Centroaﬃne
minimalsurfaces
AtsushiFujioka

Centroaﬃne minimal surfaces

Contents.

Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswithnon-vanishing
Tchebychev
operator..

Atsushi Fujioka.

Faculty of Engineering Science
Kansai University

October 26, 2012
Petite Hotel Sonntag
The 18th International Symposium on Complex Geometry

1 / 27

Contents
Centroaﬃne
minimalsurfaces
AtsushiFujioka1

Introduction

2

Centroaﬃne surfaces

3

Fundamental examples of centroaﬃne minimal surfaces

4

Examples with non-vanishing Tchebychev operator

Contents
Introduction..

Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswithnon-vanishing
Tchebychev
operator..

..

2 / 27

Centroaﬃne minimal hypersurfaces
Centroaﬃne
minimalsurfaces
AtsushiFujiokaContents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswith
non-vanishing
Tchebychev
operator

Centroaﬃne minimal hypersurfaces:◦ Hypersurfaces in the Euclidean space◦ Objects in centroaﬃne diﬀerential geometry↑Study properties of submanifolds which are invariant under
the aﬃne transformations ﬁxing the origin.
(centroaﬃne transformations)◦ Deﬁned for non-degenerate centroaﬃne hypersurfaces◦ Extremals for the area integral of the centroaﬃne metric3 / 27

BackgroundCentroaﬃne
minimalsurfaces
AtsushiFujiokaContents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswith
non-vanishing
Tchebychev
operator◦ 1994 C. P. Wang: Deﬁnition of centroaﬃne minimal
hypersurfaces◦ Proper aﬃne hyperspheres centered at the origin are
centroaﬃne minimal.
◦ Only a few essentially new examples are known even if thecase of surfaces.
◦ The integrability conditions for centroaﬃne minimal surfaces
include Tzitzéica equation:(log ψ)xy = −ψ −1
ψ2

◦ 2000 W. Schief: A generalization and a discretization ofTzitzéica transformation for proper aﬃne spheres
4 / 27

Review of Euclidean diﬀerential geometry
Centroaﬃneminimalsurfaces

Euclidean diﬀerential geometry: Study properties of submanifolds in Rn which are invariant under the Euclidean
motions.
In the following, we consider surfaces in R3 .

AtsushiFujiokaContents
Introduction

Gauss formulaCentroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswith
non-vanishing
Tchebychev
operator.

f : D → R3 : a surface(x1 , x2 ): local coordinates
h , i: the standard inner product on R3
n: the unit normal vector ﬁeld=⇒ Gauss formula:fxi xj = Γ1ij fx1 + Γ2ij fx2 + hfxi xj , nin(i, j = 1, 2)(1)

5 / 27

Deﬁnition of centroaﬃne surfaces
Centroaﬃne
minimalsurfaces

Deﬁnition

AtsushiFujiokaf : D → R3 : a surface
f : a centroaﬃne surface
m def
f : transversal to the tangent plane.
Contents
Introduction
Centroaﬃne
surfaces.

Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswith
non-vanishing
Tchebychev
operator

Gauss formulaf : D → R3 : a centroaﬃne surface(x1 , x2 ): local coordinates..

fxi xj = Γ̃1ij fx1 + Γ̃2ij fx2 − h(∂xi , ∂xj )f(i, j = 1, 2).

(2)

6 / 27

Centroaﬃne metricCentroaﬃne
minimalsurfaces

The symmetric (0, 2)-tensor ﬁeld h in Gauss formula (2) iscalled the centroaﬃne metric.

AtsushiFujiokaDeﬁnitionf : D → R3 : a centroaﬃne surface
f : non-degenerate (resp. deﬁnite, indeﬁnite)m def
h: non-degenerate (resp. deﬁnite, indeﬁnite)

Contents
Introduction
Centroaﬃne
surfaces.

