A short proof of the minimality of Simons cone

A Short Proof of the Minimality of Simons Cone

G. DEPHILIPPIS(*) - E. PAOLINI(**)

ABSTRACT- In 1969 Bombieri, De Giorgi and Giusti proved that Simons cone is a minimal surface, thus providing the first example of a minimal surface with a singularity. We suggest a simplified proof of the same result. Our proof is based on the use of sub-calibrations, which are unit vector fields extending the normal vector to the surface, and having non-positive divergence. With respect to ca- librations (which are divergence free) sub-calibrations are more easy to find and anyway are enough to prove the minimality of the surface.

Introduction.

The minimality of Simons cone is a very important step in the theory of regularity of minimal surfaces in higher dimension. In 1960s, Reifenberg, De Giorgi and Federer and Fleming, found the first regularity results for minimal surfaces inRⁿ. They found only apartial regularityresult, in the sense that minimal surfaces were proven to be regular outside a set (the singular set) of small dimension. Subsequent results of De Giorgi, Flem- ing, Almgren and Simons gave more precise limitations to the singular set, finally proving that an (n 1)-dimensional minimal surface inRⁿis regular outside a singular set whose dimension is at mostn 8.

Simons cone is an example showing that the partial regularity result is optimal: the cone

Sˆ fx2R⁸:x²₁‡x²₂‡x²₃‡x²₄ˆx²₅‡x²₆‡x²₇‡x²₈g

is minimal and has a singular point in the origin. In 1969 Bombieri, De Giorgi and Giusti [2] proved that this surface is indeed minimal, thus

(*) Indirizzo dell'A.: via Bellini 7, 50144 Firenze.

E-mail: nemo1685@yahoo.it

(**) Indirizzo dell'A.: Dipartimento di Matematica ``U Dini'', viale Morgagni 67A, 50134 Firenze.

E-mail: paolini@math.unifi.it

completing the regularity theory for minimal hyper-surfaces. Since the blow up of a minimal surface in every point is a minimal cone (if the point is non singular, the cone is actually a plane) we understand that the study of minimal cones is very important in the theory of minimal sur- faces. Actually this is true not only for the minimizer of the area func- tional but also for all those geometrical functionals whose principal term is the area.

After the first proof of the minimality of Simons cone, other simplifi- cations and generalizations were obtained. In particular we would like to mention Lawson [6], Simoes [12], Miranda [9] Massari and Miranda [7], Giusti [5], Concus and Miranda [3], Morgan [11], Benarros and Miranda [1], Davini [4].

In this paper, which originates from the Degree Thesis of the first author, we present yet another proof of the minimality of Simons cone.

The original proof of Bombieri, De Giorgi and Giusti and the fol- lowing simplifications (see [4]) use the nice tool of calibrations. A cali- bration is a divergence free unit vector fieldjwhich extends the normal field of the surface to the whole ambient space. Using the divergence theorem one finds that if such a field can be found, then the surface is minimal. Using the symmetries of the coneS, the problem of finding a calibration can be reduced to a 2-dimensional problem, and the differ- ential equation divjˆ0 can be reduced to an ODE. Then a somewhat deep study of this equation gives the existence of a global solution with the desired properties.

The proof originally given by Massari and Miranda (see [10] and [8]) uses a different approach. Over the cone Sone constructs the cylinder SRR^n‡1. This cylinder can be approximated by the graphs of an increasing sequence of functionsuk:Rⁿ!Rwhich are sub-solutions to the minimal surfaces equation divh

Du= 

1‡ jDuj²

q i

ˆ0. Hence the cy- linder is a sub-minimal surface. Reasoning in a similar way on the complementary set one finds that the cylinder SR is minimal and hence also the coneSis minimal.

We put together ideas from both these two approaches to find a sim- plified proof. In fact we consider sub-minimal surfaces i.e. oriented sur- faces which are minimal with respect to the internal deformations (Defi- nition 1.1) and we prove a sub-calibration result for the sub-minimal sur- faces (Definition 1.4, Theorem 1.5). Hence we explicitly find a sub-cali- bration for the Simons cone (Section 2) obtaining the sub-minimality of the coneS. Passing to the complementary set (i.e. changing the orientation of

S) we find that S is also super-minimal and hence it is indeed minimal (Proposition 1.2).

This method simplifies the original approach with calibrations because we are able to explicitly find the sub-calibration (we only need to check the inequality divj0 instead of proving the existence of a solution to the equation). Also this is a simplification with respect to the proof involving sub-solutions to the minimal surface equation because we don't need to pass to higher dimension, to consider the functional on the graphs, and then to go back to the original space.

1. Sub-minimal sets and sub-calibrations

In the following we consider, as ambient space, an open setVRⁿ. For a measurable setERⁿ we define theperimeter of E inV as(¹).

