}0\\ 0&{}1&{}0\\ 0&{}1&{}1\end{array}\right) ,\\ (5)&\ \gamma (x)=(e^x,\psi _1(x),\psi _2(x)),\ A=\left( \begin{array}{ccc}0&{}0&{}0\\ 0&{}1&{}0\\ 0&{}0&{}1\end{array}\right) . \end{aligned}$$
In particular, a flat proper affine sphere \(f=(X,Y,Z):M\rightarrow {\mathbf {R}}^3\) is of cohomogeneity one, since such f is a piece of the following surfaces up to centroaffine congruence:
$$\begin{aligned} (1)'\ XYZ=1,\ (2)'\ (X^2+Y^2)Z=1 \end{aligned}$$
(Magid and Ryan 1990, Theorem 2, 3).

We shall prove that any nondegenerate centroaffine surface of cohomogeneity one has the normal form in the following sense.

Lemma 3.3

Let \(f:M\rightarrow {\mathbf {R}}^3\) be a nondegenerate centroaffine surface with cohomogeneity one as in (1.1). If f is indefinite, then there exist asymptotic line coordinates (uv) with respect to the centroaffine metric h such that \(x=u+v\), \(y=u-v\), that is, we can express f by changing \(\gamma \) and A, if necessary, as
$$\begin{aligned} f(u,v)=\gamma (u+v)e^{(u-v)A} \end{aligned}$$
(3.1)
with \(h=2\varphi du dv\) for a non-vanishing function \(\varphi \).
If f is definite, then there exists a holomorphic coordinate z with respect to the centroaffine metric h such that \(x=z+\bar{z}\), \(y=\sqrt{-1}(z-\bar{z})\), that is, we can express f by changing \(\gamma \) and A, if necessary, as
$$\begin{aligned} f(z,\bar{z})=\gamma (z+\bar{z})e^{\sqrt{-1}(z-\bar{z})A} \end{aligned}$$
(3.2)
with \(h=2\varphi dz d\bar{z}\) for a non-vanishing function \(\varphi \).

Proof

First we consider the case that f is indefinite. Then changing \(\gamma \) in (1.1), if necessary, we can parametrize f as
$$\begin{aligned} f(u,v)=\gamma (u+v)e^{\alpha (u,v)A}, \end{aligned}$$
(3.3)
where (uv) are asymptotic line coordinates with respect to the centroaffine metric, and \(\alpha \) is a function of u and v. Differentiating (3.3) by u and v, we have
$$\begin{aligned} f_u=\gamma 'e^{\alpha A}+\gamma \alpha _uAe^{\alpha A},\ f_v=\gamma 'e^{\alpha A}+\gamma \alpha _vAe^{\alpha A} \end{aligned}$$
(3.4)
respectively. Since f is a centroaffine surface, from (3.3) and (3.4), we have
$$\begin{aligned} 0\not =\det \left( \begin{array}{c}f\\ f_u\\ f_v\end{array}\right) =(\alpha _u-\alpha _v)\det \left( \begin{array}{c}\gamma \\ \gamma A\\ \gamma '\end{array}\right) \det e^{\alpha A}, \end{aligned}$$
which is equivalent to
$$\begin{aligned} \alpha _u\not =\alpha _v,\ \det \left( \begin{array}{c}\gamma \\ \gamma A\\ \gamma '\end{array}\right) \not =0. \end{aligned}$$
(3.5)
Differentiating the first equation of (3.4) by u and v, we have
$$\begin{aligned} f_{uu}= & {} \gamma ''e^{\alpha A}+2\gamma '\alpha _uAe^{\alpha A}+\gamma \alpha _{uu}Ae^{\alpha A} +\gamma \alpha _u^2A^2e^{\alpha A}, \end{aligned}$$
(3.6)
$$\begin{aligned} f_{uv}= & {} \gamma ''e^{\alpha A}+\gamma '(\alpha _u+\alpha _v)Ae^{\alpha A} +\gamma \alpha _{uv}Ae^{\alpha A}+\gamma \alpha _u\alpha _vA^2e^{\alpha A} \end{aligned}$$
(3.7)
respectively. Since f is indefinite and (uv) are asymptotic line coordinates with respect to the centroaffine metric, we have
$$\begin{aligned} \det \left( \begin{array}{c}f_{uu}\\ f_u\\ f_v\end{array}\right) =0,\ \det \left( \begin{array}{c}f_{vv}\\ f_u\\ f_v\end{array}\right) =0,\ \det \left( \begin{array}{c}f_{uv}\\ f_u\\ f_v\end{array}\right) \not =0. \end{aligned}$$
(3.8)
From (3.4), the first equation of (3.5), (3.6) and the first equation of (3.8), we have
$$\begin{aligned} \det \left( \begin{array}{c}\gamma ''\\ \gamma A\\ \gamma '\end{array}\right) +2\alpha _u\det \left( \begin{array}{c}\gamma 'A\\ \gamma A\\ \gamma '\end{array}\right) +\alpha _u^2\det \left( \begin{array}{c}\gamma A^2\\ \gamma A\\ \gamma '\end{array}\right) =0. \end{aligned}$$
(3.9)
Similarly, we have
$$\begin{aligned} \det \left( \begin{array}{c}\gamma ''\\ \gamma A\\ \gamma '\end{array}\right) +2\alpha _v\det \left( \begin{array}{c}\gamma 'A\\ \gamma A\\ \gamma '\end{array}\right) +\alpha _v^2\det \left( \begin{array}{c}\gamma A^2\\ \gamma A\\ \gamma '\end{array}\right) =0. \end{aligned}$$
(3.10)
From (3.4), the first equation of (3.5), (3.7) and the third equation of (3.8), we have
$$\begin{aligned} \det \left( \begin{array}{c}\gamma ''\\ \gamma A\\ \gamma '\end{array}\right) +(\alpha _u+\alpha _v)\det \left( \begin{array}{c}\gamma 'A\\ \gamma A\\ \gamma '\end{array}\right) +\alpha _u\alpha _v\det \left( \begin{array}{c}\gamma A^2\\ \gamma A\\ \gamma '\end{array}\right) \not =0. \end{aligned}$$
(3.11)
From (3.9), (3.10) and (3.11), we have \(\det \left( \begin{array}{c}\gamma A^2\\ \gamma A\\ \gamma '\end{array}\right) \not =0\) so that
$$\begin{aligned} \alpha _u+\alpha _v= & {} -2\det \left( \begin{array}{c}\gamma 'A\\ \gamma A\\ \gamma '\end{array}\right) \Bigg / \det \left( \begin{array}{c}\gamma A^2\\ \gamma A\\ \gamma '\end{array}\right) , \end{aligned}$$
(3.12)
$$\begin{aligned} \alpha _u\alpha _v= & {} \det \left( \begin{array}{c}\gamma ''\\ \gamma A\\ \gamma '\end{array}\right) \Bigg / \det \left( \begin{array}{c}\gamma A^2\\ \gamma A\\ \gamma '\end{array}\right) . \end{aligned}$$
(3.13)
From (3.12), we have \(\alpha _{uu}=\alpha _{vv}\). Hence \(\alpha \) can be written as
$$\begin{aligned} \alpha (u,v)=\alpha _1(u+v)+\alpha _2(u-v), \end{aligned}$$
(3.14)
where \(\alpha _1\) and \(\alpha _2\) are functions of \(u+v\) and \(u-v\) only respectively. From (3.13) and (3.14), \(\alpha _2'\) should be constant, which implies that
$$\begin{aligned} f(u,v)=\gamma (u+v)e^{\{\alpha _1(u+v)+c_1(u-v)+c_2\}A}, \end{aligned}$$
where \(c_1\in {\mathbf {R}}{\setminus }\{0\}\), \(c_2\in {\mathbf {R}}\). Changing \(\gamma \) and A, if necessary, we have the normal form (3.1).