Fundamental
examples ofcentroaﬃne
minimalsurfaces

Proposition.

f : D → R3 : a centroaﬃne surface

Exampleswith
non-vanishing
Tchebychevoperator..

f : deﬁnite (resp. indeﬁnite)m
The Euclidean Gaussian curvature: positive (resp. negative).
7 / 27

Proof of Proposition
Centroaﬃne
minimalsurfaces
AtsushiFujioka

ProofRecall two kinds of Gauss formula:Contents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswith
non-vanishing
Tchebychev
operator.

fxi xj = Γ1ij fx1 + Γ2ij fx2 + hfxi xj , nin(1)

fxi xj = Γ̃1ij fx1 + Γ̃2ij fx2 − h(∂xi , ∂xj )f(2)

From (1) and (2)hfxi xj , ni = −h(∂xi , ∂xj )hf , ni.
8 / 27

Notations
Centroaﬃne
minimalsurfaces
AtsushiFujiokaContents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswith
non-vanishing
Tchebychev
operator

For simplicity, we consider indeﬁnite case.
f : D → R3 : an indeﬁnite centroaﬃne surface
K : the Euclidean Gaussian curvature < 0(x, y ): asymptotic line coordinatesψ := h(∂x , ∂y )d: the signed distance from the origin to the tangent plane()1Kρ := − log − 44d/ ff

α := ψ det  fx  det  fx fxx
fy
/ ff
β := ψ det  fy  det  fy fyyfx

9 / 27

Gauss formula in asymptotic line coordinates
Centroaﬃne
minimalsurfaces
AtsushiFujiokaGauss formula.

Contents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces()ψxα
fxx =+ ρx fx + fyψψfxy = −ψf + ρy fx + ρx fy()ψyβ
fyy =+ ρy fy + fx
ψψ

Proof
Use

Exampleswith
non-vanishing
Tchebychev
operator

Γ112 = −(3)

.

1 Ky
1 Kx
, Γ212 = −
4 K
4K

etc.
..

.
10 / 27

Integrability conditions
Centroaﬃne
minimalsurfaces
AtsushiFujioka

Proposition.

The integrability conditions for Gauss formula (3) areαβ(log |ψ|)xy = −ψ − ψ 2 + ρx ρyContents
Introduction
Centroaﬃne
surfaces(4)αy + ρx ψx = ρxx ψ
βx + ρy ψy = ρyy ψFundamental
examples ofcentroaﬃne
minimalsurfaces.

If ρ is constant and α, β 6= 0, changing the coordinates, if
necessary, we obtain Tzitzéica equation:

Exampleswith
non-vanishing
Tchebychev
operator(log ψ)xy = −ψ −..

1
ψ2.
11 / 27

Centroaﬃne scalar curvature
Centroaﬃne
minimalsurfaces
AtsushiFujiokaf : D → R3 : an indeﬁnite centroaﬃne surface(x, y ): asymptotic line coordinates
κ: the scalar curvature of the centroaﬃne metric h(the centroaﬃne scalar curvature)Contents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswith
non-vanishing
Tchebychev
operator

κ=−(log |ψ|)xy(ψ = h(∂x , ∂y ))ψIf f is ﬂat, i.e., κ = 0, we may assume that ψ = 1=⇒ The integrability conditions (4) are equivalent to equationof associativity in topological ﬁeld theory:gxxx gyyy − gxxy gxyy + 1 = 0,(5)whereρ = gxy , α = gxxx , β = gyyy .
12 / 27

Centroaﬃne Tchebychev vector ﬁeld andcentroaﬃne Tchebychev operator
Centroaﬃne
minimalsurfaces
AtsushiFujiokaContents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswithnon-vanishing
Tchebychev
operatorf : D → R3 : a non-degenerate centroaﬃne surface
˜ the connection induced by the centroaﬃne surface f∇:(The corresponding Christoﬀel symbols are Γ̃kij in (2).)∇h : the Levi-Civita connection for the centroaﬃne metric h˜ − ∇h : the diﬀerence tensor
C := ∇1
T := trh C : the centroaﬃne Tchebychev vector ﬁeld
2
hij := h(∂xi , ∂xj ), (hij ) := (hij )−1 , Cijk ∂xk := C (∂xi , ∂xj )trh C := hij Cijk ∂xk∇h T : the centroaﬃne Tchebychev operator

13 / 27

Deﬁnition of centroaﬃne minimal surfaces
Centroaﬃne
minimalsurfaces
AtsushiFujiokaContents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswith
non-vanishing
Tchebychev
operator

Centroaﬃne minimal surfaces: Extremals for the area integralof the centroaﬃne metricf : D → R3 : an indeﬁnite centroaﬃne surface(x, y ): asymptoticline() coordinates
1Kρ = − log − 44dh∇ T : the centroaﬃne Tchebychev operator
Propositionf : centroaﬃne minimal ⇐⇒ ρxy = 0⇐⇒ tr ∇h T = 0

