H. HOPF'S QUADRATIC DIFFERENTIAL AND A WEIERSTRASS FORMULA FOR GENERAL SURFACES AND SURFACES OF CONSTANT MEAN CURVATURE

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H. HOPF’S QUADRATIC DIFFERENTIAL AND A WEIERSTRASS FORMULA FOR GENERAL

SURFACES AND SURFACES OF CONSTANT MEAN CURVATURE

F. Gackstatter

To cite this version:

F. Gackstatter. H. HOPF’S QUADRATIC DIFFERENTIAL AND A WEIERSTRASS FORMULA FOR GENERAL SURFACES AND SURFACES OF CONSTANT MEAN CURVATURE. Journal de Physique Colloques, 1990, 51 (C7), pp.C7-163-C7-168. �10.1051/jphyscol:1990716�. �jpa-00231115�

COLLOQUE DE PHYSIQUE

Colloque C7, suppl6ment au n023, Tome 51, ler dE?cembre 1990

H. HOPF'S QUADRATIC DIFFERENTIAL AND A WEIERSTRASS FORMULA FOR GENERAL SURFACES AND SURFACES OF CONSTANT MEAN CURVATURE

F. GACKSTATTER

X. Mathematisches Institut der Freien Universitdt Berlin, Arnimallee 3 , D-1000 Berlin 33, F.R.G.

Abstract - Using H. Hopf's quadratic differential 4 d w 2 we find a representation formulafor genera1 surfaces in IR3 similar to the classical Weierstrass formula for minimal surfaces. For surfaces of constant mean curvature with prescribed Q d w Z an

integrability condition with the properties of a conservation equation is derived.

We continue the work of K. Kenmotsu on surfaces of prescribed mean curvature.

1 - THE CLASSICAL WEIERSTRASS FORMULA FOR MINIMAL SURFACES IN ]R3

In the theory of minimal surfaces we have the well known Weierstrass formula:

(1 ( X ~ . X ~ . X ~ ) = Re

S(+

^(1-g'), (Itg'), g) fdw ,

where ^W = u+iv are isothermal coordinates in a domain D, g is the Gauss function and fdw is an ordinary differential on D. If the complex analytic function W = w(z) describes a para- meter transformation, then f (W) =

F(=).

2 - H. HOPF'S QUADRATIC DIFFERENTIAL 4 d w 2 FOR GENERAL SURFACES Let S be a regular surface in IR3 defined by

( 2 ) x(w): D ⁺ IR', X(W) = (xl(u,v) ,x2(u,v),x3(u,v))

with isothermal coordinates W = utiv. Then the first fundamental form is (3) ds2 = E(duZ+dvZ) = E J d w J P .

Let L, M and N be the coefficients of the second fundamental form and

( 4 ) 1

6

^:= Z. (L-N) - iM

the complex valued function introduced by H. Hopf, see / l / , p. 136-8. This function is useful in surface theory, for example the Codazzi equations can be shortly written

(5)

Q,

= EH W '

a

1 2

where H is the mean curvature and

&

⁼

$ (g

^{t i}

&)

^{, =}

^-

ⁱ

6) .

If the complex analytic function W = w(z) describes a parameter transformation, then

( 6 )

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990716

C7-164 COLLOQUE DE PHYSIQUE

and @ transforms like a quadratic differential. Since

with the Gauss curvature K and the principal curvatures k,, the umbilic and flat points of S are the zeros of

Q .

For surfaces of constant mean curvature or minimal surfaces

is complex analytic in W and &dwZ is an analytic quadratic differential in the sense of the classical theory of Riemann surfaces.

Remarks: If S is a surface of class C 3 isothermal coordinates can be constructed, see /l/, p. 99. In the following existence and continuity of derivatives will be assumed without ex- p1 icit mention. Surf aces of constant mean curvature and minimal surf aces are real analytic.

3

-

g, fdw AND @dw2 IN THE THEORY OF MINIMAL SURFACES These three objects are not independent, it is

( 9 ) Qi = -fgl

.

Proof: Start with = -2Xw.Yw in /l/, p. 138. The normal vector is Y =

-+

_{+ 9} ^{( 2} Re g. 2 Im 9 , g Z - ,

for minimal surfaces we have 1

X, =

(xU-ixv)

=

4

f ((1-g'), i(l+gz), 29) , and with Xu.Y = Xv*Y = 0 and g =:a+ib we get the scalar product

1 1

4 = -2 f ((l-g'), i(l+g2), ~g)]-[-~ l+ g (aU-iav, bu-ibv, a(a,-iav)+b(bu-ibv))]

.

Using g' = au+ibu and the Cauchy-Riemann equations a, = bv, av = -bu we come to (9), q.e.d.

The product fg' is a quadratic differential. I Replace fdw in (I) by - , then

( 1 ' 1 (x1,x2,x3) =

-

~e

5 ^(f

^(I-g2). (~+g'), g)

P

dw

.

Surprisingly a similar formula holds for arbitrary surfaces S.