P(E;V)ˆsup Z

E

divg dx:g2C_c¹(V;Rⁿ);jgj 1 8<

:

9=

;:

IfEhas a regular boundary, P(E;V) is equal to the (n 1)-dimensional surface area of the boundary@E\V.

DEFINITION1.1. (minimal and subminimal sets). We say that a mea- surable setEisa local minimum of the perimeteror simplyminimalinV, if in all bounded open setsAVone has (²)

P(E;A)P(F;A) for all F such that E4 FA:

We say thatEissub-minimalinV, if in all bounded open setsAVone has

P(E;A)P(F;A) for all F E such that En FA:

PROPOSITION 1.2. If both E and E^cˆVnE are sub-minimal in V, then E is minimal inV.

(¹) We denote withC¹_c(V;R) the set ofC¹real valued functions with compact support inV.

(²) With E4F we denote the symmetric difference (EnF)[(FnE); with FEwe mean that the closure ofFis a compact subset ofE.

PROOF. Let A be a given bounded open subset of V and suppose E4FA. Then consider F⁰ˆE\F andF⁰⁰ˆ(E[F)^c. Clearly F⁰E and F⁰⁰E^c. Moreover EnF⁰ˆEnFE4F A and E^cnF⁰⁰ˆ

ˆE^cnF^cE4F A. So, by the sub-minimality ofEwe get P(E;A)P(F⁰;A)ˆP(E\F;A):

…1†

The minimality ofE^cinstead givesP(E^c;A)P(F⁰⁰;A) which can be writ- ten as

P(E;A)P((F⁰⁰)^c;A)ˆP(E[F;A):

…2†

From (1) and (2) we get

2P(E;A)P(E\F;A)‡P(E[F;A) …3†

while, as a general property of perimeter, we know that P(E\F;A)‡P(E[F;A)P(E;A)‡P(F;A):

…4†

Putting together (3) and (4) we conclude P(E;A)P(F;A)

which gives the minimality ofE. p

PROPOSITION1.3. (convergence of subminimal sets). Let Ek and E be measurable sets inVwith E_kE and suppose that(³)E_k !E in L¹_loc(V).

If E_k is sub-minimal inV for every k, then also E is sub-minimal inV.

PROOF. LetAbe an open bounded subset ofV and letF be any mea- surable set such thatFEandEnFA. Consider the setsF⁰_kˆF\Ek. ClearlyF⁰_kEk. MoreoverEknF⁰_kEnFA. So, by the minimality ofEk

we get

P(E_k;A)P(F⁰_k;A)ˆP(E_k\F;A):

…5†

Notice also thatEkEk[FE, henceEk[F converges toEinL¹. By the lower-semicontinuity of the perimeter we get

P(E;A)lim inf

k P(E_k[F;A):

(³) We say that Ek!E in L¹_loc(V) if for every bounded setAV we have j(Ek4E)\Aj !0 wherejXjis the Lebesgue measure of the setX.

Applying the inequality (4) we obtain P(E;A)lim inf

k ‰P(E_k;A)‡P(F;A) P(E_k\F;A)Š and by (5) we conclude

P(E;A)P(F;A):

HenceEis a sub-minimal set. p

DEFINITION1.4. (subcalibration). LetEVbe a measurable set such that the boundary @E\V has C² regularity. We say that a vector field j2 C¹(V;Rⁿ) is a sub-calibration of E in V if it satisfies the following properties:

(i) j(x)ˆnE(x) is the exterior normal vector to @E for all x2@E\V;

(ii) divj(x)0 for allx2E\V;

(iii) jj(x)j 1 for allx2V.

THEOREM 1.5. (subminimality of subcalibrated sets). If j is a sub- calibration of a set E with boundary of class C² in V then E is sub- minimal inV.

PROOF. LetAV be any open, bounded set and letFEbe a mea- surable set with EnFA. Choose a sequence of functionsh_j2 C¹_c(A;R) such thath_j(x)ˆ1 forx2EnF, 0h_j(x)1 for allx2Aand such that the sequence of setsA_j ˆ fx2A:h_j(x)ˆ1gis increasing andS

A_j ˆA.

If we definej_jˆh_jj, we have Z

E\A

divj_j Z

F\A

divj_jˆ Z

EnF

divj_jˆ Z

EnF

divj0

which means that

Z

E\A

divj_j Z

F\A

divj_j: …6†

Sincej_j2 C¹_c(A;Rⁿ), we have Z

F\A

divj_jsup Z

F\A

divc:c2 C¹_c(A;Rⁿ);jcj 1 8<

:

9=

;ˆP(F;A):

…7†

On the other hand (⁴) Z

E\A

divj_jˆ Z

@E\A

hj_j;n_EidH^{n 1}ˆ Z

@E\A

h_jhj;n_EidH^{n 1}

ˆ Z

@E\A

h_jdH^{n 1} H^{n 1}(@E\A_j) …8†

where A_jˆ fx2A:h_j(x)ˆ1g. Since we have chosen h_j so that A_j "A, passing to the limit in (8) we obtain

lim inf

j!1

Z

E\A

divj_j H^{n 1}(@E\A)ˆP(E;A):

