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## S CUOLA N ORMALE S UPERIORE DI P ISA Classe di Scienze

e

o

### 1 (1976), p. 139-155

<http://www.numdam.org/item?id=ASNSP_1976_4_3_1_139_0>

© Scuola Normale Superiore, Pisa, 1976, tous droits réservés.

L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale.

Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

(2)

### Regularity

JOHANNES C. C. NITSCHE

### (*)

dedicated to Hans

### Lewy

1. - This paper deals with the

### regularity properties

of the free boundaries

for minimal surfaces under

### assumptions

which are weaker than those ac-

cessible heretofore.

Consider a

in Euclidean

### 3-space consisting

of a surface T

and of a rectifiable Jordan arc .h =

its end

on

but no other

### points

in common with T. Denote

P the semi-disc

in the

### ( u, v ) -plane {~~;~-)-~1~&#x3E;0}~ by

a’ P and a" P its

and

and

### by

P’ the domain P U a’P. A surface S =

=

### X(u, v) ; (u, v)

E is

said to be bounded

the

or

if its

vector

=

satisfies the

conditions :

### ii) v)

maps the arc a’P onto the open arc

0 t

### yr} monotonically

in such a way that

lim sin

_

lim

sin

_

o-+o 0-n-0

that

### is,

there exists a continuous

function t =

### t(8)

which maps the interval 0 0 onto the interval 0 t n such that

sin

=

The relation n--

### vn)]

= 0 holds for every sequence of

in P’

to a

on a’ P.

Here tET

### I

denotes the distance between the

### point X

and

the surface T.

Pervenuto alla Redazione il 15 Settembre 1975.

(3)

### Although

the distance function

### v)]

is continuous in the closure

the same cannot

In

the

trace of S on

that

the set of limit

### points

on T for all sequences

as in

may well look

bizarre.

We shall denote

### by %

the collection of all surfaces S =

E

bounded

the chain

whose

vector

to

n

If the end

### points

of F can be connected on T

### by

a recti-

fiable Jordan arc, then the solution of Plateau’s

for the

closed contour

### represents

a surface of class %. In 1938 R. Courant

### proved

that whenever the class 9t is not

### empty

there exists in 52( a surface S mini-

### mizing

the value of Dirichlet’s

see

pp. 87-96 and

### [3],

pp. 201-203. It was later shown in

### [13]-and

this is crucial here-that the solution surface S has also the smallest

area among all

surfaces bounded

the chain

The

### posi-

tion vector of S possesses the

The

of a’ P onto

is

### iv) v)

is harmonic in P and satisfies in P the conditions 0.

In the

of

pp.

is a

minimal surface.

### Naturally,

there may be more than one solution.

For the last

### thirty-five

years it has been a

of

to

the

### regularity

of the solution surface on its free

### boundary

and the

nature of its trace.

a

### large body

of results exists

these

A detailed

and a

can be found

in

### chapter

VI.2 and on p. 707 of

In the

case, if T is a

that

### part

of the trace which

### corresponds

to the open arc a"P is an

### analytic

curve, and the solution vector

an

### analytic

extension across a"P.

This has been

I. F. Ritter

The case of an

surface T was

studied

H.

### Lewy [8].

It is obvious that the smoothness of the trace will

on the

of the

surface T. From

and

### [13]

we know that the solution vector

### v)

which minimizes Dirichlet’s

### integral

in the class 9t has a continuous

extension to the closure of each domain

=

v; u2 + v2 C

v &#x3E;

### 0 a 1, belonging

to the Holder class

if the

### supporting

surface

satisfies a chord-arc condition with constant c &#x3E; 0. Here y =

(4)

For a

### regular

surface T of class C’ it follows that

to

where the

can be

### arbitrarily

chosen in the open interval

and is

of a. In

W.

a

### Kellogg-type

theorem:

If T is an «admissible » surface of class

then

E

E if a &#x3E;

and

E for

if a = 0.

