Planar harmonic mappings and curvature estimates

PLANAR HARMONIC MAPPINGS AND CURVATURE ESTIMATES

SOaK HEUI JUN

1. Introduction

Let E be the class of all complex-valued, harmonic, orientation- preserving, univalent mappings defined on

= {z : Izl > I} that map

00

to

Then fEE has the representation (1.1)

where

fez) = h(z) + g(z) + A log Izl

00 00

h(z) = az + L

^{z -}

^{and g(z) = I3z +} L ^bkz-

k=O k=l

are analytic in

and 0

1131 < lal (see [3]). In addition, f can be viewed as a solution of the partial differential equation

(1.2) h=afz

where the function

= -t is analytic in ~ and satisfies la(z)1 < 1 [2]. The function a is called the second complex dilatation function of f. Conversely, any univalent solution of (1.2) with

analytic and

lal < 1 is an orientation-preserving harmonic mapping on

and can be expressed in the form f = HoD, where D is a diffeomorphism of

~

and H is analytic in

For more details, see [7].

There is a geometric interpretation of the analytic function a. Let

n be a doubly connected domain in the extended w-plane having the

Received October 22, 1994.

1991 AMS Subject Classification: 30C45, 53AI0, 30C50.

Key words: harmonic mapping, minimal surface, curvature.

This work was supported by NON DIRECTED RESEARCH FUND, Korea Research Foundation, 1993.

804 Sook Heui Jun

point w =

as one of its boundary continua. Consider the nonpara- metric surface S over n whose cQordinates are u, v and G = 4>( u, v).

Then S is a minimal surface if and. only if there is a univalent harmonic mapping f = h +9 + A log Izl (EE) from .6. onto n such that the third component satisfies the differential relation

2 - 2

G z = ^-Jzfz = ^-afz·

The normal direction to the minimal surface at a point (u, v, G) is given by the relation

N = (-2Im{J!i}, -2Re{ J!i}, I-la!) 1+ lal

which depends only on the dilatation function a.

In the second section of this article we discuss the relation between harrrionic mappings f = h+9+Alog Izl in.6. and the analytic functions h+f.g, where kl = 1, when C\f(.6.) is convex or convex in one direction.

In the third section we give some applications of univalent harmonic mappings fEE to nonparametric minimal surfaces over n.

2. Convex and convex in one direction mappings

DEFINITION

2.1. A set D is called convex in the direction

(0

<p

< '7t") if every line parallel to the line through 0 and e iq, has a con-

nected intersection with D.

DEFINITION

2.2. A function f defined on .6. = {z : Izl > I} is convex, convex in the qirection

(0

< 11") if f maps .6. onto a domain whose complement is convex, convex in the direction

(0

<p

< 11"), respectively.

DEFINITION

2.3. A function fez) in a domain D is said to be

valent in D if for each

(infinity included) the equation fez) =

has at most

roots in D and if there is some

such that the equation

fez) =

has exactly

roots in D.

LEMMA

2.4. Let D (00 E D) be a domain whose complement is convex in the direction of the real axis, and let p( w) be a continuous extended real-valued function in D. H the mapping W

W + p(w) is locally 1-1, then it is at most 2-valent in D. In this case the complement of the image of D is convex in the direction of the real axis.

Proof. Let WllW2 E D. HWI +p(wd = W2+P(W2)

W2), then writing

=

+

W2 =

+ iV2 we have

= V2 = c, say, and

+

+ ic) =

+ ^P(U2 + ic).

The complement of the image of D is convex in the direction of the real axis, since the mapping

+ p(w) maps horizontal lines into themselves.

CASE

1: H

W2 n Dc = 4J, then the real function

+ ^p(

+

ic), which is defined on some interval, is not strictly monotonic and therefore not locally I-I.

CASE

2: If

W2 n DC

4J, then we have two different cases such that

W2 n DC has only one point, say a, or 'Wi'W'2 n DC has more than one point (i.e. line segment), say ab, because DC is convex in the direction of the real axis (see figure below).

Suppose there exists W E

W2 n D such that w + p( w) =

+

p(wd = W2 + P(W2), then Im{w} = c. Without loss of generality let's assume

E

a, then a contradiction is obtained as in Case I.

So for each Wo the equation W + p( w) = Wo has at most 2 roots in D. Therefore

+ p( w) is at most 2-valent in D by the definition 2.3.

