Minimal immersions of spheres and the Almgren-Calabi theorem

Introduction

The follm·Ting 1:-asic ::-e~ult ^\-Taf::

proved

ipdepcndent.ly by P>-lmgren ( 1) ar:.d Calabi (2): ~"'l:Y :a-1:;1phero which is

minimally

iP.mersed in the

standard 3-sphere

~s~

be

an e~uat.or.

This is an analogue

in

sphe- :r:ical

geometry

of the fa,.-nous Bernstein theore.r:1 on rr.i.nimal hyper-

surfaces

in three-dimensional

Euclidean

geometry. After

the

solu-

t.ion of thE;! Bernst~in probler.t in Euclidean

geometr..t

of 'higher

~ ' . (d G' ' "1 .... . B b' ' ~"'' ""' ') ....,..

a.1mensJ.ons _ e ~J.Ct'9l.• .f'.-r.v:;;re:n, .,:)l.'7!0ns, om ;.erl., '-=~us

....

^{1. ,} ~..,;uern

raised the question of

generalizing this result to higher-dimcn-

sional spheric~l

geometry (3). Tne following

easy

(i.e. homogene-

ous) example of a minimal

irnmersion

of

s

³

into the standard

4.,..

sphere S

4· (

1)

wa~ 'known (

4}; Let SO ( 3)

operate on

the symmetric (3x3)-matrices of trace zero

by

conjugation, let s

⁴

^{be the in-}

variant uni,t

spher~. The principal orbit

:is of type

S0(3)/z

2

+ z

2, so the ope of maxima+ .

vol~e .

defines a non-equatorial minimal

immersion of s3 into

S~(1

). The spherical Bernstein

problem WqS

therefore formulated as a question about the existence of non- equatorial empeddingl$ of hyperspheres; there has been much recent progress

on th~s (5), (6), (10).

However, the isolated nature

of

the above counterexample in s

⁴

(1 ), (it is the only such example which is homogeneous), still leaves open the natural question: To

which e;<tent ooes. the Almgren-Calabi theorem

f.;1il in

highe;r dimen..-

sions?- in view of the significance of this

for

the study

of

sin- gularities of min;i.lnc:tl

nypers~1rfaces

it is a basic proolem

in

Rie- mannian geometry. !n t'llis paper we give the complete solution by

proving the

following theorem:

T'l;eorem 1.

be the EucliGean t111.i t. sphere of dir:1ension n, n)4., Then there exist infinitely many non-congruen-t minimal immersions of

s

n-1

'ITiese sp11eres ere invar:~2.nt under a s il7lple .f~rans forrnation group ^4>

The reJ.evant rr.ethcds f:::-om equivaria.::-1~.:. differer.tial georaetr.f to analyze the reduced minimal equation

in

orb:.t space, a spherical

lune of angle ~, have been discussed in dctc.il in (6), (1 0); \ve sum!i'.arize the results in section 1. The existence of mini.TTI.al

hyperspheres is reduced to the existence

of

solution curves of the reduced equation i.vhich oscilla:te between two points on the singu- lar boundaries. This is established at one stroke :=or all dinten- sions by a technique of (1 0), which analyzes t11e variation of the pattern of critical points of auxilary funcitons along solution curves which e.vnanate £:-om the singular boundar.t of the spherical lune; this is discussed Section 2. This nethod requires less exp- li.cit computation than (6), (1 O); the fe.v necessary estimates and the proof of the main theorem are completed in Section 3.

The important question of stability of the vertex singularity of the cones on these hypersphe=es may be cheCked by Simons'

criterion: If the norm of their second fundamental form is uni- forrnly bounded by ~, ~ne ^{n-2 .... ~} cone is st~ble in Rn+l (8).

By

compu- ter assisted appro:dmations i t can be sho....rn that in many concrete cases, stability does in fact occur {11).

1, Orbital geometry_and the reduced minimal eauation.