In case that f is definite, by similar argument as above, we have the normal form (3.2). \(\square \)

4 Gauss Formula for the Normal Form

For an indefinite centroaffine surface \(f:M\rightarrow {\mathbf {R}}^3\) with the centroaffine metric h, we can take asymptotic line coordinates (uv) and the Gauss formula becomes as follows (cf. Schief 2000, Theorem 1, Fujioka 2009):
$$\begin{aligned} \begin{aligned} f_{uu}&=\left( \dfrac{\varphi _u}{\varphi }+\rho _u\right) f_u+\dfrac{a}{\varphi }f_v,\ f_{vv}=\left( \dfrac{\varphi _v}{\varphi }+\rho _v\right) f_v+\dfrac{b}{\varphi }f_u,\\ f_{uv}&=-\varphi f+\rho _vf_u+\rho _uf_v \end{aligned} \end{aligned}$$
(4.1)
with the integrability conditions:
$$\begin{aligned} \begin{aligned} (\log \vert \varphi \vert )_{uv}&=-\varphi -\dfrac{ab}{\varphi ^2}+\rho _u\rho _v,\\ a_v+\rho _u\varphi _u&=\rho _{uu}\varphi ,\ b_u+\rho _v\varphi _v=\rho _{vv}\varphi , \end{aligned} \end{aligned}$$
(4.2)
where
$$\begin{aligned} \varphi= & {} h(\partial _u,\partial _v) =-\det \left( \begin{array}{c}f_{uv}\\ f_u\\ f_v\end{array}\right) \Bigg / \det \left( \begin{array}{c}f\\ f_u\\ f_v\end{array}\right) \not =0, \end{aligned}$$
(4.3)
$$\begin{aligned} a= & {} \varphi \det \left( \begin{array}{c}f\\ f_u\\ f_{uu}\end{array}\right) \Bigg / \det \left( \begin{array}{c}f\\ f_u\\ f_v\end{array}\right) ,\nonumber \\ b= & {} \varphi \det \left( \begin{array}{c}f\\ f_v\\ f_{vv}\end{array}\right) \Bigg / \det \left( \begin{array}{l}f\\ f_v\\ f_u\end{array}\right) \end{aligned}$$
(4.4)
and
$$\begin{aligned} \rho =-\frac{1}{4}\log \left( \det \left( \begin{array}{c}f_{uv}\\ f_u\\ f_v\end{array}\right) ^2\Bigg / \det \left( \begin{array}{c}f\\ f_u\\ f_v\end{array}\right) ^4\right) . \end{aligned}$$
(4.5)
Then, the difference tensor C, the Tchebychev vector field T and the Tchebychev operator \({\mathcal {T}}\) are computed as follows:
$$\begin{aligned} C(\partial _u,\partial _u)= & {} \rho _u\partial _u+\frac{a}{\varphi }\partial _v,\ C(\partial _v,\partial _v)=\frac{b}{\varphi }\partial _u+\rho _v\partial _v, \nonumber \\ C(\partial _u,\partial _v)= & {} \rho _v\partial _u+\rho _u\partial _v, \nonumber \\ T= & {} \frac{\rho _v}{\varphi }\partial _u +\frac{\rho _u}{\varphi }\partial _v,\ \end{aligned}$$
(4.6)
$$\begin{aligned} {\mathcal {T}}(\partial _u)= & {} \frac{\rho _{uv}}{\varphi }\partial _u +\frac{a_v}{\varphi ^2}\partial _v,\ {\mathcal {T}}(\partial _v) =\frac{b_u}{\varphi ^2}\partial _u +\frac{\rho _{uv}}{\varphi }\partial _v. \end{aligned}$$
(4.7)
Moreover, the Pick function J and the square norm of the Tchebychev vector field \(\Vert T\Vert _h^2\) are given by
$$\begin{aligned} J=\frac{3\rho _u\rho _v}{\varphi }+\frac{ab}{\varphi ^3},\ \Vert T\Vert _h^2=\frac{2\rho _u\rho _v}{\varphi }. \end{aligned}$$
We also note that the centroaffine curvature \(\kappa \), which is the curvature of h, is given by
$$\begin{aligned} \kappa =-\frac{(\log \vert \varphi \vert )_{uv}}{\varphi }. \end{aligned}$$
(4.8)
From (3.1), (4.3), (4.4) and (4.5), we have the following:

Proposition 4.1

Let \(f:M\rightarrow {\mathbf {R}}^3\) be an indefinite centroaffine surface of cohomogeneity one given in the normal form (3.1). Then the functions \(\varphi \), a and b as above depend only on \(u+v\), while
$$\begin{aligned} \rho (u,v)=\alpha (u+v)+\frac{1}{2}(u-v) \mathop {\mathrm {tr}} A, \end{aligned}$$
(4.9)
where \(\alpha \) is a function of \(u+v\) only.
On the other hand, for a definite centroaffine surface \(f:M\rightarrow {\mathbf {R}}^3\) with the centroaffine metric h, we can take a holomorphic coordinate z and the Gauss formula becomes as follows:
$$\begin{aligned} f_{zz}=\left( \dfrac{\varphi _z}{\varphi }+\rho _z\right) f_z+\dfrac{a}{\varphi }f_{\bar{z}},\ f_{z\bar{z}}=-\varphi f+\rho _{\bar{z}}f_z+\rho _zf_{\bar{z}} \end{aligned}$$
with the integrability conditions:
$$\begin{aligned} (\log \vert \varphi \vert )_{z\bar{z}}=-\varphi -\dfrac{\vert a\vert ^2}{\varphi ^2}+\vert \rho _z\vert ^2,\ a_{\bar{z}}+\rho _z\varphi _z=\rho _{zz}\varphi , \end{aligned}$$
where
$$\begin{aligned} \varphi= & {} h(\partial _z,\partial _{\bar{z}}) =-\det \left( \begin{array}{c}f_{z\bar{z}}\\ f_z\\ f_{\bar{z}}\end{array}\right) \Bigg / \det \left( \begin{array}{c}f\\ f_z\\ f_{\bar{z}}\end{array}\right) \not =0, \end{aligned}$$
(4.10)
$$\begin{aligned} a= & {} \varphi \det \left( \begin{array}{c}f\\ f_z\\ f_{zz}\end{array}\right) \Bigg / \det \left( \begin{array}{c}f\\ f_z\\ f_{\bar{z}}\end{array}\right) \end{aligned}$$
(4.11)
and
$$\begin{aligned} \rho =-\frac{1}{4}\log \left( -\det \left( \begin{array}{c}f_{z\bar{z}}\\ f_z\\ f_{\bar{z}}\end{array}\right) ^2\Bigg / \det \left( \begin{array}{c}f\\ f_z\\ f_{\bar{z}}\end{array}\right) ^4 \right) . \end{aligned}$$
Then the difference tensor C, the Tchebychev vector field T, the Tchebychev operator \({\mathcal {T}}\) and the centroaffine curvature \(\kappa \) are computed as follows:
$$\begin{aligned}&\displaystyle C(\partial _z,\partial _z) =\rho _z\partial _z+\frac{a}{\varphi }\partial _{\bar{z}},\ C(\partial _z,\partial _{\bar{z}}) =\rho _{\bar{z}}\partial _z+\rho _z\partial _{\bar{z}}, \\&\displaystyle T=\frac{\rho _{\bar{z}}}{\varphi }\partial _z +\frac{\rho _z}{\varphi }\partial _{\bar{z}},\ {\mathcal {T}}(\partial _z) =\frac{\rho _{z\bar{z}}}{\varphi }\partial _z +\frac{a_{\bar{z}}}{\varphi ^2}\partial _{\bar{z}},\ \kappa =-\frac{(\log \vert \varphi \vert )_{z\bar{z}}}{\varphi }. \end{aligned}$$
Moreover, the Pick function J and the square norm of the Tchebychev vector field \(\Vert T\Vert _h^2\) are given by
$$\begin{aligned} J=\frac{3\vert \rho _z\vert ^2}{\varphi }+\frac{\vert a\vert ^2}{\varphi ^3},\ \Vert T\Vert _h^2=\frac{2\vert \rho _z\vert ^2}{\varphi }. \end{aligned}$$
Similar to Proposition 4.1, we have the following:

Proposition 4.2

Let \(f:M\rightarrow {\mathbf {R}}^3\) be a definite centroaffine surface of cohomogeneity one given in the normal form (3.2). Then the functions \(\varphi \) and a above depend only on \(z+\bar{z}\), while
$$\begin{aligned} \rho (z,\bar{z})=\alpha (z+\bar{z})+\frac{\sqrt{-1}}{2}(z-\bar{z}) \mathop {\mathrm {tr}}A, \end{aligned}$$
where \(\alpha \) is a function of \(z+\bar{z}\) only.

5 Proper Affine Spheres

As stated in Example 3.2, flat proper affine spheres are of cohomogeneity one. In the following, we shall consider non-flat proper affine spheres of cohomogeneity one.

Theorem 5.1

Let \(f:M\rightarrow {\mathbf {R}}^3\) be a cohomogeneity one indefinite centroaffine surface. Take \(\gamma \) and A as in the normal form (3.1), and \(\varphi , a, b\) as in (4.3), (4.4). Suppose that f is a non-flat proper affine sphere. Then (1) ab are constant. (2) \(\varphi \) satisfies
$$\begin{aligned} (\varphi ')^2=-2\varphi ^3+c\varphi ^2+ab \end{aligned}$$
(5.1)
for some \(c\in {\mathbf {R}}\), and
$$\begin{aligned} 2\varphi '+a+b\not =0. \end{aligned}$$
(5.2)
(3) The minimal polynomial of A is
$$\begin{aligned} t^3-\frac{c}{4}t+\frac{a-b}{8}. \end{aligned}$$
(5.3)
(4) \(\gamma \) satisfies
$$\begin{aligned} (2\varphi '+a+b)\gamma '=\gamma \{4\varphi A^2+(a-b)A-2\varphi ^2E\} \end{aligned}$$
(5.4)
with
$$\begin{aligned} \det \left( \begin{array}{c}\gamma \\ \gamma A\\ \gamma A^2\end{array}\right) \not =0. \end{aligned}$$
(5.5)
Conversely, for given \(a, b, c \in {\mathbf {R}}\), take a function \(\varphi \) and \(A \in \mathfrak {gl}(3, \mathbf {R})\) with the properties (2) and (3). Find a curve \(\gamma \) satisfying the linear ordinary differential equation (5.4) with the condition (5.5), and set f as in (3.1). Then f is a cohomogeneity one indefinite non-flat proper affine sphere.

Proof

Since f is a proper affine sphere, the Tchebychev vector field vanishes, which is equivalent to that \(\rho \) is constant from (4.6). Then by Proposition 4.1 the Gauss formula (4.1) becomes
$$\begin{aligned} f_{uu}=\dfrac{\varphi '}{\varphi }f_u+\dfrac{a}{\varphi }f_v,\ f_{vv}=\dfrac{\varphi '}{\varphi }f_v+\dfrac{b}{\varphi }f_u,\ f_{uv}=-\varphi f, \end{aligned}$$
(5.6)
where \(\varphi \), a and b are functions of \(u+v\) only. Moreover, the integrability conditions (4.2) become
$$\begin{aligned} (\log \vert \varphi \vert )''=-\varphi -\dfrac{ab}{\varphi ^2},\ a'=0,\ b'=0, \end{aligned}$$
(5.7)
which implies that a and b are constant, so that the first equation of (5.7) can be integrated once as (5.1) for some \(c\in {\mathbf {R}}\).
Substituting (3.1) into (5.6), we have the Gauss formula:
$$\begin{aligned} \begin{aligned} \gamma ''+2\gamma 'A+\gamma A^2&=\dfrac{\varphi '}{\varphi }(\gamma '+\gamma A)+\dfrac{a}{\varphi }(\gamma '-\gamma A),\\ \gamma ''-2\gamma 'A+\gamma A^2&=\dfrac{\varphi '}{\varphi }(\gamma '-\gamma A)+\dfrac{b}{\varphi }(\gamma '+\gamma A),\\ \gamma ''-\gamma A^2&=-\varphi \gamma , \end{aligned} \end{aligned}$$
which is equivalent to (5.4) and
$$\begin{aligned} \gamma '\left\{ 4\varphi A-(a-b)E\right\} =(2\varphi '-a-b)\gamma A,\ \gamma ''=\gamma (A^2-\varphi E). \end{aligned}$$
(5.8)
From (5.4) and the first equation of (5.8), we have
$$\begin{aligned}&\gamma \left\{ 4\varphi A-(a-b)E\right\} \left\{ 4\varphi A^2+(a-b)A-2\varphi ^2E\right\} \\&\quad =(2\varphi '+a+b)(2\varphi '-a-b) \gamma A, \end{aligned}$$
which is equivalent to
$$\begin{aligned} 8A^3-2cA+(a-b)E=O \end{aligned}$$
(5.9)
from (5.1) and the second equation of (3.5). Since a and b are constant and f is non-flat, i.e., the centroaffine curvature \(\kappa \not =0\), from (4.8) we have (5.2). Note that from the second equation of (3.5), (5.4) and (5.2), we have (5.5). Hence (5.3) is the minimal polynomial of A. This completes the proof of the first half.
To prove the latter half, from the above argument, we have only to check that (5.4) restores the Gauss formula, namely (5.8). The first equation of (5.8) can be deduced from (5.4) under the condition (5.9). Moreover, from (5.4) we have
$$\begin{aligned} \begin{aligned} \gamma ''&=-\frac{2\varphi ''}{(2\varphi '+a+b)^2}\gamma \left\{ 4\varphi A^2+(a-b)A-2\varphi ^2E\right\} \\&\quad +\frac{1}{(2\varphi '+a+b)^2}\gamma \left\{ 4\varphi A^2+(a-b)A-2\varphi ^2E\right\} ^2\\&\quad +\frac{1}{2\varphi '+a+b}\gamma (4\varphi 'A^2-4\varphi \varphi ' E). \end{aligned} \end{aligned}$$
(5.10)
On the other hand, from (5.1), we have
$$\begin{aligned} \varphi ''=c\varphi -3\varphi ^2. \end{aligned}$$
(5.11)
From (5.1), (5.9) and (5.11), a direct computation shows that (5.10) is equivalent to the second equation of (5.8). \(\square \)

Remark 5.2

For a cohomogeneity one indefinite non-flat proper affine sphere \(f(u,v)=\gamma (u+v) e^{(u-v)A}\), \(\{e^{yA}\}_{y \in \mathbf {R}}\) is a one-parameter subgroup of the special linear group \(SL(3, \mathbf {R})\). Moreover, 0 is an eigenvalue of A if and only if \(a=b\), in which case f is an equiaffine rotation surface. Indeed, from (5.9), we have that \(\mathrm {tr} A=0\) and \(\det A=-(a-b)/8\).

Yang et al. (2014) classified equiaffine-rotational centroaffine surfaces with vanishing Pick function as well as centroaffine minimal surfaces with constant square norm of the Tchebychev vector field.

