曲面の微分幾何 - kobe...

曲面の微分幾何—計算から眺めるいくつかの話題—

Masatoshi KOKUBU

Tokyo Denki University

January 09 2009

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One of (my) motivations:Wish to discover new surfaces which are interesting from

the (differential-)geometric viewpoint.

MenuPart 1 From Classical Minimal Surface Theory

Before Weierstrass-Enneper

After Weierstrass-Enneper

After Osserman

Part 2 Flat fronts and their analogue

Flat fronts in H3

W-fronts in H3

ª¢ 1(Â)

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Part 1 From Minimal Surface Theory

Definition

A surface S ⊂ E3 is said to be minimal if∀p ∈ S, ∃ a nbd U s.t. U has least area

among surfaces whose boundary equals ∂U.

A plane is a trivial example.

—18th century—

1740s Euler catenoidy = cosh(x)(catenary)

ª¢ 2(Â)


Theorem 1 (Meusnier 1770s)

minimal ⇐⇒ mean curvature H = 0

1770s Meusnier catenoid, helicoid

(s cos t, s sin t, t)

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–the first half of 19th century–

1835 Scherk Scherk’s surface, Scherk’s second surface

ez cos x − cos y = 0

sin z − sinh x sinh y = 0

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Weierstrass-Enneper formula–the latter half of 19th century–

Theorem 2

Suppose a surface is given by an isometric immersion

x = (x1, x2, x3) : (M2, ds2) → E3.x is minimal ⇐⇒ each xi is harmonic.

∃ local isothermal parameter (u, v), i.e., ds2 = λ(du2 + dv2),Setting z = u + iv, a minimal immersion x is given by a

real part of a null holomorphic immersion w.r.t z.

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Theorem 3 (Weierstrass, Enneper 1860s)

x(z) = Re∫ z

z0

(1 − g2, i(1 + g2), 2g

)f dz

where g, f holomorphic in z.

Crucial formula for pieces of minimal surface!

g is the Gauss map, i.e.,

g = (stereographic projection) ◦ N

pN

N(p)

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1864 Enneper

g = z, f = 1

Re

z − z3

3i(z + z

3

3 )z2

1878 Henneberg

g = z, f = 1 − z−4

x(z) = Re

−z3

3 + z −1z +

13z3

iz33 + iz +

iz +

i3z3

z2 + 1z2

x(−1/z̄) = x(z) is satisfied!x defines x̌ : RP2 \ {[0]} → E3well. (branched at [1], [i].)

ª¢ 7(Â)

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Global study due to Osserman (1960s —)–the latter half of 19th century–

Theorem 4 (Osserman)

M: Riemann surface, g: meromorphic function on M,ω: a holomorphic 1-form on M.(i) {zeros of ω} = {poles of g} where ordpω = 2ordpg(ii)

∫1 − g2ω,

∫i(1 + g2)ω,

∫2gω have no real period

⇒Re

∫ (1 − g2, i(1 + g2), 2g

)ω (⋆)

is a conformal minimal immersion M → E3.“⇐” also holds.

(i′) {zeros of ω} ⊃ {poles of g} where ordpω ≥ 2ordpg(ii)⇒ (⋆) gives a branched minimal immersion M → E3.ª¢ 8(Â)



x : M → E3 a complete minimal surface of finite totalcurvature

⇒M ∼= M̄ \ {p1, . . . , pn}Gauss map g extends meromorphically on M̄total curvature = −4π deg(g)


total curvature ≤ 4π(1 − genus(M) − #{ends})

When one wishes to construct an example of a complete

minimal surface of finite total curvature, one has to take

care about genus and punctured points of M, degree of g,and, of course, completeness.

ª¢ 9(Â)


1983 Jorge-Meeks M = Ĉ \ {1, ζ, ζ2} (ζ3 = 1)g = z2, ω = (z3 − 1)−2dz

(i) At z = 0, ω = 0 of order 4At z = 0, g = 0 of order 2

(ii)(

1−z4(z3−1)2 , i(

1+z4(z3−1)2),

2z2(z3−1)2

)has residue

(−49, 0, 0) at z = 1(29,

23√

3, 0) at z = ζ

(29,−2

3√

3, 0) at z = ζ2

(iii) ds2 = (|z|4+1)2

|z3−1|2

∃k-noid for ∀k ≥ 2.

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℘: Weierstrass’ ℘ function on C/{Z ⊕ Zi} (square torus)1981 Chen-Gackstatter

g =√

3π/2g2 ℘′/℘, ω = ℘ dz

M = T2 \ {1 point}tot. curv. = −8πexplict formula

1982 Costa

g = 2√

2π e1℘′ , ω = ℘ dz

M = T2 \ {3 points}tot. curv. = −12πembedded

explict formula

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1980s – 90s After Jorge-Meeks, Chen-Gackstatter and

Costa, So many new minimal surfaces are discovered

(Hoffman, Meeks, Karcher, Kusner, Rosenberg, Lopez,

Ros, Rossman, Miyaoka, Sato, ..........)

2000s – Discovery is continuing! (Fujimori, Shoda, Traizet,

Weber, ..........)

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Part 2 Flat Front in H3H3: hyperbolic 3-space, i.e., simply-connected, complete

Riemannian 3-manifold of constant sectional curvature −1H3 is represented as Poincaré ball, upper half-space, etc.

flat front · · ·{flat = “Gauss curvature = 0”,front = “surface with ‘good’ singularities”

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Remarks

Fact [Volkov-Vladimirova(1972); S. Sasaki(1973)]

A complete flat surface immersed in H3 is a horosphereor a (hyperbolic) cylinder.