,.. _~e~ · G -

=

0 { ) ^{p X} 0 ( ) ^{q ac~} · ^on :-:p+q ·.-^{~ -} ~p ^~ e. '""' .. · ^~ c_. ·""'-^a R ¹

^0Y

^{~ne • ' represen-} t;;:tim'! pp

e

pq ^1.3

e,

^{pp =} standard repres,.7!ntation o£ 0 ( p) ,

e ·-

trivial representation, {{:c,y,::;) E R' x R+x R}, defines the orbit space

!l ^:::::

the

p q. The space of ir-variants

is

additional equation ^~

.,

'J

::: .c::. ^v~

...

~-- ;::::

of the restriction a= this ~ction to

1 •

n ^{' $}

s ;

nence is a spherical lune of angle

2·

^11: Ir:.t=ccqcing spher- ical polar coordi:1ates ^{(r, e)} E [0, ^it

J

^x [0. ~

L

wo have= The

orbital distance metric ds²

=

^{dr2 + sin2}

^rde

^{2 • ~~e} volume func-

· · · h 1 f · · , b · · o+q-2

t1on, reg1ster1ng t e vo ume o pr1nc1pa~ or 1ts: v = s1n~ r s inp-l

e

cos q-l

e.

From standard formulas the mean curva '.: ure of a hypersurface represented by the curve y(s) ;:::: (r(s), e(s)) ir..

orbit space, parameterized by a.rc ler:gth is H(y(s)) == J.:(y(s))-

~

ln v(y(s)) where k is the geodesic curvature and n is the an

unit normal of the curve. By straightforward computation i t fol,- lO\V's that the hyper surface is minir:~al iff i t.s generating curve satisfies the differential equation:

(*) ..

r

=

^{cos a}

..

_sina

e =

_sinr

..

_{sin a cos a}

a

=

^-(p+q-1) _sinr ^{cosr +} _sinr

Ke.

where K9

=

(p-1)cote -, (q-1)tane, and a is the angle from to the tangent vector.

~nis equation becomes singular at the boundaries

e=O

^and

We have the following main reduction theorem:

Theorem 2.

0

or

e

⁼

2·

it

Let y(s) = (r(s),

e(s)),

s E t(a,b) ·be a smooth curve in int X,

pc.rametc:z:i::::ed by arc length, 'tlhich satisfies ( *). Assume tnat.

r(b-) E (O,n), e(a+)::::O, e(b-):::: ;. a:(a+)

=;

^mod(2n),

a (b-)= ~ (mod2 rt). Then y ( s) , sE [a, b] is the projection to X of a smooth, minimally L~mersed hypersphere of sn.

·ne observe that the condi·tion that y(s) enter the boundary orthogonally i•·nplies that the in·v·erse image is a smooth hypersur- face in Sn, its topological type is determined as the union of two napping cylinders det:ernined by the orbital data along ·r{ s) • For more details, see (1 0) .

We need more precise information about

(*)

at the singula;r boun- dary. ~IJe summarize:

Proposition 1.

Let (b,O), bE(O,n), be any point in the interior of the singular boundary 9=0. Then there {s a unique analytic solution curve yb(s)

-

^(rb(s), eb ( s)) of ( *) with rb(O) = b,

eb(o)

⁼

o.

Here c. ( 0+) n

and (rb (sL ^{" ( \} ~ {s)) depends analytically

=

^{2i tlb s} , on}

D

( s 'b) as long as s is restricted to an interval where yb(s) does not intersect the singular boundary again. A similar result holds for

e

^{= 'It} _2"

Proof:

This is proved by formal power series substitution and majoriza- tion. See (7).

Proposition 2.

Let bE(O, r.). J16ny solution curve (r{s), B(s)) of (*) which approaches (b,O) as s ⁺ 0-, must be analytic and hence coincide with the above unique solution (reversed in s).

/

.-·.

Proof:

It. is not hard to shO'Itl thc.t ^c:{O-)

= - -

^)i; ₂ ^{(see (10)),} hence the l i f t t.o is a smooth, r:1.inirnc:.l hypersuriace; i t follo<.vs from st.andard regularity theorews t.hat i t is analytic.