THEOREM

2.5. Let f ELand A E R. H f is convex in the direction

of the real axis, then h - 9 is confonnal, at most 2-valent, and convex

in the direction of the real axis in

806 Sook Heui Jun

Proof. Since f is univalent, there exists a function z = z(w) such . that f(z(w» = ^{w and z(J(z»} = ^z. Thus we have h - 9 = ^{f -}

Aloglzl- 2Re{g} and h(z(w» - g(z(w» = w + ^{pew) where pew)} =

-Aloglz(w)I-2Re{g(z(w»}. Sincea(z) = _;;~:{:~t1 satisfies la(z)1 <

1, we have h'(z) - g'(z) 1= ⁰

Since h'(z) - g'(z) 1= ^{0 and} z(w) is 1-1, the mapping w + p(w) is locally 1-1. Therefore w + ^{p(w) is at}

most 2-valent and convex in the direction of the real axis by Lemma 2.4.

THEOREM 2.6. Let f = h + ⁹ + ^{A log Izl, A E R., be} harmonic and locally univalent in Ll. If h - 9 is

conformal univalent mapping of Ll and convex in the direction of the real axis, then the function f is at most 2-valent and convex in the direction of the real axis.

Proof Writing w = h(z) - g(z), z = z(w), we have f(z(w» =

+ 2Re{g(z(w»)-} + Aloglz(w)l

w + pew) is locally 1-1 and so at most 2-valent and convex in the direction of the real axis by Lemma 2.4.

THEOREM 2.7. If a function f ^{= h} + 9 E :E is convex, then the analytic functions

(0::; 4> < 271) are convex in the direction 4>/2 and at most 2-valent.

Proof Since f is convex, e -it/>/2 f = e -it/>/2 g + e -it/>/2h is convex. So e-

e

g is convex in the direction of the real axis and at most 2-valent by Theorem 2.5. Hence h - eit/>g is convex in the direction 4>/2

and at most 2-valent.

3. Curvature estimates for some minimal surfaces

Let n be a doubly connected domain in the extended w-plane hav- ing the point

=

as one of its boundary continua. Let S be a nonparametric surface over n ^{given by}

S={(u,v,4>(u,v»: u+ivEn}.

(3.1) Let

·J,(u v)

= J _{J1 ...} ^{</>u dv -}

^{</>v du}

+

+

. d

F =

+ ltP, an w = J

1 + ¹ +

+ </>v

Then S is a minimal surface if and only if S admits a conformal reparametrization of the form

S={(u(z),v(z),G(z»

where

U(z)=Re{~JF'(~-w)dz}

v(z) = Re{ -~ J ^F' (~+ ^{w )dz}, and}

G(z) = Re{F(z)}.

The function

1 J ^{F' 1J}

f = u + iv = - -dz - - wF'dz

2 w 2

is a univalent harmonic mapping from

onto n with f(

=

and G is a real-valued harmonic function satisfying

= -af; = w

f;

where a is defined in (1.2). In

the variables F and w considered as functions of z are regular analytic, _~~ and w are single-valued and

t ^{~~ "I 0,}

The function w( z) is regular at z =

and Iw(

I < 1.

Furthermore, ~(W is regular and different from zero at

Observe also that we may assume f is orientation-preserving and that we may obtain any other set of isothermal parameters by applying a conformal mapping to

[1]. Since f(z) E E, f is of the form (1.1).

REMARK.

The meromorphic function -l/w has geometric signifi- cance. It is the stereographic projection of the Gauss map. That is, the trace of the unit normal vector to the surface

(3.2) N = (-2Re{w}, 2Im{w}, 1 - Iw1

1 + Iwl

808 Sook He.ui .Jun

w

F'

has stereographic projection -l/w.

Here is the strategy. Use knowledge about harmonic. mappings of

onto Q to gain infonnation about nonparametric minimal surfaces S that lie over

The Gaussian curvature K at each point of S is given by

see [6]. From (3.1) and (1.1), we have that 2(h'-g'+¥)

1+w

= -

Hence,

Furthennore the estimate 1~iJ12 :s; Iz1L 1 from Schwarz's lemma for

Izl > 1 implies

(3.3)

where

In this section we discuss the estimates of IK(p) I for the following two cases:

(1) Q = C\[a, b], arbitrary

E Qj (2)

= C\{O}, arbitrary p E Qj

where [a, b] is a real line segment in the complex plane.