Remark 5.3

If \(A \in \mathfrak {gl}(3, \mathbf {R})\) satisfies the condition (3) in Theorem 5.1 and has a curve \(\gamma \) with (5.5), then it is given as one of the followings up to similarity:
$$\begin{aligned} \begin{aligned}&(\text{ i })\ \left( \begin{array}{ccc}p&{}0&{}0\\ 0&{}q&{}0\\ 0&{}0&{}r\end{array}\right) ,\ p+q+r=0,\ (p-q)(q-r)(r-p)\not =0,\\&(\text{ ii })\ \left( \begin{array}{ccc}-2q&{}0&{}0\\ 0&{}q&{}r\\ 0&{}-r&{}q\end{array}\right) ,\ r \not =0,\ (\text{ iii })\ \left( \begin{array}{ccc}-2q&{}0&{}0\\ 0&{}q&{}0\\ 0&{}1&{}q\end{array}\right) ,\ q \not =0,\\&(\text{ iv })\ \left( \begin{array}{ccc}0&{}0&{}0\\ 1&{}0&{}0\\ 0&{}1&{}0\end{array}\right) , \end{aligned} \end{aligned}$$
where \(p,q,r\in {\mathbf {R}}\). In particular, if A has an eigenvalue 0, then it is given by (iv) or one of the followings:
$$\begin{aligned} (\text{ i })'\ \left( \begin{array}{ccc}0&{}0&{}0\\ 0&{}q&{}0\\ 0&{}0&{}-q\end{array}\right) ,\ q\not =0,\ (\text{ ii })'\ \left( \begin{array}{ccc}0&{}0&{}0\\ 0&{}0&{}r\\ 0&{}-r&{}0\end{array}\right) ,\ r\not =0. \end{aligned}$$
In the case that A is as in (i), from (5.5) for \(\gamma =(\gamma _1, \gamma _2, \gamma _3)\), we have
$$\begin{aligned} 0 \ne \det \left( \begin{array}{c}\gamma \\ \gamma A\\ \gamma A^2\end{array}\right) =\gamma _1 \gamma _2 \gamma _3 \det \left( \begin{array}{ccc} 1 &{} 1 &{} 1 \\ p &{} q &{} r \\ p^2 &{} q^2 &{} r^2 \end{array} \right) , \end{aligned}$$
which implies the second condition in (i). Similarly, we obtain the condition for each case.

Considering the definite case, similar to Theorem 5.1, we have the following:

Theorem 5.4

Let \(f:M\rightarrow {\mathbf {R}}^3\) be a cohomogeneity one definite centroaffine surface. Take \(\gamma \) and A as in the normal form (3.2), and \(\varphi , a\) as in (4.10), (4.11). Suppose that f is a non-flat proper affine sphere. Then (1) a is constant. (2) \(\varphi \) satisfies
$$\begin{aligned} (\varphi ')^2=-2\varphi ^3+c\varphi ^2+\vert a\vert ^2 \end{aligned}$$
(5.12)
for some \(c\in {\mathbf {R}}\), and
$$\begin{aligned} 2\varphi '+a+\bar{a}\not =0. \end{aligned}$$
(5.13)
(3) The minimal polynomial of A is
$$\begin{aligned} t^3+\frac{c}{4}t+\frac{\sqrt{-1}(a-\bar{a})}{8}. \end{aligned}$$
(5.14)
(4) \(\gamma \) satisfies
$$\begin{aligned} (2\varphi '+a+\bar{a})\gamma '=\gamma \{-4\varphi A^2+\sqrt{-1}(a-\bar{a})A-2\varphi ^2E\} \end{aligned}$$
(5.15)
with (5.5).

Conversely, for given \(a \in {\mathbf {C}}\) and \(c \in {\mathbf {R}}\), take a function \(\varphi \) and \(A \in \mathfrak {gl}(3, \mathbf {R})\) with the properties (2) and (3). Find a curve \(\gamma \) satisfying the linear ordinary differential equation (5.15) with the condition (5.5), and set f as in (3.2). Then f is a cohomogeneity one definite non-flat proper affine sphere.

Remark 5.5

Similar to Remark 5.2, in Theorem 5.4, f is an equiaffine rotation surface if and only if \(a\in {\mathbf {R}}\).

6 Indefinite Centroaffine Minimal Surfaces of Constant Curvature

In this section, we use the same notation as in §4.

Lemma 6.1

Let \(f:M\rightarrow {\mathbf {R}}^3\) be a cohomogeneity one indefinite centroaffine minimal surface of constant curvature which is given by the normal form (3.1). Then changing u, v, if necessary, we have one of the followings:
$$\begin{aligned} \mathrm {(I)}&\ \kappa =0,\ \varphi \in {\mathbf {R}}{\setminus }\{0\},\ \rho =c_1u+c_2v+c_3,\ a,b\in {\mathbf {R}},\\ \mathrm {(II)}&\ \kappa =1,\ \rho =c_1u+c_3,\ a=-c_1\varphi +c_4,\ b=0,\\ \mathrm {(III)}&\ \kappa =1,\ \rho =c_1u+c_2v+c_3,\ a=-c_1\varphi ,\ b=-c_2\varphi ,\ c_1,c_2\not =0, \end{aligned}$$
where \(c_1,\dots ,c_4\in {\mathbf {R}}\).

Proof

Note that \(\varphi \), a and b are functions of \(u+v\) only by Proposition 4.1. Moreover, since f is centroaffine minimal, i.e., \(\text{ tr } {\mathcal {T}}=0\), we have \(\alpha ''=0\) from (4.7) and (4.9). Hence there exist \(c_1,c_2,c_3\in {\mathbf {R}}\) such that
$$\begin{aligned} \rho (u,v)=c_1u+c_2v+c_3. \end{aligned}$$
Then from (4.8), the integrability conditions (4.2) become
$$\begin{aligned} (\kappa -1)\varphi -\dfrac{ab}{\varphi ^2}+c_1c_2=0,\ a'+c_1\varphi '=0,\ b'+c_2\varphi '=0. \end{aligned}$$
(6.1)
From the second and the third equations of (6.1), there exist \(c_4,c_5\in {\mathbf {R}}\) such that
$$\begin{aligned} a=-c_1\varphi +c_4,\ b=-c_2\varphi +c_5. \end{aligned}$$
(6.2)
Substituting (6.2) into the first equation of (6.1), we have
$$\begin{aligned} (\kappa -1)\varphi ^3+(c_1c_5+c_2c_4)\varphi -c_4c_5=0. \end{aligned}$$
(6.3)
In the case that \(\kappa \not =1\), from (6.3) we have \(\varphi \in {\mathbf {R}}{\setminus }\{0\}\), which implies that \(\kappa =0\) from (4.8), and \(a,b\in {\mathbf {R}}\) from (6.2). Hence we have the case (I).
In the case that \(\kappa =1\), from (6.3) we have
$$\begin{aligned} c_1c_5+c_2c_4=0,\ c_4c_5=0 \end{aligned}$$
(6.4)
since \(\varphi \) is not constant. Then exchanging u with v, if necessary, we have \(c_5=0\) from the second equation of (6.4) so that \(c_2=0\) or \(c_4=0\) from the first equation of (6.4). If \(c_2=0\), we have the case (II). If \(c_4=0\), from (6.2) we have \(a=-c_1\varphi \), \(b=-c_2\varphi \). Since the case that \(c_1c_2=0\) can be deduced to the case (II), we may assume that \(c_1c_2\not =0\). Hence we have the case (III). \(\square \)

In the case (I) in Lemma 6.1, we have the following:

Theorem 6.2

Let \(f:M\rightarrow {\mathbf {R}}^3\) be a cohomogeneity one indefinite centroaffine minimal surface which belongs to the case (I) in Lemma 6.1. Then f is a flat centroaffine surface with vanishing Tchebychev operator as in Example 3.2 which is indefinite.