4

-

4 dw2 AND A WEIERSTRASS FORMULA FOR GENERAL SURFACES

Let S be a regular surface defined by (2) with isothermal coordinates. The normal vector Y = (nl,n2,n$ describes the Gauss map and by stereographic projection we get the complex

valued Gauss function g =

--

Theorem l: Let S be a regular surface in IR3 defined by (2) with isothermal coordinates

W = u+iv. Let g be the Gauss function and QdwZ Hopf's quadratic differential. Then following representation formula holds:

For minimal surfaces g is meromorphic, g, = g', and we come back to (1 ^{l ) .}

Proof: The following equalities are due to Kenmotsu /2/:

(12)

4

g, = EHg W '

Attention: K. Kenmotsu uses a @-function different from Hopf 'S 9-function. If we denote Kenmotsu's function with

B K

, the relation OK = @/E holds. We have introduced Hopf 'S func- tion above.

Using (12), (13) and =

gw

we find (14) gwgw = ¹ H @ ( I + I ~ I ~ ) ~ , which will be important in the foll0wing.

Now we can prove ^(10). Using (11) and (14) we get

and the same for x2 and x3 , q.e.d.

ema ark;:

According to (13) g W = 0 iff

@

= 0 and +lgw is not singular at points and lines where g,. = 0. - If g, 0 in D, then @ E 0, all points are umbilic points or flat points and S lies on a sphere or in a plane. These two special cases must be excluded in theorem 1.

5

-

THE FORMULA OF KENMOTSU

Using (1 1) K. Kenmotsu proved (see 121, (4.4))

2gw

(15) '(X 1.X .X 2 3 =

-

^{Re ((I-g'),} i(l+g2), 29) T.z _{+ 9} dw

.

6

-

COMPARISON OF (10) AND (15)

With his important formula (15) K. Kenmotsu can construct surfaces of prescribed mean curva- ture. By introducing Hopf's differential $dw2 into (15) we come to formula (10), which is closer to the formula of Weierstrass. See also 121, (3.6).

If g,= 0, then '$ E 0 and S lies on a spere. The plane case will be excluded now. We cannot use (10) but (15). Problem 1: Set g = i7 and H = 1 in (15) and construct the unit sphere.

If S is a minimal surface (H = 0) we cannot use (15) but (10). If you want to construct mini- mal surf aces and their constant mean curvature companions ( 10) is preferable.

7 - INTEGRABILITY CONDITIONS FOR SURFACES OF CONSTANT MEAN CURVATURE

In the preceding sections we started with surfaces S and gave representation formulas to re- construct S. For applications it is more interesting to start with H or $ and g and to con- struct new surfaces S. Now we treat the constant mean curvature case.

C7-166 COLLOQUE DE PHYSIQUE

Let S, be a surface of constant mean curvature defined by (2) with isothermal coordinates

W = u+iv , g the Gauss function and cf' dwZ Hopfls analytic quadratic differential. The inte- grability condition of Kenmotsu is

(16) SW$

-3-

l+gg SwSV = 0 ⁹

see 121, (4.3). This means: The Gauss function g of S, fulfils (16), and contrary, if g is a solution of (16) with gg $ 0, then you can construct a branched surface S, using (15), see 121, theorem 4. In /3/, p. 40, there are further remarks on branch points.

To construct surfaces S, of prescribed m d w Z we prove

Theorem 2: Given S,, g and @dwZ as above. If w c D is a point with Gauss curvature K(w)

+

^0,

then (14) and (16) are equivalent.

Proof: Since

Q-

_W = 0, (14) is equivalent to

(17) - 2

gwmgw + gwvgw - l+gTj gw~w(g~w+aJ,) = 0.

If A+iB = 0 is an abbreviation for (16) and g = a+ib , then

(17') (A+iB)Bw

+

(A-iB)gw = 0 and (Aau+~bu) - i(Aav+Bbv) = 0

are abbreviations for (17). From the first part in (17') we see that (17) or (14) is a conse- quence of (16), from the second part we see that (16) is a consequence of (171, if the Jaco- bian determinant of the g-mapping

Since J measures the area distortion of the g-mapping, J = 0 iff K = 0, q.e.d.

Remarks: To find the Gauss function g of a surface Ss we have to solve the harmonic equation (16) or the nonlinear first order PDE (14) with complex analytic

.

⁽¹⁴⁾ may be considered as a first integral (conservation equation) belonging to (16). Compare with the oscillator and his equations Y + d x = 0 and k 2 + 3 x 2 = 2E. Instead of the energy E we take H@.

-

^In

/2/, (4.1), we find the integrability condition of Kenmotsu for general surfaces. D. Hoffman and R. Osserman 131 gave a third order PDE as integrability condition, which surprisingly only depends on g. Problem 2: Derive a conservation equation for the general case.

8 - CONSTRUCTION OF SURFACES OF CONSTANT MEAN CURVATURE

Since surfaces S, with vanishing K lie on right circular cylinders, we have

Theorem 3: Take any complex domain D or any Riemann surface and an analytic quadratic diffe- rential 9 d w Z $ 0 on D. Construct a solution g with 3

+

⁰ of the integrability condition (14) with constant H. Then the representation formula (10) gives a branched surface S H of constant mean curvature H. On the contrary, every surface SU not lying on a sphere or a right circular cylinder can be constructed on this way. The isolated zeros of ^L? are umbilic points of S,.