Hyperbolic Gauss maps G, G∗is defined for fronts (across the

singularities).

f (M2)

p

G(p)

G∗(p)

Theorem 7

flat ⇐⇒ G, G∗ : holomorphic w.r.t. holomorphicstructure compatible to the 2nd fundamental form

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Representation formula

Theorem 8 ((G, G∗)-formula)f : M2 → H3 = SL(2, C)/ SU(2) : a flat front with hyper-bolic Gauss maps G, G∗.⇐⇒ f = EE ∗ where

E =(

G/ξ ξG∗/(G − G∗)1/ξ ξ/(G − G∗)

)with ξ := exp

(∫ dGG − G∗

)

period condition for f = EE ∗

⇐⇒∫

γ

dGG − G∗

∈√−1R, ∀[γ] ∈ π1(M)

ª¢ 15(Â)


Example 9 (flat fronts of revolution)

horosphere

hyperbolic cylinder

hourglass

snowman

M :=

{Ĉ \ {0} if α = 0Ĉ \ {0, ∞} otherwise,

G(z) = z,G∗(z) = αz (α ∈ R \ {1})

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Example 10 (n-noid)M := Ĉ \ {z|zn = 1}, G(z) = z, G∗(z) = z−n+1 (n = 2, 3, . . . )

Example 11 ((n + 2)-noid)M := Ĉ \ {0, ∞, zn = 1}, G(z) = z, G∗(z) = zn+1 (n = 1, 2, . . . )

Theorem 12 (KUY)

For a weakly complete flat front with regular ends,

deg G + deg G∗ ≥ #{ends}‘ =′ ⇐⇒ All ends are embedded.

ª¢ 17(Â)


Construction of ”with ends at arbitrary positions”

Let p1, . . . , pn−1, ∞ be distinct points in ∂H3 = C ∪ {∞}.Choose non-zero real numbers a1, . . . , an−1 such that a1 +· · · + an−1 ̸= 0, 1.

M = C \ {p1, . . . , pn−1}G = z

G∗ =

(z ∑n−1k=1

{ak ∏j ̸=k

(z − pj

)}− ∏n−1j=1 (z − pj)

)∑n−1k=1

{ak ∏j ̸=k

(z − pj

)}⇒ a weakly complete flat front with embedded regular

ends at p1, . . . , pn−1, ∞.

Problem ∃ flat front having two different ends accu-mulating at the same point in ∂H3?

ª¢ 18(Â)


Examples of higher genus

Example 13

Let ℘ denote the Weierstrass ℘ function on T = C/{Z1⊕Zi}(squared torus).

M = T \ {z ; ℘(3℘2 − e21) = 0}

G =1℘′

G∗ = −8e213

℘

℘′

⇒ a weakly complete flat front of genus one, with fiveembedded regular ends.

Figure

ª¢ 19(Â)

file:figures/torusfront.nbdviout: jfdviout: d6dviout: je

Example 14

Consider a hyperelliptic Riemann surface

R : w2 = z(

z2k − 2czk−1 − 1)

=: zϕ(z)

where c =

{0 if k = 1k/(k − 1) if k > 1

of genus k.Set M := R \ {z ; z(zϕ(z))′ = 0}(= R \ {4k + 1 pts}).Then

G = w, G∗ =1

(2k + 1)w(2kzϕ(z) − z2ϕ′(z))

⇒ a weakly complete flat front of genus k, with 4k + 1embedded regular ends.

ª¢ 20(Â)


Open Problem

Consider weakly complete flat fronts of genus k with nregular ends.

➀ For each k, What is the least number nmin(k) of n?➁ How about the same problem assuming ends must be

embedded?

In particular,

∃ genus-1 front with four embedded ends?

Genus-0 case is solved.

For positive genus case, we know nmin(k) ≤ 4k + 1 byprevious example; moreover, we know nmin(k) ≤ 2k + 3 byexistence therem (non-computational argument).

ª¢ 21(Â)


An analogueWe consider Weingarten surfaces in H3 satisfying

α(H − 1) = βK for some [α : β] ∈ RP1, (⋆)called a W-surface, for short, in this talk.

The class of W-surfaces includes

• flat surfaces (K = 0)• CMC-1 surfaces (H = 1)

It is natural to consider W-fronts rather than W-surfaces.

But CMC-1 fronts necessarily CMC-1 (regular) surface.

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Theorem 15 (Representation formula)M : Riemann surface; G : meromorphic functionds2ϵ : metric on M with constant curvature ϵ⇒∃ W-front f : M → H3 with α(H − 1) = βK, α/(α − 2β) = ϵ

More precisely,

∃ a holomorphic map h : M̃ → S2 or C or D s. t. ds2ϵ =4|dh|2

(1+ϵ|h|2)2Using this h, define

G = (−Gh)−3/2[−GGh GGhh/2 − G2h−Gh Ghh/2

]and H =

[1+ϵ2|h|21+ϵ|h|2 −ϵh̄−ϵh 1 + ϵ|h|2

],

and set

f = GHG∗.Then f : M′(⊂ M) → H3 = PSL(2, C)/ PSU(2) is a Weingarten frontwith α(H − 1) = βK, α/(α − 2β) = ϵ.Here, M′ = M \ {π(pi), π(qi)} ;pi is the common zero of dh and {G; h}dh, qi is a pole of dh or{G; h}dh;and vise versa.

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End of slides. Click [END] to finish the presentation.

Thank you.

ª END Bye ¢
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