Proof

It is obvious from (4.7) and (4.8). \(\square \)

We proceed to the cases (II), (III) in Lemma 6.1.

Lemma 6.3

Let \(\varphi \) be the function in (4.1) for a cohomogeneity one indefinite centroaffine minimal surface with \(\kappa =1\). Then there exists \(c \in \mathbf {R}\) such that
$$\begin{aligned} (\varphi ')^2=-2\varphi ^3+c\varphi ^2. \end{aligned}$$
(6.5)
Furthermore, \(\varphi \) is given as
$$\begin{aligned} \varphi (s)= {\left\{ \begin{array}{ll} \frac{c}{2\cosh ^2\frac{\sqrt{c}}{2}s},\ c>0,\\ -\frac{2}{s^2},\ c=0,\\ \frac{c}{2\cos ^2\frac{\sqrt{-c}}{2}s},\ c<0 \end{array}\right. } \end{aligned}$$
up to translation of the variable s.

Proof

From (4.8), we have
$$\begin{aligned} \left( \frac{\varphi '}{\varphi }\right) '=-\varphi , \end{aligned}$$
(6.6)
which implies (6.5). \(\square \)

It is known that an indefinite centroaffine minimal surface is ruled if the curvature \(\kappa =1\) and the Pick function \(J=0\) (Fujioka 2009, Remark 5.3). If we consider the case (II) in Lemma 6.1, we have the following:

Theorem 6.4

Let \(f:M\rightarrow {\mathbf {R}}^3\) be a cohomogeneity one indefinite centroaffine minimal surface which belongs to the case (II) in Lemma 6.1. Then we can choose \(\gamma \) and A in (1.1) up to centroaffine congruence as follows:
$$\begin{aligned} \mathrm {(i)}&\ \gamma (x)=(x-p,x-q,x-r),\ A=\left( \begin{array}{ccc}p&{}0&{}0\\ 0&{}q&{}0\\ 0&{}0&{}r\end{array}\right) ,\\&\ (p-q)(q-r)(r-p)\not =0,\\ \mathrm {(ii)}&\ \gamma (x)=(x-p,x-q,-r),\ A=\left( \begin{array}{ccc}p&{}0&{}0\\ 0&{}q&{}r\\ 0&{}-r&{}q\end{array}\right) ,\ r\not =0,\\ \mathrm {(iii)}&\ \gamma (x)=(x-p,-1,x-q),\ A=\left( \begin{array}{ccc}p&{}0&{}0\\ 0&{}q&{}0\\ 0&{}1&{}q\end{array}\right) ,\ p\not =q,\\ \mathrm {(iv)}&\ \gamma (x)=(0,-1,x-p),\ A=\left( \begin{array}{ccc}p&{}0&{}0\\ 1&{}p&{}0\\ 0&{}1&{}p\end{array}\right) , \end{aligned}$$
where \(p,q,r\in {\mathbf {R}}\).

Proof

By Lemma 3.3 we can use the normal form (3.1). By Proposition 4.1 and Lemma 6.1, from (4.1), we have
$$\begin{aligned} f_{uu}=\left( \dfrac{\varphi '}{\varphi }+c_1\right) f_u+\frac{-c_1\varphi +c_4}{\varphi }f_v,\ f_{vv}=\dfrac{\varphi '}{\varphi }f_v,\ f_{uv}=-\varphi f+c_1f_v . \end{aligned}$$
(6.7)
From the second equation of (6.7), we have
$$\begin{aligned} f_v=\varphi \Psi (u), \end{aligned}$$
(6.8)
where \(\Psi \) is an \({\mathbf {R}}^3\)-valued function of u only. Substituting (6.8) into the third equation of (6.7), we have
$$\begin{aligned} f=\left( c_1-\frac{\varphi '}{\varphi }\right) \Psi -\Psi '. \end{aligned}$$
(6.9)
Differentiating (6.9) by u, we have
$$\begin{aligned} f_u=\varphi \Psi +\left( c_1-\frac{\varphi '}{\varphi }\right) \Psi '-\Psi '' \end{aligned}$$
(6.10)
from (6.6). Differentiating (6.10) by u, we have
$$\begin{aligned}&\varphi '\Psi +\varphi \Psi '+\varphi \Psi '+\left( c_1-\frac{\varphi '}{\varphi }\right) \Psi ''-\Psi '''\\&\quad =\left( \frac{\varphi '}{\varphi }+c_1\right) \left\{ \varphi \Psi +\left( c_1-\frac{\varphi '}{\varphi }\right) \Psi ' -\Psi ''\right\} + \left( -c_1\varphi +c_4\right) \Psi \end{aligned}$$
from the first equation of (6.7), (6.6), (6.8) and (6.10), which is equivalent to
$$\begin{aligned} \Psi '''-2c_1\Psi ''+\left\{ c_1^2-\left( \frac{\varphi '}{\varphi }\right) ^2- 2\varphi \right\} \Psi '+c_4\Psi =0. \end{aligned}$$
(6.11)
Note that from (6.6) the coefficient of \(\Psi '\) in the left hand side of (6.11) is constant. Hence if we put \(x=c_1-\frac{\varphi '}{\varphi }\), \(y=u\), depending on the roots of the characteristic equation of (6.11), we have the cases (i)–(iv). \(\square \)

It is known that if \(f:M\rightarrow {\mathbf {R}}^3\) is a proper affine sphere of constant curvature \(\kappa \), then \(\kappa =0,1\). In particular, in the case that \(\kappa =1\), f is a ruled surface (Simon 1991).

Corollary 6.5

Let \(f:M\rightarrow {\mathbf {R}}^3\) be a cohomogeneity one indefinite proper affine sphere with \(\kappa =1\). Then we can choose \(\gamma \) and A in (1.1) up to centroaffine congruence as follows:
$$\begin{aligned} \mathrm {(i)}&\ \gamma (x)=(x-p,x-q,x-r),\ A=\left( \begin{array}{ccc}p&{}0&{}0\\ 0&{}q&{}0\\ 0&{}0&{}r\end{array}\right) ,\\&\ p+q+r=0,\ (p-q)(q-r)(r-p)\not =0,\\ \mathrm {(ii)}&\ \gamma (x)=(x+2q,x-q,-r),\ A=\left( \begin{array}{ccc}-2q&{}0&{}0\\ 0&{}q&{}r\\ 0&{}-r&{}q\end{array}\right) ,\ r\not =0,\\ \mathrm {(iii)}&\ \gamma (x)=(x+2q,-1,x-q),\ A=\left( \begin{array}{ccc}-2q&{}0&{}0\\ 0&{}q&{}0\\ 0&{}1&{}q\end{array}\right) ,\ q\not =0,\\ \mathrm {(iv)}&\ \gamma (x)=(0,-1,x),\ A=\left( \begin{array}{ccc}0&{}0&{}0\\ 1&{}0&{}0\\ 0&{}1&{}0\end{array}\right) , \end{aligned}$$
where \(p,q,r\in {\mathbf {R}}\).