(1) The case of Q = C\[a, b].

If n = C\[a, b], thfm f is of the form

b

+

{3 - b

f(z)=oz+ao+ +{3z+-=

z z

00

""" ak

+ 2Re LJ zk + ^A log Izl,

k=2

where ao is real and 0 - {3 is positive real [4J. SO h - 9 = (0 - {3)(z + z-1) + ao. In [4], the author has observed the following result.

LEMMA

3.1 [4,

2.4]. H S is a nonparametric minimal sunace over n, and if tbe unit normal to tbe surface at

is (0,0,1), tben we bave

(3.4) (3.5)

wbere p = P1 + iP2 E n ^and d is tbe distance from p to [a, b].

REMARK.

w( (0) = 0 if and only if the unit normal vector to the surface with the standard orientation at

is (0,0,1).

In this article, we want to consider the case w( (0) =F O. Since w and F' are analytic in ~ ^and ~ I- 0,

we have series expansions

(3.6)

co

""" X k

w(z) = L- zk'

k=O

co

F'(z) = L ~:'

k=O

where XoYo I- 0, Ixol < ^1. Substitute these two series into (3.1) and obtain

f(z) = -2 1

[YO z + (lL!. _ _YO~1) ^logz + ...]

xo xo xo

- 2' 1 [xoYoz + (YOX1 + XOY1) log z +...]+ constant.

810 Sook Heui Jun

Since j(z) must be a single-valued function we must have

YI YOXI ( )

- - - 2 -

= - YOXI +XOYI .

Xo Xo

Hence

j(z) = ;:oz+ O(lzl-I) + [_X~oz+O(lzl-I)]

( YI YOXI)

+ - -

log Izl + constant.

Xo Xo

That is, we have

(3.7) A = - - ^YI

YOXI = - ( ^YOXI + XOYI . )

Xo Xo

Also YI is real because Re{ F} = if> and if> is single-valued.

Let P = ^{PI + iP2 E flj} then there exists z = ^,e

^E ~ such that j(z) =

^{Since a - (3 is real, P2} = (a- (3)(r_,-I) sin8+Im{Alog,}.

From (3.3), if A is real we have

(3.8) 4

IKI ~

(, - ~) ^{(a - (3)2} + 4p~

The necessary and sufficient condition for P2 = ^{0 is sin 8 =} 0 since a - (3 and ,_,-I are positive real for, > 1. Thus we obtain the curvature estimates

and

IKI ~ ⁴

^{if P2 = O.}

(a - (3)2 (, - ~ ⁾

Although the parameter, in (3.8) is not a geometric quantity, the estimate (3.8) gives us a very important result near

E S1. Let 1-+

in the estimate (3.8) to conclude that K = O. Mter a translation we find that the sum and product of the principal curvatures are zero.

This implies that the minimal surface S is plane near

since f(

=

00 E

S1.

Now we want to discuss what A real means in the case of w(00) =J o.

THEOREM

3.2. A is real if and only if Xo is real or Xl = Y1 = o.

Proof. From (3.7), we obtain (3.9)

- - 2

X1YO - X1YOXO Y1 =

xo(l + Ixol2) because Y1 is real.

Suppose A is real ; then from (3.7), we have -(YOX1 + ^XOY1) = -!L -

Since 0 < Ixol < 1, we obtain

0

(3.10) X1Y0(1 - _X~) and

= Xo(1 + x~)

Substitute (3.10) into (3.9) and obtain

This implies that Y1 = 0 or Xo is real. If

= ^{0, then Xl} = ^{0 from}

(3.7). Thus A is real implies that Xl = ^Y1 = 0 or Xo is real.

Assume that Xo is real or Xl = ^Y1 = ^O.

If Xl = ^Y1 = 0, then A = 0, so that A is real.

If Xo is real, then from (3.7) we have A = -X1YO - XOY1 because Y1 is real. Substitute (3.9) into above equation and obtain A = -~~r

Thus A is real.

The following remarks give some geometric meaning to the condi- tions in Theorem 3.2.

REMARKS.

1. Let Nco be the unit normal vector to the surface at

00.