…9†

Putting together (6), (7) and (9) we obtain P(E;A)P(F;A)

which is the sub-minimality ofE. p

2. Simons cone.

Letnˆ2m, and consider the cone

Cˆ f(x;y)2R^mR^m:jxj jyjg Rⁿ:

This cone is the zero sub-levelCˆ ff 0gof the functionf:Rⁿ!R f(x;y)ˆjxj⁴ jyj⁴

4 ; (x;y)2R^mR^m …10†

hence the following sequences of sets

E_kˆ (x;y)2R^mR^m:f(x;y) 1 k

; Fkˆ (x;y)2R^mR^m:f(x;y)1

k

both converge toCinL¹_loc(Rⁿ), withE_k CandF_k C. More precisely, jA\(CnE_k)j !0 andjA\(F_knC)j !0 for every bounded setA. We are

(⁴) The Hausdorff measureH^{n 1}on a regular set corresponds to the (n 1)- dimensional surface area of the set.

going to prove that all the setsEkandF_k^c ˆRⁿnFk are sub-minimal. To achieve this we will show that the vector field

jˆ Df jDfj …11†

is a sub-calibration.

PROPOSITION2.1. If m4, the vector fieldj defined in(10), (11)is a sub-calibration of the sets E_kinV ˆRⁿn f0gwhile the vector field jis a sub-calibration of the sets F_k^c inV.

PROOF. Clearlyjjj ˆ1 everywhere inV. Also we know that the gradient Df is orthogonal to the level sets@Ekand@Fk, and points outwards the sub- level. SojˆDf=jDfjis the exterior normal vector toE_kin@E_k and jis the exterior normal vector toF_k^cin@F_kˆ@F_k^c.

We want to compute the divergence ofjˆDf=jDfj. We recall that f ˆ1

4(jxj⁴ jyj⁴);

f_xiˆ jxj²x_i; f_yiˆ jyj²y_i; jDfj²ˆ jxj⁶‡ jyj⁶; f_xixj ˆ2x_ix_j‡d_ijjxj²; f_yiyj ˆ 2y_iy_j d_ijjyj²; f_xiyj ˆ0:

So

fxi

jDfj

xi

ˆ

(2x²_i‡ jxj²)(jxj⁶‡ jyj⁶) P

j jxj²x_ijxj²x_j(2x_ix_j‡d_ijjxj²)

jDfj³ ;

X

i

f_xi jDfj

xi

ˆ(2‡m)jxj²(jxj⁶‡ jyj⁶) 2jxj⁸ jxj⁸ jDfj³

ˆ(m 1)jxj⁸‡(m‡2)jxj²jyj⁶ jDfj³ :

The derivatives with respect toy_iare the same, with signs changed. So we have

jDfj³div Df

jDfjˆ(m 1)jxj⁸‡(m‡2)jxj²jyj⁶ (m 1)jyj⁸ (m‡2)jyj²jxj⁶

ˆ(jxj⁴ jyj⁴)

(m 1)jxj⁴ (m‡2)jxj²jyj²‡(m 1)jyj⁴ : We want to prove that the sign of this quantity is the same as the sign of

jxj⁴ jyj⁴, so we have only to prove that the expression in square brackets is non-negative. With the substitutiontˆ jxj²=jyj²we obtain the relation

(m 1)t² (m‡2)t‡(m 1)0

which is true for everytif the discriminantDis non-positive, i.e.

Dˆ(m‡2)² 4(m 1)²ˆ3m(4 m)0 which holds true form4.

So we have proved that divjhas the same sign ofjxj⁴ jyj⁴in particular divj0 onCEkand divj0 onC^cF^c_k. This was the last condition for checking thatjis a sub-calibration for everyE_kandF^c_kinV. p THEOREM2.2. (minimality of the cone). The cone C is a minimal set in Rⁿ.

PROOF. Proposition 2.1 ensures that we can apply Theorem 1.5 to obtain that the setsE_k andF^c_k are sub-minimal in VˆRⁿn f0g. IfAis any open subset ofRⁿ then

P(E;A)ˆP(E;An f0g) for every measurable set E

and this is enough to conclude thatE_k is sub-minimal in the whole space Rⁿ (notice also that 062@E_k). The same reasoning gives the sub- minimality of F^c_k in Rⁿ.

So we have a sequence of sub-minimal setsEkwhich converge inL¹_locto the cone C. Hence, by Proposition 1.3 we conclude that C is itself sub- minimal. On the other hand F^c_k are sub-minimal sets converging to C^c. Hence also C^c is sub-minimal. By Proposition 1.2 we conclude that C is

minimal. p

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Manoscritto pervenuto in redazione il 10 marzo 2008.