Proof

In Lemma 6.1, if \(\kappa =1\) and f is a proper affine sphere, then we have the case (ii) with \(c_1=0\). Then from (6.11), we have \(\text{ tr }\,A=0\) in (i)–(iv) in Theorem 6.4. \(\square \)

Remark 6.6

An equiaffine rotation surface in the list of Corollary 6.5 is a piece of a hyperboloid of one sheet (Fig. 1). Indeed, we note that 0 is an eigenvalue of A if and only if \(c_4=0\), which implies that f is a quadric.
Fig. 1

(1) hyperboloid constructed as (i) of Corollary 6.5; (2) hyperboloid constructed as (ii) of Corollary 6.5; (3) hyperboloid constructed as (iv) of Corollary 6.5

As stated in Remark 5.2, f in Theorem 6.4 is an equiaffine rotation surface if \(\mathrm {tr} A=\det A =0\). For example, if \(p=0, \ q=1/5, \ r=1\) in (ii), we get a centroaffine rotation surface (Fig. 2) but not equiaffine rotational, which is a deformation of Fig. 1(2).

Fig. 2

(ii) of Theorem 6.4

Theorem 6.7

Let \(f:M\rightarrow {\mathbf {R}}^3\) be a cohomogeneity one indefinite centroaffine minimal surface which belongs to the case (III) in Lemma 6.1. Set a function \(\lambda \) for \(c_1, c_2, c \in \mathbf {R}\) in Lemmas 6.1 and 6.3 as
$$\begin{aligned} \lambda (s)= {\left\{ \begin{array}{ll} c_1+c_2+\sqrt{c}\tanh \frac{\sqrt{c}}{2}s, \ c>0,\\ c_1+c_2+\frac{2}{s}, \ c=0,\\ c_1+c_2-\sqrt{-c}\tan \frac{\sqrt{-c}}{2}s, \ c<0. \end{array}\right. } \end{aligned}$$
(6.12)
Then in the normal form (3.1) we can choose \(\gamma \) and A up to centroaffine congruence as follows:
\(\mathrm {(i)}\)
\(c>4c_1c_2\), \(c\not =(c_1+c_2)^2\),
$$\begin{aligned} \begin{aligned} \gamma (s)&= {\left\{ \begin{array}{ll} \left( \lambda (s),\frac{e^{\frac{c_1+c_2}{2}s}}{\cosh \frac{\sqrt{c}}{2}s},\frac{e^{\frac{c_1+c_2}{2}s}}{\cosh \frac{\sqrt{c}}{2}s}\right) ,\ c>0,\\ \left( \lambda (s),\frac{e^{\frac{c_1+c_2}{2}s}}{s},\frac{e^{\frac{c_1+c_2}{2}s}}{s}\right) ,\ c=0,\\ \left( \lambda (s),\frac{e^{\frac{c_1+c_2}{2}s}}{\cos \frac{\sqrt{-c}}{2}s},\frac{e^{\frac{c_1+c_2}{2}s}}{\cos \frac{\sqrt{-c}}{2}s}\right) ,\ c<0,\ \end{array}\right. }\\ A&=\left( \begin{array}{ccc}0&{}0&{}0\\ 0&{}\frac{c_1-c_2+\sqrt{c-4c_1c_2}}{2}&{}0\\ 0&{}0&{}\frac{c_1-c_2-\sqrt{c-4c_1c_2}}{2}\end{array}\right) , \end{aligned} \end{aligned}$$
\(\mathrm {(ii)}\)
\(c<4c_1c_2\), \(\gamma \) is defined as in the case \(\mathrm {(i)}\), and
$$\begin{aligned} A=\left( \begin{array}{ccc}0&{}0&{}0\\ 0&{}\frac{c_1-c_2}{2}&{}-\frac{\sqrt{4c_1c_2-c}}{2}\\ 0&{}\frac{\sqrt{4c_1c_2-c}}{2}&{}\frac{c_1-c_2}{2}\end{array}\right) , \end{aligned}$$
\((\mathrm {iii})_1\)
\(c=4c_1c_2\), \(c_1\not =c_2\), \(\gamma \) is defined as in the case \(\mathrm {(i)}\), and
$$\begin{aligned} A=\left( \begin{array}{ccc}0&{}0&{}0\\ 0&{}\frac{c_1-c_2}{2}&{}0\\ 0&{}1&{}\frac{c_1-c_2}{2}\end{array}\right) , \end{aligned}$$
\((\mathrm {iii})_2\)
\(c=(c_1+c_2)^2\), \(c_1\not =c_2\),
$$\begin{aligned} \begin{aligned} \gamma (s)&= {\left\{ \begin{array}{ll} \lambda (s)\left( \frac{c_1^2-c_2^2}{c}\left( s+\frac{1}{\sqrt{c}} \sinh \sqrt{c}s\right) -\frac{2(c_1-c_2)}{c}\cosh ^2\frac{\sqrt{c}}{2}s,1,1\right) ,\ c>0,\\ \lambda (s)\left( \frac{c_1-c_2}{2}s^2,1,1\right) ,\ c=0,\\ \end{array}\right. }\\ A&=\left( \begin{array}{ccc}0&{}0&{}0\\ 1&{}0&{}0\\ 0&{}0&{}c_1-c_2\end{array}\right) , \end{aligned} \end{aligned}$$
\(\mathrm {(iv)}\)
\(c=(c_1+c_2)^2\), \(c_1=c_2\),
$$\begin{aligned} \begin{aligned} \gamma (s)&= {\left\{ \begin{array}{ll} \lambda (s)\left( \frac{2}{c}\cosh ^2\frac{\sqrt{c}}{2}s-\frac{2c_1}{c}\left( s+\frac{1}{\sqrt{c}} \sinh \sqrt{c}s\right) ,1,1\right) ,\ c>0,\\ \lambda (s)\left( -\frac{1}{2}s^2,1,1\right) ,\ c=0,\\ \end{array}\right. }\\ A&=\left( \begin{array}{ccc}0&{}0&{}0\\ 1&{}0&{}0\\ 0&{}1&{}0\end{array}\right) . \end{aligned} \end{aligned}$$

Proof

By Lemma 3.3 we can use the normal form (3.1). By Proposition 4.1 and Lemma 6.1, from (4.1), we have
$$\begin{aligned} \begin{aligned} f_{uu}&=\left( \dfrac{\varphi '}{\varphi }+c_1\right) f_u-c_1f_v,\ f_{vv}=\left( \dfrac{\varphi '}{\varphi }+c_2\right) f_v-c_2f_u,\\ f_{uv}&=-\varphi f+c_2f_u+c_1f_v . \end{aligned} \end{aligned}$$
(6.13)
Substituting (3.1) into (6.13), we have
$$\begin{aligned} \begin{aligned} \gamma ''+2\gamma 'A+\gamma A^2&=\dfrac{\varphi '}{\varphi }(\gamma '+\gamma A)+2c_1\gamma A,\\ \gamma ''-2\gamma 'A+\gamma A^2&=\dfrac{\varphi '}{\varphi }(\gamma '-\gamma A)-2c_2\gamma A,\\ \gamma ''-\gamma A^2&=-\varphi \gamma +(c_1+c_2)\gamma '-(c_1-c_2)\gamma A, \end{aligned} \end{aligned}$$
which is equivalent to
$$\begin{aligned} \left\{ \frac{\varphi '}{\varphi }-(c_1+c_2)\right\} \gamma '= & {} \gamma \{2A^2-2(c_1-c_2)A-\varphi E\},\nonumber \\ 2\gamma 'A= & {} \left( \frac{\varphi '}{\varphi }+c_1+c_2\right) \gamma A, 2\gamma ''=\left( \frac{\varphi '}{\varphi }+c_1+c_2\right) \gamma '-\varphi \gamma .\nonumber \\ \end{aligned}$$
(6.14)
Then similar argument to the proof of Theorem 5.1 shows that the second and the third equations of (6.14) can be deduced from the first equation of (6.14), and we have (5.5).
Moreover, the minimal polynomial of A is given by
$$\begin{aligned} t^3-(c_1-c_2)t^2+\frac{1}{4}\{(c_1+c_2)^2-c\}t, \end{aligned}$$
(6.15)
which is classified into the following five cases: it has (i) distinct real roots, (ii) one real root and two non-real complex conjugate roots, \((\mathrm {iii})_1\) two distinct real roots (zero is a single root), \((\mathrm {iii})_2\) two distinct real roots (zero is a multiple root), (iv) one multiple root. Up to similarity A is given as in (i)–(iv). Then it is straightforward to see that \(\gamma \) given as in (i)–(iv) satisfies (6.14) and (5.5). \(\square \)