The equation { *) is sym&;;etric

tn

^respect to reflection of para- rr:cte:::-; hence any solution curve \.:hich hi t.s the boundary 1 can be continued back along the s~~e trajectory with a discontinuous jump J..n at the boundary; hence ^-' ci..L

,

a!. solution curves may be conside-

as defined for all s. Closeby solution curves will generically avoid the boundary, but we have the phenomenon of "sharp turning"

close to the boundary:

Proposition 3.

Let bE(O,~) and g>O. There exists a 6>0 such that for any solution curve y(s) = (r(s), e{s)) with r(O) = b, 9(0) <

o,

there is an s 0 E (O,c:) pectively ; - 9(0) <

&,

with 2!. +

2

8 (So ) < € ¹

cr. (

r;o

>

I

^< c:

(res- at the other bounda~J).

Here

the details

become a

delicate estimate, see

(10) for

this

argument:..

Proposition 4.

Special solution curves of

(*)

are: (a) b)

c) sin²r cos²

e

=

g-1

p+l-1 d) sin²r sin 2

e

⁼ _p+q-1 ^o-1

Here (a) corresponds to t.he equator z=O of

sn.

The "meridian solution" (b) is the suspension of the principal orbit

of

maximal volume, and does·not l i f t to a smooth hypersurface of· sp+q. The solutions (c) and (d) can be checked by careful computatiton, they

:'.\

represent principal orbits of the larger sym:rr.etry groups 0 (p+1) x

O{q) and O(p) x O(q+l) on Sn, respectively.

Prooosit.ion 5.

Lcet ( r { s) ,

e (

^{s) 1} o: ( s) ) be any solution of ( *} • Then:

( .:~) any relative maximurn (minimum) of

r(s)

occurs \•lith ->--^~{ T'"(

... 2 -

2) • n

(b) any relat.ive maximum ( rnin.imtun) of ^e(s) occurs with 6>90(6<90).

(c) any relative maximu.tn (ninimum) a(s) occurs v:it.h

in

t11e first or third (second or fourth) quadrant,

Proof:

(a) and (b) are immediate

by

differentiating (*)· Computi~g o:

and subtituting the relation between

..

a, e, r defined by a = 0 yields:

a =

^{K ( r, e) sin a} cos a: a·t a critical point of a:, wher~

K(r1 9)

=

p+1-q-sin- 2 r[p+q-2+(p-1 )cot²

e

+ (q-1) tan2e]is always negative

2. Qualitative features of solution curves under deformations.

Definition 1 . Let the regions I-IV

in

orbit space be defined by:

I: ( r, e) ^,. n:

it)

(eo· ^{!:) ( r,}

^{e) ,.. (0 1 1t) (90, 1t}

'C: ( 2 ^{8 X} 2 . ^{II: 0:} ₂ ^X

2).

III: ( r, e) E ( 0 ^{1 2!)}

2 ^X (0, eo) ^{IV: ( r, e)} E

<;.

^{it) X (0 1}

^eo).

of ( *) with

r

b( 0) = (b u 0, ; ) e and (rb(s), eb(s)) to orbit space. Let

be the projection ri(b) be the i-th

r:l

maximum and minimum of rb(s), s>O, respectively, and define their

i i i i

corresponding arguments by rH(b)=rb(sMb), rm(b)

=

^{rb(smb). The}

and Definition 3. Let

a!

^{(b) = a:.0 (}

^tm~)

^are defined simil<:J.rly.

yb be the largest segment of + yb(s), s>O, vfl1ich does not touch the singular boundary.

::

Pronosition 6.

~~·---~~----

r: { )

Let. b E {0 1 2-). Then all critical points for rb s

along yb are non-degenerate and vary continuously with b. Ncne of them can coincide.

Proof:

By t.he uniqueness theorem for differential equations

l.

) ,

ri (b) ^-'-

-

^"

,...

.,.