Then from (3.2) we have

N _ (-2Re{Xo} 2Im{xo} 1-lxoI2)

00 -

1 + I ^xol

1 + ^{Ix ol}

1 + ^{Ix o/}

Thus (i) Nco C xz-plane if and only if Xo is real.

(ii) The unit normal vector Nco has real stereographic projection

-l/xo if and only if Xo is real.

812 SookH~uiJun

2. If

= YI = 0, then from (3.7) .and (3.6), we have A = ^{0 and}

F( z) = YoZ -1I:j- -

- ... + constant. Recall that Re {F( z)} = 4J(u, v).

We want to find 4J( u, v). Consider the equation

. f( Yo XoYo_ ()

w = ^u + ^xv = ^z) = - z - - - z + 0 1

2xo 2 (z

Then we obtain

Yo + xoYo-,- = I.YoI

I xoYol

+ 0(1)

2xo w 2 w 41 xoF ^{z 4 z •}

This implies that z =

+0(1). Since Re{F(z)} = 4J(u v)

we have

4J( u, v) = Re{yoz -

2 - •••

+ constant}

z 2z

=Re {YO (2(x o w + I ^xo I

'X'(jW)) _

Yo(1-l xoI 4

) ) _ ...}.

Yo(1-lxoI4) 2(xow + Ixol2xow)

Th us'P ,,/.,(

) =

v+ 0(1) as u +v 2 2 T h

ere

. no logarithmic term·in above equation. Since YI = 0, F(z) is single-valued function of z, ^{that is, 'l/J} is single valued. By the elementa,ry calculation, we obtain K = 0 near

E n. So the minimal surface S is plane near

00.

If 4J(u, v) is interpreted as the potential of a flow of a hypothetical

"Chaplygin gas" whose density

aild speed v are connected by the relation p2(1 + v

= 1, then 'l/J is the stream-function, F the complex potential and 4Ju- i<pv the conjugate complex velocity (cf. [1]).

(2) The case of n = {w : Iwl > O}.

LEMMA

3.3 15,

2.4]. H f E

and f extends to be of bounded variation on Izl = 1, then

- L L

la + bll ::; ^271"' Ibnl::; 2n7l" for n ~ 2,

L L

1,8 + all::; ^-2' 71" lanl::; . ^{- 2}

^{for n} ~ 2,

where L is the length of f(lzl = 1).

THEOREM

3.4. H fEE and f(f),,) = ^C\{p} for some pE C, then f

has the representation

p -

f(z) = ^o:z +

_z + ^{{1z -} = _z ^{+ A log Izl·}

Proof. Since f(Ll)

C\{p}, the length of f(lzl = 1) is zero. There- fore it follows from Lemma 3.3 that

0:

= -b}, ^{1 = -ab and

an

= b

= 0 for n

2.

Substitute these into (1.1) and obtain

p -

f(z) = ^o:z + ao - - + {1z - = + A log Izl.

z z

f(lzl = 1) = ^{p} implies that ao = ^p.

THEOREM

3.5. H S is a nonparametric minimal surface over n =

C\{O} and if the unit normal to the surface at

is (0,0,1), then S is a plane.

Proof. Let b - a

° in the curvature estimates (3.4) and (3.5) of Lemma 3.1. Then

= O. Since S is a minimal surface, we find that the sum and product of the principal curvatures are zero at every point.

This implies that S is a plane.

References

1. L. Bers, Isolated singularities of minimal surfaces, Ann. Math. 53 (1951), 364-386.

2. W. Hengartner and G. Schober, Harmonic mappings with given dilatation, J.

London Math. Soc. 33 (1986), 473-483.

3. W. Hengartner and G. Schober, Univalent harmonic junctions, Thans. Amer.

Math. Soc. 299 (1987), 1-31.

4. S. H. Jun, CurtJature estimates for minimal surfaces, Proc. Amer. Math. Soc.

114(1992), 527-533.

5. S. H. Jun, Univalent harmonic mappings on

a =

Izl >

Math. Soc. 119 (1993), 109-114.

6. R. Ossennan, "A surtJey of minimal surfaces", Dover, 1986.

814 Sook Heui Jun

7. W. L. Wendland, "Elliptic systems in the plane", Pitman, London, 1979.

Department of Mathematics Seoul Woman's University

126 Kongnung 2-dong, Nowon-Gu

Seoul 139-774, Korea