Remark 6.8

In Theorem 6.7, \(\text{ tr }\,A=0\) if and only if \(c_1=c_2\), in which case f is an equiaffine rotation surface, and up to similarity we have (iv) and the followings:
\(\mathrm {(i)}'\)
  \(c>4c_1^2\),
$$\begin{aligned} \begin{aligned} \gamma (s)= \left( \lambda (s),\frac{e^{c_1s}}{\cosh \frac{\sqrt{c}}{2}s},\frac{e^{c_1s}}{\cosh \frac{\sqrt{c}}{2}s}\right) ,\ A=\left( \begin{array}{ccc}0&{}0&{}0\\ 0&{}\frac{\sqrt{c-4c_1^2}}{2}&{}0\\ 0&{}0&{}-\frac{\sqrt{c-4c_1^2}}{2}\end{array}\right) , \end{aligned} \end{aligned}$$
\(\mathrm {(ii)}'\)
  \(c<4c_1^2\),
$$\begin{aligned} \begin{aligned} \gamma (s)&= {\left\{ \begin{array}{ll} \left( \lambda (s),\frac{e^{c_1s}}{\cosh \frac{\sqrt{c}}{2}s}, \frac{e^{c_1s}}{\cosh \frac{\sqrt{c}}{2}s} \right) ,\ c>0,\\ \left( \lambda (s),\frac{e^{c_1s}}{s}, \frac{e^{c_1s}}{s} \right) ,\ c=0,\\ \left( \lambda (s),\frac{e^{c_1s}}{\cos \frac{\sqrt{-c}}{2}s}, \frac{e^{c_1s}}{\cos \frac{\sqrt{-c}}{2}s} \right) ,\ c<0, \end{array}\right. }\ A&=\left( \begin{array}{ccc}0&{}0&{}0\\ 0&{}0&{}-\frac{\sqrt{4c_1^2-c}}{2}\\ 0&{}\frac{\sqrt{4c_1^2-c}}{2}&{}0\end{array}\right) . \end{aligned} \end{aligned}$$

Images of orbits of type (i), (ii) and (iv) in Theorems 6.4 and 6.7 can be observed in Figs. 1 and 2. To understand ones of type (iii), see Fig. 3. If \(c_1=1, \ c_2=0\) in \((\mathrm {iii})_1\) of Theorem 6.7, we get a centroaffine rotation surface \(x^{-1} (x+2, (y+2) e^{(x+y)/2}, e^{(x+y)/ 2})\). If \(c_1=1, \ c_2=-1\) in \((\mathrm {iii})_2\), we get \(2x^{-1}(x^2+y, 1, e^{2y})\).

Fujioka classified indefinite centroaffine minimal surfaces such that \(a=b\) and \(\kappa \) is constant, and obtained the surface given by \(\mathrm {(ii)}'\) with \(c=0\) (Fujioka 2006, Theorem 2.4), although the other surfaces are dropped.

Fig. 3

\(\mathbf{a}\,\, x^{-1}\big (x+2, (y+2)e^{(x+y)/2}, e^{(x+y)/2}\big ); \mathbf{b}\,\, 2x^{-1}\big (x^{2}+y, 1, e^{2y}\big )\)

7 Definite Cohomogeneity One Centroaffine Minimal Surfaces of Constant Curvature

Similar to Lemma 6.1, we have the following:

Lemma 7.1

Let \(f:M\rightarrow {\mathbf {R}}^3\) be a cohomogeneity one definite centroaffine minimal surface of constant curvature which is given by the normal form (3.2). Then changing z, \(\gamma \) and A, if necessary, we have one of the followings:
$$\begin{aligned} \mathrm {(I)}&\ \kappa =0,\ \varphi \in {\mathbf {R}}{\setminus }\{0\},\ \rho =c_1z+\bar{c}_1\bar{z}+c_3,\ a\in {\mathbf {C}},\\ \mathrm {(II)}&\ \kappa =1,\ \rho =c_3,\ a=0,\\ \mathrm {(III)}&\ \kappa =1,\ \rho =c_1z+\bar{c}_1\bar{z}+c_3,\ a=-c_1\varphi ,\ c_1\not =0, \end{aligned}$$
where \(c_1\in {\mathbf {C}}\), \(c_3\in {\mathbf {R}}\).

Then similar to Sect. 6, we have the following:

Theorem 7.2

Let \(f:M\rightarrow {\mathbf {R}}^3\) be a cohomogeneity one definite centroaffine minimal surface. Then one of the following holds: (I) f is a flat centroaffine surface with vanishing Tchebychev operator as in Example 3.2 which is definite.(II) f is a piece of an ellipsoid. (III) In the normal form (3.2), we can choose \(\gamma \) and A up to centroaffine congruence as follows:
\(\mathrm {(i)}\)
\(4\vert c_1\vert ^2>c\), \(c\not =(c_1+\bar{c}_1)^2\),
$$\begin{aligned} \begin{aligned} \gamma (s)&= {\left\{ \begin{array}{ll} \left( \lambda (s),\frac{e^{\frac{c_1+\bar{c}_1}{2}s}}{\cosh \frac{\sqrt{c}}{2}s},\frac{e^{\frac{c_1+\bar{c}_1}{2}s}}{\cosh \frac{\sqrt{c}}{2}s}\right) , c>0,\\ \left( \lambda (s),\frac{e^{\frac{c_1+\bar{c}_1}{2}s}}{s},\frac{e^{\frac{c_1+\bar{c}_1}{2}s}}{s}\right) ,\ c=0,\\ \left( \lambda (s),\frac{e^{\frac{c_1+\bar{c}_1}{2}s}}{\cos \frac{\sqrt{-c}}{2}s},\frac{e^{\frac{c_1+\bar{c}_1}{2}s}}{\cos \frac{\sqrt{-c}}{2}s}\right) ,\ c<0,\ \end{array}\right. }\\ A&=\left( \begin{array}{ccc}0&{}0&{}0\\ 0&{}\frac{-\sqrt{-1}(c_1-\bar{c}_1)+\sqrt{4\vert c_1\vert ^2-c}}{2}&{}0\\ 0&{}0&{}\frac{-\sqrt{-1}(c_1-\bar{c}_1)-\sqrt{4\vert c_1\vert ^2-c}}{2}\end{array}\right) , \end{aligned} \end{aligned}$$
\(\mathrm {(ii)}\)
\(4\vert c_1\vert ^2<c\), \(\gamma \) is defined as in the case (i), and
$$\begin{aligned} A=\left( \begin{array}{ccc}0&{}0&{}0\\ 0&{}\frac{-\sqrt{-1}(c_1-\bar{c}_1)}{2}&{}-\frac{\sqrt{c-4\vert c_1\vert ^2}}{2}\\ 0&{}\frac{\sqrt{c-4\vert c_1\vert ^2}}{2}&{}\frac{-\sqrt{-1}(c_1-\bar{c}_1)}{2}\end{array}\right) , \end{aligned}$$
\(\mathrm {(iii)_1}\)
\(c=4\vert c_1\vert ^2\), \(c_1\not =\bar{c}_1\), \(\gamma \) is defined as in the case \(\mathrm {(i)}\), and
$$\begin{aligned} A=\left( \begin{array}{ccc}0&{}0&{}0\\ 0&{}\frac{-\sqrt{-1}(c_1-\bar{c}_1)}{2}&{}0\\ 0&{}1&{}\frac{-\sqrt{-1}(c_1-\bar{c}_1)}{2}\end{array}\right) , \end{aligned}$$
\(\mathrm {(iii)_2}\)
\(c=(c_1+\bar{c}_1)^2\), \(c_1\not =\bar{c}_1\),
$$\begin{aligned} \begin{aligned} \gamma (s)&= {\left\{ \begin{array}{ll} \lambda (s)\left( \frac{\sqrt{-1}(c_1^2-\bar{c}_1^2)}{c} \left( s+\frac{1}{\sqrt{c}}\sinh \sqrt{c}s\right) -\frac{2\sqrt{-1}(c_1-\bar{c}_1)}{c}\cosh ^2\frac{\sqrt{c}}{2}s, 1,1\right) ,\\ \ c>0,\\ \lambda (s)\left( \frac{\sqrt{-1}(c_1-\bar{c}_1)}{2}s^2,1,1\right) ,\ c=0,\\ \end{array}\right. }\\ A&=\left( \begin{array}{ccc}0&{}0&{}0\\ 1&{}0&{}0\\ 0&{}0&{}-\sqrt{-1}(c_1-\bar{c}_1)\end{array}\right) , \end{aligned} \end{aligned}$$
\(\mathrm {(iv)}\)
\(c=(c_1+\bar{c}_1)^2\), \(c_1=\bar{c}_1\),
$$\begin{aligned} \begin{aligned} \gamma (s)&= {\left\{ \begin{array}{ll} \lambda (s)\left( -\frac{2}{c}\cosh ^2\frac{\sqrt{c}}{2}s +\frac{2c_1}{c} \left( s+\frac{1}{\sqrt{c}}\sinh \sqrt{c}s \right) , 1,1 \right) ,\ c>0,\\ \lambda (s)\left( \frac{1}{2}s^2,1,1\right) ,\ c=0,\\ \end{array}\right. }\\ A&=\left( \begin{array}{ccc}0&{}0&{}0\\ 1&{}0&{}0\\ 0&{}1&{}0\end{array}\right) , \end{aligned} \end{aligned}$$
where
$$\begin{aligned} \lambda (s)= {\left\{ \begin{array}{ll} c_1+\bar{c}_1+\sqrt{c}\tanh \frac{\sqrt{c}}{2}s,\ c>0,\\ c_1+\bar{c}_1+\frac{2}{s},\ c=0,\\ c_1+\bar{c}_1-\sqrt{-c}\tan \frac{\sqrt{-c}}{2}s,\ c<0. \end{array}\right. } \end{aligned}$$