_{2 •}

-1'1 m ^{From (*) \ve} have

r

^{= {p+q-1 }cotr at a critical}

90ir:t of r( s). Hence these are all non-degenerate and stable;

frorn continuous dependence on initial conditions (Prop.l) i t fol- lows that i

5Hb'

s i

rnb vary continuously \vi th b. From Prop. 5 i t follc::t.rs that a critical point a is non-degenerate unless

s cosa = 0. Ass~~e ~{s

1

^{) = 0, cos~{s}

1

^{) = 0; from}

^(*)

^{we have}

cosrb(s 1 )

=

^{0, i.e.} _{rb(s 1 )}

= ;,

contradicting uniqueness of the equator solution. This argument also proves that critical points of rb(s) and ~

0

^(s) cannot coincide. If ~(s

1

⁾

=

^{O, sin~(s}

₁

⁾

= Og i t follows that K8 (s 1 ) ⁼ 0, i.e. eb(s1 )

=

e₀ ^s contradicting

·the uniqueness of the meridian solution. Hence. all critical points on yb.

+

are non-degenerate and stableff

q.e.d.

Definition

4.

The N-th (r,a)-pattern of yb, P (b), ^N is the finite sequence of symbols constructed as follows: we assign symbols

i i i i

in the order the critical points

r~(b),

r r _m~ aH, ^c:: rn ^{same .as}

i i i .,..

until

rm (b), aM (b), am(b) occur along _yb, s>O, 'lfle have passed the N-th critical point or Yb (s) has reached the boundary.

Note: PN(b) is well-defined

~J

Proposition 6.

Corollary.

AssumE: that +

y b has at least N i

critical points among the rN(b) ¹

is locally constant around b=b1 . Proof:

....

By P::ropostion ^{1, yb'} has at least ^N cri"'.:.ical p-oints for ^b

:::;uf:fic.iently close to b, . By P:=oposition 6 these must :wove conti- '

nuously c.nd cannot change: type under s~na.ll vc.:::iations of bi furthermorf.!e since they never coincide, they cc.nnot pass each other, and PN(b) is rigid

q.e.d.

Since any change in PN(b) must result from hitting the boundary, this i~ediately gives a criterion for establishing the

existence of "closed" solutions yb, oscillating bc.ck and forth between b and another boundary point. Let b E (06 ~) ^and

a: • <

0

at

11: ~

a

= 2;

hence ~(s) < 0 for small positive ^s, ~ ^{(s) >0} for small

D .

negative s, and c::b(s) may be considered to have a (not well- defined) maximum at s=O, coinciding \·lith a rnini."TTurn for rb ( s) •

s

irnilarly1 at an intersection ^IN"i th the boundary \ve have:

In I: A rr~ximum for r(s) coincides \vith a maximum for a:(s).

In I I: A minimum for r { s) co inc ides with a minimum for a: ( s) • In IV: A :ruaximum for r( s) coincides with a minimum for a:(s).

We need to examine what happens under small perturbatons of the intersection situation.

Prooosition 7.

Let yb ( s) intersect the bounda::y at s=s 1 fer b=b1 E ( 0, ; ) . Variation of b around b

1 then all~1s a continuous crossing of critical points of a:b(s) and rb(s) at the intersection point according to the following scheme:

"."

.-J

·~,

Intersection pcint in

^{I: An} rM(b)

c:-osses

an a., (b) •

J.."'l

I I : " 1\.,'1"1

.

^r _m ^{(b) cress-es an am~ ,. o •}

⁾

III: An

r

m {b) c::osses an a:H(b).

IV:

A.."'!. r ~1^{(b) crosses}

an

a: (b) • :m

Proof:

-

.

^{( )}

^,.,

^{( )}

Assume ca.se IV,

l.~e.

r 0 s 1

^>

2' _{8b s 1 .}

⁼

0 By Pro- position

1:

For any

c:>O there

is

a 6>0 and an s 0E(O,s 1 ) "lith

e:, 0 < a. ( s_{0 )+ ;} < g

D . - when

Jl..nalyzing

the sign of a:

.. in combination

with

Proposi ton

5, yields the following possibilities:

(i) reaches

a relative

r:tinimum

a: J (b) > - n2. , by Proposi-

m

..

tion

3

i t then

turns

sharply

up

to

_{2 -}^7t e:. By estirna

ting

c:

it reaches ; , .