Problem 3: Determine g and (O of the right circular cylinder and use (10).

9 - CONSTRUCTION OF CONSTANT MEAN CURVATURE COMPANIONS OF MINIMAL SURFACES

Let S, be a minimal surface with g and

4

dwZ (instead of fdw). To construct constant mean cur- vature companions we take the same D and mdw2 and a new Gauss function G(w) = g(w)+&(w). G and E are not complex analytic. Since g,- = 0 and gw = g', the integrability condition (14) is

Theorem 4: If E,, is a one parameter family of solutions of (19) with E, --0 for H -- 0 in D, then D, g+&, @ dwZ and the representatio; formula (10) give a one parameter family of compa- nlons of S,. Since there is the same 3 d w , the umbilic points of the companions come from the flat points of S,.

Problem 4: If a parabola rolls on a stright line, the focus describes a catenary, the meridian of the catenoid. With a hyperboloid you can get on this way the hyperbolic surfaces of Delau- nay of constant mean curvature. In this Delaunay class we may find companions of (a part of) the catenoid. Determine the &-functions.

Approximation: Linearising (19) we get

(20) - 6, = 4 l H

7

@ (1+1g12)2 =

- a

^{1 H} f (l+jg12)2

with f =

m .

The inhomogeneous Cauchy-Riemann equation (20) has the solution

- 7

Now theorem 4 can be used.

Problem 5: D.M. Anderson has constructed constant mean curvature companions of the periodic minimal surfaces of H.A. Schwarz, see 141, p. 240. Use the method above to construct the com- panions.

10 - GAUSS AND TOTAL CURVATURE

Let S be a regular surface defined by (2) with isothermal coordinates, g the Gauss function and @dw2 Hopf's differential. From (7) and (14) we get

Note: I g w J 2 - J g G J 2 = J.

The curvatura integra is

We see the area distortion J of the g-mapping and the spherical area distortion

&

of the Gauss map.

For minimal surfaces gg = 0, g, = g', @ = -fgt and we get the well known formula

1 1 - FINAL REMARKS

Complete abelian minimal surfaces in 01' can be constructed with compact Riemann surfaces D.

Two subcl'asses are the complete minimal surfaces of finite total curvature C and the periodic surfaces, which are of interest in this workshop. A geometric characterization is given in 151.

To every compact Riemann surface D you can construct complete minimal surfaces of finite C, see 161. Only to special Riemann surfaces D you can construct periodic minimal surfaces with- out boundary components. For example the genus p of D must be 2 3 , since there are three in- dependent differentials of first kind, fdw, gfdw and g2fdw, see 171, p. 35. The most simple Riemann surfaces of genus p = 3 are the hyperel liptic surfaces generated by {(W-a, ).

. .

^(W-a8)'

.

So it is not surprising that the first Schwarz surfaces are constructed with

{ W .

C7-168 COLLOQUE DE PHYSIQUE

Since g, fdw and 4 d w 2 are analytic functions and differentials on compact Riemann surfaces 0 , we have the formulas of Riemann-Hurwitz and Riemann of the classical function theory, for example

(25) number of zeros of @dw2 - number of poles = 4p-4

.

If we translate these formulas into geometric language, we find the Gauss-Bonnet theorem and a flat point formula. For abelian minimal surfaces this was done in /5/. For periodic minimal surfaces without boundary components we deduce the well known formula C = 21c)C for the to- tal curvature of a fundamental part and

for the sum of the flat points, counted with multiplicity, see /5/, (5), and /7/, p. 33.

According to Hopf this can also be deduced from Poincar6's index theorem.

If S, is a surface of constant mean curvature, then

for the sum of the umbilic points in a fundamental part, counted with multiplicity.

The results of this last section I had in mind when I wrote my abstract with the title 'Glo- bal formulas for compact Riemann surfaces, minimal surfaces and surfaces of constant mean curvature" at the end of July. Then I came to the investigations of the preceeding sections via Hopf 'S quadratic differential.

REFERENCES

/l/ Hopf, H., Differential Geometry in the Large, Lecture Notes in Math. 1000, 1983.

121 Kenmotsu, K., Weierstrass Formula for Surfaces of Prescribed Mean Curvature, Math. Ann.

245 (1979), 89-99.

/3/ Hoffman, D.A. and Osserman, R., The Gauss Map of Surfaces in jR3 and IR', Proc. London Math. Soc. (3),

50

(1985), 27-56.

141 Nitsche, J.C.C., Lectures on Minimal Surfaces, Vol. 1, Cambridge Univ. Press 1989.

151 Gackstatter, F . , uber abelsche Minimalf lachen, Math. Nachr. 74 (1976), 157-165.

/6/ Gackstatter, F. und Kunert, R., Konstruktion v01 lstandiger G i m a l f lachen von end1 icher Gesamtkrummung, Arch. Rat. Mech. Anal. 65 ( 19771, 289-297.

171 Wohlgemuth, M., Abelsche MinimalflachenTDiplomarbeit Univ. Bonn 1988.

Berlin, im September 1990,