Proof

We just remark that the formulas corresponding to the first equation of (6.14) and (6.15) are given as
$$\begin{aligned}&\left\{ \dfrac{\varphi '}{\varphi }-(c_1+\bar{c}_1) \right\} \gamma ' =\gamma \left\{ -2A^2-2(c_1-\bar{c}_1)\sqrt{-1}A-\varphi E \right\} ,\\&\quad t^3+\sqrt{-1} (c_1-\bar{c}_1)t^2+\dfrac{1}{4}\{c-(c_1+\bar{c}_1)^2\}t, \end{aligned}$$
respectively. \(\square \)

Remark 7.3

In the last case in Theorem 7.2, \(\text{ tr }\,A=0\) if and only if \(c_1\in {\mathbf {R}}\), in which case f is an equiaffine rotation surface, and up to similarity we have (iv) or the followings:
\(\mathrm {(i)}'\)
\(4c_1^2>c\),
$$\begin{aligned} \begin{aligned} \gamma (s)&= {\left\{ \begin{array}{ll} \left( \lambda (s),\frac{e^{c_1s}}{\cosh \frac{\sqrt{c}}{2}s},\frac{e^{c_1s}}{\cosh \frac{\sqrt{c}}{2}s}\right) , c>0,\\ \left( \lambda (s),\frac{e^{c_1s}}{s},\frac{e^{c_1s}}{s}\right) ,\ c=0,\\ \left( \lambda (s),\frac{e^{c_1s}}{\cos \frac{\sqrt{-c}}{2}s},\frac{e^{c_1s}}{\cos \frac{\sqrt{-c}}{2}s}\right) ,\ c<0,\ \end{array}\right. }\\ A&=\left( \begin{array}{ccc}0&{}0&{}0\\ 0&{}\frac{\sqrt{4c_1^2-c}}{2}&{}0\\ 0&{}0&{}-\frac{\sqrt{4c_1^2-c}}{2}\end{array}\right) , \end{aligned} \end{aligned}$$
\(\mathrm {(ii)}'\)
\(4c_1^2<c\),
$$\begin{aligned} \begin{aligned} \gamma (s)= \left( \lambda (s), \frac{e^{c_1s}}{\cosh \frac{\sqrt{c}}{2}s}, \frac{e^{c_1s}}{\cosh \frac{\sqrt{c}}{2}s}\right) , \ A=\left( \begin{array}{ccc}0&{}0&{}0\\ 0&{}0&{}-\frac{\sqrt{c-4c_1^2}}{2}\\ 0&{}\frac{\sqrt{c-4c_1^2}}{2}&{}0\end{array}\right) . \end{aligned} \end{aligned}$$

Fujioka classified definite centroaffine minimal surfaces such that \(a=\bar{a}\) and \(\kappa \) is constant, and obtained the surface given by \(\mathrm {(i)}'\) with \(c=0\) (Fujioka (2006), Theorem 3.5), although the other surfaces are dropped.

References

  1. Fujioka, A.: Centroaffine minimal surfaces with constant curvature metric. Kyungpook Math. J. 46, 297–305 (2006)MathSciNetzbMATHGoogle Scholar
  2. Fujioka, A.: Centroaffine minimal surfaces with non-semisimple centroaffine Tchebychev operator. Results Math. 56, 177–195 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Furuhata, H., Vrancken, L.: The center map of an affine immersion. Results Math. 49, 201–217 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Lee, I.C.: On generalized affine rotation surfaces. Results Math. 27, 63–76 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Liu, H., Wang, C.: The centroaffine Tchebychev operator. Results Math. 27, 77–92 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Magid, M.A., Ryan, P.J.: Flat affine spheres in \({\mathbf{R}}^3\). Geom. Dedicata 33, 277–288 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Manhart, F.: Affine rotational surfaces with vanishing affine curvature. J. Geom. 80, 166–178 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Ohdera, Y.: Centroaffine Rotation Minimal Surfaces (in Japanese). Master Thesis, Hokkaido University (2014)Google Scholar
  9. Schief, W.K.: Hyperbolic surfaces in centro-affine geometry. Integrability and discretization. Chaos Soliton Fract. 11, 97–106 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Simon, U.: Local classification of two-dimensional affine spheres with constant curvature metric. Differ. Geom. Appl. 1, 123–132 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Wang, C.: Centroaffine minimal hypersurfaces in \({\mathbb{R}}^{n+1}\). Geom. Dedicata 51, 63–74 (1994)MathSciNetCrossRefGoogle Scholar
  12. Watanabe, M.: Tchebychev Vector Fields of Centroaffine Rotation Surfaces (in Japanese). Master Thesis, Hokkaido University (2015)Google Scholar
  13. Yang, Y., Yu, Y., Liu, H.: Centroaffine geometry of equiaffine rotation surfaces in \({\mathbb{R}}^3\). J. Math. Anal. Appl. 414, 46–60 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2018

Authors and Affiliations

  1. 1.Department of MathematicsKansai UniversitySuitaJapan
  2. 2.Department of MathematicsHokkaido UniversitySapporoJapan

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