(i.e.

and

rM (b) ) ⁱ

after

a short

s-interval (which tends

to zero

with

6) •

(ii}

decreases to

as approaches an

intersec-

tion

with

the

bounda~.

decreases

past ^'it

2 (i.e.

and · r~(b))

and turns

h 1 3'lt h .

s arp

y

to - ~

+

e.

By

c oos~ng e: small,

we

may ass~e that we remain

in

!V. The a:b ( s)

now

reaches a relative

minimum

aftel;"

a a-small

(i.e.

decreases

to zero

with o)

s-interval, otherwise,

an estimate

of

a • shows

that

:::.-eaches

while still in IV, contradicting Proposition S(a). Hence, in

case

(i)

an a (b)

m

precedes an

rMi(b) , and in

(iii) and r~(b)

precedes and a.mj(b),

approaching

each

other as o~o. In case (ii) they coincide. This proves the propositon in region IV. The /

region

III foll(JWs

by reflection around

the symmetry

line

a similar

argument

at

the boundary

I

and

II:

.,..

e =

₂

concludes the proof for

q.e.d.

3. Small perturbations: of solutions and oroof of the main theorem.

'D • t 8

,_ropes~ on .

Let r(s) =

^(r(s),

S{s),

~(s))

be a

~elution of

(x)

with

0

< a ( 0) < ~,

a (

^{0) < 0, ( r ( 0) ,}

^{e (}

0) ) E ! I I. Then

y (

s}

=

^{(:de) ,}

e(s)) escapes the region III by crossing 9 ;::: e₀ fo::.- an s :::; s0

> 0; furthermore ~(s) < 0 for s E (0, s 0

j.

In partucular this holds for yb, b E (0, ~).

Proof.

..

_"

In I I.I we have a > 0 at ^~=0 and a < 0 at . c:

= ·:2u

^':1: hence ^cx(s) E (0, ~)until y(s) escapes III.

By

Prcpositon 5(c), c:(s) has no relative· minimum, henc::e ~(s) rerr.ains negative until the es-

..

( ) 'it h

cape. By

*

we have a > 0 at r =

2 ,

^{ence the} escape must be across 9= 9 • Since

<;

^{~ n) u} we must have 0 follows.

yb(s) starts in III and ~(s) < 0 for a E

·~(s) < ~· to begin with, and the result q.e.d.

We need to exacine the (r,a)-patterns of yb(s) close to the kncwn solutions b = ; and b

= ( P'~~~ 1 ^)~.

^{Since the case n}

⁼

^p

+ q < 9 of Theore.'n 1 is covered in (51 6), we assume n ;;. 9: for technical simplicity we also assu1ne, without loss of generality, that q - 1 .;;; p

<

q + 1 •

Proposition 9.

With the above assumptions we have: For any N there exists

an

~

2N ( ) ( 1 1 2 2 .,..N N )

such that P b = rM, rm, rH, rm, ••• , ""M' r~ for ^b~ _{... 2} ^(.!£

-e:, 2 .

^'It)

Proof:

Let b E (

~-e:, ~)

and

let yb(s) cross from III to II for· s=s1 • ^{From (*)} we have ~(s) < 0 for a E (0, ~} in II. By continuous dependence on initial conditions: For any

o

^{we may}

choose e sosrnallthat lrb(s)-~1 < 5u IO:b(s) -~1 < 5 until eb < s 3 )

=

_{e 0 -} 5 •

Lernma.

For sufficiently s:rnall o, yb ( s) crosses r

= 2

^1t before ^s~s₃ ^•

Proof:

Define the auxilary function Differentiation gives:

t: ( s) -- cos o:(s)

cos r(s)"

(1)

u

^{= sin-1r[(p+q-1 )sin2c, - usinai\0 + u2sinr]. By} Proposition

v

5 (a), cos ^c:(s) > 0 for r(s) < ^it 2' so u(s) > 0 and K 9 (s) < ⁰ for s E (sl,s2) as long ^as y- ( s) does not cross into

r.