### Jager’s proof,

as well as similar methods

### employed

in connection with related

among other

### things,

that the normal vector of the

### supporting

surface T have continuous second

and thus lose

their

### applicability

if T does not possess a certain

### «starting regularity » - C3

at least. If one wants to go

### further,

new difficulties have to be overcome.

it would be of

### great

interest to settle the cases m == 2 and

m = 1. After

### all,

in the related situation of Plateau’s

a

### Kellogg- type

theorem is known to be valid for all see

### [11], [12].

It is the purpose of the

### present

paper to extend the

of

### Jager’s

theorem to the case m = 2. Our result

a

formulated

921 of

The

consists of two

In

the first

an initial

### regularity property

is ascertained which then per- mits an

of the

condition in its

form. The

second

### part

concerns itself with the

Here the ideas

of

are

in a

### suitably

modified form with the effect that any reference to the

for solutions of

a formidable

not amenable to

### transparent

demonstrations-can be

### entirely

avoided. Our result is as follows

surface will be

in section

THEOREM.

class C’,’

### (m 2, 7 0 a 1)

then the solution vector

### v)

has a continuous extension to the closure

each

domain

### Pa, 0 a 1, belonging

to the H61der class

0 a

to the

Hilder class

every

c

### (0, 1) i f

0153 = 1 and to the HOlder class

### Cm-1,0 for (o, 1 ) i f

a = 0.

REMARK 1. For m &#x3E; 2 the cases

and are

alent. In

the

### conclusion X

E is an immediate consequence of the

T E

a

of our

### proofs

will show that the

### C’,’-character-and then,

as a consequence, the

the trace is

assured

the

that T

to class

It goes without

that the

from

to is less

than the

from to

### Cl~°‘

would be. Since our demonstration of the

rests in

### part

on the function theoretic methods

in

### [11],

one could also resort to further facts known from the

of

functions and

### replace

the various Holder conditions

### by

(5)

suitable Dini conditions. For Plateau’s

### problem

this has been done

### by F. D. Lesley [7].

REMARK 2. It can be

that the vector

### ~(u, v)

is Holder continuous in the closure of all of P

the chain

### (7~, T)

itself satisfies a chord-

arc

so

in

### particular,

the arc .1~ meets the sur-

face T

### subject

to a « lift-off» » condition

p.

### 134).

It is not pos-

sible to achieve a

for

near the

=

### 0)

even if rand T themselves should possess such a

unless h

meets T

### orthogonally.

This can

be observed in the

case

where the

surface is a

Here the vector

an

### analytic extension,

and the extended vector appears as a solution of Plateau’s

for a contour

of .r and its

### image

under reflec- tion across T. Since this contour will have two corners, unless h meets T

the vector

### g(u, v)

can at best be Holder continuous in

The trace

is

the theorem of

its

is seen to be

### majorized by

that

of T itself. A similar estimate holds

also for more

sup-

### porting

surfaces T. A detailed discussion of this

will be the

of an

### independent investigation.

2. - A surface T imbedded in 3-dimensional Euclidean space is said to be admissible of class

### (m&#x3E;2,0~ocl)

if it satisfies the

conditions:

For every

zo = yo,

### ~0}

of T there exist an open

### sphere containing

go and a function E=

y,

E with

### non-vanishing

gra- dient such that the

E

= 0

E T m 8 are

There is a

### positive

number p = pT such that every

in

the

set

can be

in the form

The vectors

### a*(~), 9è*(!),

as well as the functions

are of class

in

defines a

of T

foot

on

### T ), 91*(X)

is the unit normal vector of T in this

There is a

constant

### Co oo

such that

(6)

Note that an admissible surface which is not

### compact

cannot have finite

### boundary points.

The reader may find it convenient to

T

### locally

with the aid

of isothermal Gaussian

in the

=

where

If

### * (\$o , qo)

denotes the unit normal vector of the surface T at one of its

every

r =

y,

in a

has a

This

### representation

establishes a one-to-one

### correspondence

between the

coordinates x, y, z of the

and the

=

y,

77

y,

y,

A

the differential

N

### (coefficients

of the second fundamental form of

and

K

### (mean

and

Gaussian curvatures of

### T)

shows that

Here

Consider a vector

### X(u, v)

which has continuous first derivatives in a domain of the

### (u, v)-plane

and maps this domain into the

set

In the

### representation

then the abbreviations

=

==

and

==

### v))

will be used.

(7)

3. - We now assume that the

### supporting

surface T is admissible of class C2

for

The set W of

surfaces is

not

### empty

in this case. Let S =

=

E

### P’}

be a solution sur-

face whose

### position

vector minimizes Dirichlet’s

to the

conditions

of section 1.

iv = w =

we shall

### interchangeably

use the notations

or

or

later

also

### a(w) etc.)-whichever

is most convenient.