14 / 27

Quadrics
Centroaﬃne
minimalsurfaces
AtsushiFujiokaContents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswith
non-vanishing
Tchebychev
operator

Ellipsoids centered at the originHyperboloids of one sheet centered at the origin
Hyperboloids of two sheets centered at the origin=⇒ T = 0
Elliptic paraboloids removing the vertex which is the origin
Hyperbolic paraboloids removing the saddle point which isthe origin=⇒ T 6= 0, ∇h T = 0

15 / 27

Proper aﬃne spheres
Centroaﬃne
minimalsurfaces
AtsushiFujiokaContents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswith
non-vanishing
Tchebychev
operator

Proper aﬃne spheres: Blaschke surfaces whose aﬃne shapeoperator is a non-zero scalar operator
The center: The point where the aﬃne normals of proper aﬃnespheres meetf : D → R3 : a non-degenerate centroaﬃne surface1|K |ρ := − log 44d
T : the centroaﬃne Tchebychev vector ﬁeld
Propositionf : a proper aﬃne sphere centered at the origin ⇐⇒ ρ: constant⇐⇒ T = 0

16 / 27

Flat proper aﬃne spheres
Centroaﬃne
minimalsurfaces
AtsushiFujiokaf : D → R3 : a proper aﬃne sphere centered at the origin
In the following, we put f = (X , Y , Z ), if necessary.
We consider centroaﬃne surfaces modulo centroaﬃnecongruence.

Contents
Introduction
Centroaﬃne
surfaces

Theorem (cf. M. A. Magid-P. J. Ryan 1990)

Fundamental
examples ofcentroaﬃne
minimalsurfaces

If f is ﬂat, we have the following:
1: XYZ = 1 (negative deﬁnite)
2: (X 2 + Y 2 )Z = 1 (indeﬁnite)

Exampleswith
non-vanishing
Tchebychev
operator.
17 / 27

Proper aﬃne spheres with constant centroaﬃnescalar curvature
Centroaﬃne
minimalsurfacesf : D → R3 : a proper aﬃne sphere centered at the origin
κ: the centroaﬃne scalar curvature

AtsushiFujioka

Theorem (cf. U. Simon 1991)
Contents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswith
non-vanishing
Tchebychev
operator.

If κ is constant, then κ = 0, 1.
If κ = 1, we have the following:
1: Ellipsoids centered at the origin (positive deﬁnite)
2: Hyperboloid of two sheets centered at the origin(negative deﬁnite)
3: f = A0 (u) + vA(u), A is any R3 -valued function s.t.
Adet  A0  is non-zero constant. (indeﬁnite)
A00.18 / 27

Centroaﬃne surfaces with vanishing centroaﬃne
Tchebychev operator
Centroaﬃne
minimalsurfaces
AtsushiFujiokaContents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswithnon-vanishing
Tchebychev
operator

Theorem (H. L. Liu-C. P. Wang 1995).

If ∇h T = 0, except the above examples, we have the following:In 1∼3, a, b, c ∈ R.
1: X a Y b Z c = 1, abc(a + b + c) 6= 0{()}X−1
2: exp −a tan(X 2 + Y 2 )b Z c = 1,Yc(2b + c)(a2 + b 2 ) 6= 0
3: Z = −X (a log X + b log Y ), b(a + b) 6= 0Y2
4: Z = ±X log X +X
5: f = (e x , A1 (x)e y , A2 (x)e y ), A1 and A2 are any linearly
independent solutions to the diﬀerential equation:
A00 − A0 − a(x)A = 0 for any function a = a(x).
.
19 / 27

Solutions to equation of associativityCentroaﬃne
minimalsurfaces

All the examples 1∼5 in Theorem due to Liu-Wang are ﬂat.
Proposition

AtsushiFujiokaContents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswithnon-vanishing
Tchebychev
operator.

Solutions to equation of associativity (5) corresponding toindeﬁnite ﬂat centroaﬃne surfaces with ∇h T = 0 are one of
the following:
βc1c2
α1: g = x 3 + y 3 + x 2 y + xy 26622+(any polynomials of x and y with degree ≤ 2)α, β ∈ R \ {0}, c1 , c2 ∈ R, αβ − c1 c2 + 1 = 0
2: By changing x and y , if necessary,c1c2
g = x 2 y + xy 2 + c3 xy + (any function of x)22+(any polynomial of y with degree ≤ 2)
c1 , c2 , c3 ∈ R, c1 c2 = 1.