Hence

D

"

u > (p+q-l)sin²a: + ^u2 in this region, and, by choosing E small:

u

^{> (p+q-2) + u2. By} comparison \vi th the solution ^u = (p+q-2)~ ^l

tan ((p+q-2);2(s-s1 ) +C) of the equation (2)

u

^{= (p+q-2) +.u 2 , wa}

that u(~) blows infinity, i.e. yb(s) ^':t

see up to crosses r =

2

before s =

.

₅₁ + ^'lt 2(p+q-2) ^-~ < ₅₁ + 216' ^it and the Lemma

follows.

We ^na.ll' have ~b(s) < 0 for s E (O,s 2 ), '.vhere rb(s_{2 )} = ;, s_{2 E}

(s

_{1 ,} s_{3 ).} Furth~=more, by Proposition S(c), ~(s)

..

< 0 also

for

s E

[s

_{2 ,} s_{3 ].} Combining this argument with.the sharp turning of

"

Proposition 3 and a sig~ discussion for a from

(*),

we see that

~ (s) decreases rapidly through 0 to _~(s4) ^:::::

- 2

^1t +

&.

Then u( s_{4 )} ^{< 0,} rb(s) _{.,. 2'} ^1t ab(s) ^<

-2

^{1t until} c:b(ss) ⁼

^-2,

^{1t i.e. rM(b)= 1} rb(ss). By checking signs of ⁽¹⁾ '"e

may

compare with ⁽²⁾ as above, and conclude that while is still in I. By choosing e small we have: rb(s 5 ) - ~ <

o.

By the argument

of

Proposition 8 yb(s) must now escape I by crossing

e

⁼

e

₀ ^into

IV: By combining Propositions 1, 3, S(c), sign discussion for from

(*),

and the aux;ilary differential equation (1) as above,

a •

we show: yb(s) must cross r

=

^1t ₂ on its way dm.;n, turn sharply near the boundary 9=0 of I!Io a continues to decrease

1 •

rrn(b) = _{rb(s 6 ),} ab(s) < 0 for s E (O,s 6 ]. Here

E-small, so we are in the same situation as we started with.

.

. -·~

'•

;

Hencet the argument may be =epeated N times to conclude the

proof of Propositon

9.

q.e.d.

Propositon

10.

For b

sufficiently close to Arcs.l.n( .

^{' q-1} ₁ ^)~ , P"' (b) ^,

p-rq-

is either

1) ( 1 . 1)

a _m

or

a , _:m.

rM •

Proof: Straightforward computation shows

that

for

the

solution

. q-1 ~2 ..

yb(s) with

b =

ArcsJ.n(p+q;.,.l),,

ab(s) < 0

until its first inter- section with the boundary of IV. By propositons 1 and 7 the

~esult

follows.

Proof of Theorem 1.

for

c E (b, ~)}

and for

c E (b, ;)}.By

q-1 ~ ^'it

(p+q-l ) ,

_{2 ). By}

Proposi- tion

7

rc. and

_Yb.

intersect

the

boundaries of

^{I I}

and IV at

.J. l.

r! ^and

^ri

respectivey (wee must 11ave P2i+1

^(c) =

M

1 1

i

¹

ri)

(rM, rm,

• • • I

rM,

^a;

m' m ^for

^{c E}

^(ci-e:,

^{c.) l.}

^{for some}

^{£) • By}

Theore..tn

2

the rc. represent an infinite sequence of generating

l.

curves for minimally immersed hyperspheres of

q.e.d.

Remark 1.

From propositions

9

and

10

it follows that the initial points

c .

~

of

the generating curves

val [Arc in( q-l )~ n ]

must accumulate somewhere in the inter-

5

p+q-1 , 2 .

We observe that

if

does not

intersect the boundary

for

s>O when

b

varies

in

some interval

J,

this would imply that PN(b) is constant for b in

J

for

all N.

By refining

the above

techniques it should

then

be

possible to prove that yb lifts to a minimally immersed hyper- sphere fo~

a

dense set of b in ( 0, 1t) ,

Remark 2

The Yb. Fepresent an infinite sequence of

miniw~lly

immersed

.l.

generalized Clifford

tori.

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