### Denoting by C(wo; e)

the

domain we introduce the abbreviation

From the remarks in the introduction it is clear that

### v)

has a

Hlöder continuous extension to every domain

1 0 a

so that

for Wl,

and

stant

on a and

### C, = C1(a, {3).)

We shall show here that

### X(u, v)

satisfies in fact a

condition:

class C2-or

class see

remark 1

### of

section 1-then the solution vector

### v)

has a continuous extension to each domain

0 C a C

### 1, belonging

to class

In view of a well-known lemma of C. B. MORREY

pp.

### 134-135)

this

lemma will be a consequence of the

### following

assertion :

For every number a in 0 a 1 there exist

### positive

constants d and M

such that

all

We

### proceed

to prove this assertion.

Choose a number d in the interval

### 0 d (1 - a) /2

which is so small

that the domain

is

into the

set

### Tp

and that Now consider a

in

For and

we have

p.

### Assuming

now that 0 vo d and vo

we find

(8)

For a fixed

### point

uo on 8"P and 0 r 1-

### luo we

shall introduce the ab- breviation

=

uo,

virtue of

Since

the

### following

is true for almost all values of r in 1-

### 1)

There are no branch

of S on the arc

=

0 0 that

+

0 for 0 0 n.

### 2)

The derivative exists and is

### equal

to

For such values of r let

be the

of the curve

Schwarz’s

y we find

Vve shall next

### modify

our surface S. The

of S which

to the semi-disc

=

v;

will be

a

chosen surface

bounded

the chain

### Ar, T). (It

is immaterial here whether

### ~lr

is a Jordan curve or

or has

### points

in common with is a ruled surface

the

normals to T

the

of

### (Remember

the abbreviations

10 - Annali della Scuola Norm. Sup. di Pisa

(9)

and thus

Now

Here a is an

the

=

we obtain

### and,

since ao and 9to are both vectors

to

so that

and

### finally

A combination of the above

### inequalities

where the number a is

### again arbitrary

but restricted to the interval 0 a 1.

Since the

= uo has its

on the

surface

we

(10)

our

+

### 2d, -1

for 0 r 2d. We now choose

or =

whenever

whenever

-~

The function

in the last

is defined

the relation

For the area

of

### ~

we now obtain the estimates

The new

constant

in a

way on

and d.

Since

0 for 0 0 and

is a square

function of

### 0,

we can introduce the arc

on

as

in the

Then

(11)

y

### ,us ~ 1.

We set and define a new function

### (O(s)

denotes the inverse function of

For the

on the

### right

hand

side we now obtain

The function

is

### analytic

for and continuous for

The derivative

is square

Since the end

of

lie on

also the

conditions

### v(O)

= = 0 are satisfied. Under these as-

the

see

### [5],

p. 185. It now follows that

The

surface

with the

vector

is bounded

the chain

The vector is

### linearly

absolute con-

tinuous in P’ with square

### integrable

first derivatives so that the area

curves to

(12)

the

of the surface

theorem

### A~

for

which

we are led to the

This

### inequality

holds for almost all r in vo r vo

d.

### Integrating

be-

tween the limits r = vo

and r = vo

and

our current as-

0 vo

### d,

we obtain

and therefore

where

We have now dealt with the cases

and 0 vo

### e

d. For the last

case combination of the

Since

### Mo M,

we have in all cases

Our

### assertion,

and with it our lemma are

4. - At the

### present stage

we know that the vector

which is har-

monic in

has

### uniformly

bounded derivatives in each

### domain Pa,

0 a 1. From this fact it can be concluded that the limits lim v-+ o U

(13)

lim

### ,(u, v)

as well as the derivatives

exist and are

v +0

### respectively,

y for almost all u in - 1 u 1.

to W.

### Jager [6],

the solution surface =

E

### P’l

satisfies a certain weak

### transversality

condition: For -1 u 1 and suf -

small

the vector

can be

in the form

Let now be an

test vector with

### compact support

in P V a" P. Then

Consider a

1 u

### 1,

for which the limit lim

### v)

as

well as the derivative

exist and are

all u have this

In the

of the

on T we

the

Here the

### arguments in X,

are u and v and the

in

are

and

From the

condition

### (10)

it can be concluded that the vector

must be

### parallel

to the normal vector 91 of T at the

In other words :