20 / 27

Examples with constant centroaﬃne scalar
curvature
Centroaﬃne
minimalsurfaces
AtsushiFujiokaContents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswithnon-vanishing
Tchebychev
operator◦ Centroaﬃne minimal ⇐⇒ tr ∇h T = 0◦ All the above examples: ∇h T = 0◦ 2006 F: Classiﬁcation of centroaﬃne minimal surfaces withconstant κ under the assumption on some cubic diﬀerentials◦ Obtained new examples. (∇h T 6= 0)◦ Indeﬁnite case:
( −u)
ee −u
1f =cos v ,sin v , 1 −uuu
◦κ=1
◦ T is an eigenvector of ∇h T (cf. 2004 L. Vrancken)

21 / 27

Ruled surface
Centroaﬃne
minimalsurfaces
AtsushiFujiokaContentsIntroduction
Centroaﬃne
surfaces
Fundamentalexamples ofcentroaﬃne
minimalsurfaces
Exampleswithnon-vanishing
Tchebychev
operator◦ 2009 F: Classiﬁcation of centroaﬃne minimal surfaces withconstant κ such that ∇h T is not diagonalizable◦ 2010 F: Classiﬁcation of centroaﬃne minimal surfaces withconstant κ and constant Pick function◦ Pick function:
11r
J = kC k2 = hkr hip hjq Cijk Cpq(C : the diﬀerence tensor)
22◦ In both classiﬁcation, obtained new examples with κ = 0, 1.
◦ One is a ruled suface:f = A0 (u) + vA(u), A is an R3 -valued function such thatA
det  A0  6= 0A00
◦ κ = 1, J = 0
22 / 27

Flat example
Centroaﬃne
minimalsurfaces
AtsushiFujiokaContents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswith
non-vanishing
Tchebychev
operator◦ Another example is ﬂat. (κ = 0))
(∞∞∞∑
∑∑
◦f =
ϕn,1 (x)y n ,ϕn,2 (x)y n ,ϕn,3 (x)y nn=0n=0n=0(x, y ): asymptotic line coordinates around (x0 , 0) suchthat x0 6= 0
ϕ0,1 , ϕ0,2 , ϕ0,3 : Linearly independent solutions to thediﬀerential equation:
xϕ000 + ϕ00 − ϕ = 0
ϕn+1,i =◦ J = −1xϕ00
n + 1 n,i(6)(i = 1, 2, 3)

23 / 27

Solution to equation of associativity
Centroaﬃne
minimalsurfaces
AtsushiFujioka

Proposition
By changing the coordinates, if necessary, solution to equationof associativity (5) corresponding to the above example isContents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswith
non-vanishingTchebychev
operator.

11
g = x 3 y − y 2 log y62+(any polynomials of x and y with degree ≤ 2).

24 / 27

Solution to the third order diﬀerential equaiton
Centroaﬃne
minimalsurfaces
AtsushiFujioka

The diﬀerential equation (6) can be solved by using Meijer
G-functions:Contents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswithnon-vanishing
Tchebychev
operator( 2))( 2xx——
1,0+ ic2 G0,3 −8 12 , 12 , 0
8 12 , 12 , 0( 2)x—1,0+ c3 G0,3−(c1 , c2 , c3 ∈ R)
8 0, 21 , 122,0
ϕ = c1 G0,3

The second and the third terms can be written by using thegeneralized hypergeometric function 0 F2 .

25 / 27

Meijer G-function
Centroaﬃne
minimalsurfaces
AtsushiFujiokaContents
Introduction
Centroaﬃne
surfaces
Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswith
non-vanishing
Tchebychev
operator◦ Meijer G-function:()
a1 , . . . , apm,nGp,q z
b1 , . . . , bqmn∏∏
Γ(bj + s)Γ(1 − aj − s)∫
1j=1
j=1z −s ds=qp∏
2πi L ∏
Γ(1 − bj − s)
Γ(aj + s)
j=m+1

j=n+1

◦ m, n, p, q ∈ Z, 0 ≤ m ≤ q, 0 ≤ n ≤ p◦ Satisﬁes a linear diﬀerential equation of order max{p, q}.

26 / 27

Centroaﬃne
minimalsurfaces
AtsushiFujiokaContents
Introduction
Centroaﬃne
surfacesThank you for your attention.

Fundamental
examples ofcentroaﬃne
minimalsurfaces
Exampleswith
non-vanishing
Tchebychev
operator27 / 27