The solution

,~S is

### orthogonal

to T in almost all

the

### trace I

5. - We can now turn to the

### proof

of our theorem for which we shall scrutinize the

of the

vector

### v)

in a fixed domain

### Pa,

I

0 a 1. Let T(a) be a

of the

### supporting

surface T which contains the subarc

### (g = X(u, 0) ; lul

of the trace of S on T. Consider

an

### arbitrary point

in T(a) and choose a coordinate

for which this

is the

while the

to T becomes the

There exist

### positive

constants r and C such that the connected

of T

the

and

in the

of

has a

=

### 1p(x, y).

If T is an admissible sur-

face of

the function

### 1p(x, y)

possesses continuous

(14)

derivatives up to those of order m and satisfies the relations

=

=

= 0 as well as

for

### x 2 -~- y 2 c r 2, x ~ -~- y 2 c r 2.

Here ak stands for any

### partial

derivative of order k. The numbers r and C can be chosen to be the same for all

in T~a~ and

### depend

on the selection of T~a~

Let be a

on a"P in whose

we

wish to

the

### regularity properties

of the trace of S on T. In the

of the

on T we

### represent

the surface T in the form

z =

as

### explained

above. The transition to the new

achieved

### by

a translation and a

### rotation,

does not affect the

or the

of

A number E =

### 8(a)

can be chosen

so small that the

### following

conditions are satisfied:

For

the

### points

are contained in the

set

### (the neighborhood

of T in which the

is

### valid) .

Since we can now restrict the

of

### v)

to the closure of the domain

### C(u,,; 8)

r1 P it is convenient to map this domain

onto

the unit disc

in a new w’=

### u’+ iv’-plane

in such a way that the

to the

w’

w’

The

= uo -

and w = uo

will then be

onto certain

and

The

number

### 0, 0 9 yr/2, depends

on a, but does not

on the

location of the

### point (uo, 0)

on a"P. The vector

### l(u, v)

becomes a

harmonic vector of the new variables

and retains all

in the

restrict our

of the

### boundary

behavior to the subarc

of

For the sake of

we shall

### again

denote the new variables

### by u

and v

and the new vector

coordinates

### according

to

w = u -~- iv = we

as

use the notation

or

or

The harmonic

of the vector

are

real

of

(15)

which

### satisfy

the relation

The fact that the vector

satisfies a

condition

the

for

### ~~9.

The constant N is inde-

### pendent

of the location of the

on the

of a’P

and

on ac

on the choice of

see «

in

### [11],

pp. 313-314. It then follows from the

of

### complex

functions that the derivatives as well as the radial limits

### lim

exist for al- most all values of 8 in -

0

### ~/2

and that

0-1 i

for these values.

Let

such a

### good »

value of 6. In the

of the

on T the

### supporting

surface T will now be

in the

form z =

### 1p(x, y). Again

we retain the notation

for the trans-

formed

a

vector. We then

in

=

=

### z(eiOo)

= 0. Note that in view of condition

The

condition

the relation

= 0 for

almost all 8

### in 2013~/20~/2.

Since the normal vector 9Z is

to the vector

1pan - y~~ ,

### 1 11

we see that

for almost all 8 For the same values of 0 a differen- tiation of the

=

Since

=

=

=

we find that

=

=

= = 0.

### Consequently,

for almost all 0 in

(16)

and

Since

it follows as in

esp. p.

that

The constant

Due to the

of the

functions

in

the

### inequality (14) holds,

with the same constant

for all

esp. p.

the vector

to class for

is an

number

### subject

to the restric-

tion 0 1. Thus the first

of our theorem

### (Ol’P-regularity

of the trace

as consequence of the C2-character of

is

### proved.

Assume next that the

surface T

to class

### 02,rx, Using

facts

known we find

and

Let us first consider the case oc 1. We then choose and have

etc. and

### finally

where

and similar relations for the differences

(17)

From these relations it can be concluded as in

esp. p.

that the

three functions

the conditions

### Therefore,

the earlier conclusions

### g§(eei°)

and the first derivatives of the vector

### ~(w) apply

now also to the radial limits of and the second derivatives of

Differentiation of the relation

for a

value

### 8o

in an interval

From the identities

and

we obtain

and

### Considering

that

it follows that the functions

and

in a fashion

after the

in

and

### by

now familiar to us, that the vector

to class

for

If a

we can

### choose fl arbitrarily

and conclude that

to

class in

for any

E

The

cases m =

can be treated

### analogously by repeated

differentiation of the identities

The

research was

in

### part by

the

National Science Foundation

### through

Grant No. MPS 75-05418.

(18)

BIBLIOGRAPHY

R.

The existence

a minimal

### surface of

least area bounded

### by

pre- scribed Jordan arcs and

24

pp. 97-101.

### [2]

R. COURANT, The existence

minimal

structure

under

### prescribed boundary

conditions, Acta Math., 72

pp. 51-98.

### [3]

R. COURANT, Dirichlet’s

and minimal

### surfaces,

Interscience, New York, 1950.

[4] K. GOLDHORN - S. HILDEBRANDT, Zum Randverhalten der

zweidimen-

sionaler

mit

### freien Randbedingungen,

Math. Z., 118 (1970),

pp. 241-253.

### [5]

G. H. HARDY - J. E. LITTLEWOOD - G.

2nd ed.,

### Cambridge

Univ. Press, 1952.

### [6]

W. JÄGER, Behavior

minimal

with

### free

boundaries, Comm. P.

### Appl.

Math., 23 (1970), pp. 803-818.

F. D. LESLEY,

minimal

on the

Pacific

J. Math., 37

pp. 123-140.

### [8]

H. LEWY, On minimal

with

Comm. P.

### Appl.

Math., 4 (1951), pp. 1-13.

[9] C. B. MORREY, On the solutions

### of quasi-linear elliptic partial differential

equa- tions, Trans. A.M.S., 43

pp. 126-166.

### [10]

J. C. C. NITSCHE, On new results in the

minimal

### surfaces,

Bull.

A.M.S., 71 (1965), pp. 195-270.

[11] J. C. C. NITSCHE, The

behavior

minimal

### surfaces. Kellogg’s

theorem and branch

on the

### boundary,

Inventiones Math., 8

### (1969),

pp. 313-333.

[12] J. C. C. NITSCHE,

my paper on the

behavior

minimal

### surfaces,

Inventiones Math., 9 (1970), p. 270.

[13] J. C. C. NITSCHE, Minimal

with

### partially free boundary.

Least area property and Hölder

boundaries

### satisfying

a chord-arc condition,

Arch. Rat. Mech. Anal., 39 (1970), pp. 131-145.

[14] J. C. C. NITSCHE,

über

Berlin-Heidel-

### berg-New

York, 1975.

[15] I. F. RITTER, Solution

Schwarz’

minimal

### surfaces,

Univ. Nac. Tucumán Rev., Ser. A, 1 (1940), pp. 40-62.

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Once a baseline mean breakage intensity is established at the group level, analysis at the pipe level is performed using those covariates that are deemed to have a differential

The first topological obstructions for complete, embedded minimal surfaces M of finite total curvature were given by Jorge and Meeks [22], who proved that if M has genus zero, then

In Part I of this paper, we prove the existence of properly embedded minimal surfaces in S 2 ×R of prescribed finite genus, with top and bottom ends asymptotic to an end of a

Since E(gi) is uniformly bounded, the last number is arbitrarily small when r is small enough. If the former case occurs for all x E G, then g i is equicontinuous on G

Serie IV - Vol. We refer to 16l for a more detailecl reference on the inequality. tr'einberg l2) later generalized this to doubly-connectecl minimal surfaces in lRn

Then either ~/ bounds at least two distinct stable embedded minimal disks in M or a has the following strong uniqueness property: The only compact branched

Remark 1.3 The result is not true in general if we remove the assumption of positive Ricci curvature, as Colding and Minicozzi [7] constructed an open set of metrics on any

On the other hand, we obtain that as long as the metric component remains regular, the behaviour of solutions to (1.1a) is similar to the one of solutions of the harmonic map flow

THEOREM: Suppose A is a bounded open subset of a three-dimensional Riemannian manifold X, A being diffeomorphic to the unit ball, and suppose aA is a class C2 and

FEINBERG, The Isoperimentric Inequality for Doubly, connected Minimal Sur-. faces in RN,

theorem of Cheng [2] gives a computable sharp upper bound in terms of the diameter, and lower bound of the Ricci curvature; while Yau ([8]).. derived a computable

In this paper we determine the differentiability properties of the conformal representation of the surface restricted to that are V in the boundary of the

If M is a properly embedded minimal surface in a flat non- simply connected 3-manifold, and if M has finite topology then M has finite type [4]... Then dg/g extends meromorphically

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