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J JOURNAL OF SCIENCE, Nat Sci , t.XII, n°1- 1996

ON THE COMASS OF FORMS-PRODUCTS

Nguyen Duy Binh

Vinh Teacher’s Training college 1 INTRODUCTION

The calibration method was studied systematically by Dao Trong Thi in (3, 4] and R Harvey,

B Lawson in [7] Various concrete calibrations were used by many authors to find minimal

tfaces, see Federer [5] Berger [1], Dao Trong Thi (3, 4], Harvey- Lawson [7], Dadok- Harve-

organ [2], Le Hong Van [9] Hoang Xuan Huan [8] etc For applying this method, determining

+ comase and the maximal directions of a p-covector is the main obstacle In the field mentioned pve, it is still open the question whether the equality || A ý||* = |le|l*-|l0||* holds where ø and

are forms on orthogonal subspaces of R" Let p € A‘ R™, p € At R", then @A Ú € A*†t gm+n,

p note that although the inequality || A yl|* 2 ||Âl|*.||Ơl|" is obvious the equality || A pl = I|Ì* ||w||* had been proved only for some concrete cases Morgan [11] has showed that the equality lds if k < 2, orm-k<2,ork=1=3, orm—k=n-—1=3 Recently, Hoang Xuan Huan (8]

8 proved it for an arbitrary E-separable form œ In this paper we prove the equality when ¢ is her a torus form, or a certain averaged form by a group

2 THE COMASS OF A PRODUCT WITH A FACTOR BEING A TORUS FORM

First we recall some notions and facts of exterior algebra

Let R” be the n-dimensional Euclidean space, Ay R" and A* R" the dual spaces of the k- tors and the k-covectors respectively The inner product and the norm on R" induce the inner

duct and the norm on A, R” and A* R", Consider an orthonormal basis ¢1, ¢n of A" and

dual basis ¢}, , ¢, of A‘ R” (from now on, the symbol e* means the dual covector of ¢),

mn an arbitrary p-covector in R® has an unique expression g = )> ay e7, where J = (41, , tp),

ty <o+tp Sn and ej = 7, A - Ae} The comass of a p-covector ý Ìs defined by loll’ = sup{o(¢) : €€ G(p, R")},

iere the Grassmannian G(p, R") consists of all oriented p-planes in R” and it may be identified th the collection of unit simple p-vectors in R” ?

For any p-covector in R” the set of maximal directions of y is defined by

G(ø) = {€ € G(p, R") : (£) =llell"}-

Let y be a p-covector in a subspace V C R" then y can be considered as a p-covector in R™

identifying with II* @ where II is the orthogonal projection of R® on V

Because every p-covector in R" can be considered as a parallel differential p-form in R”, from w on, we shall call every p-covector in R” to be a p-form in R” unless otherwise stated

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Now we recall the notion of torus form which was introduced and its comass was com| in several papers, for example, see (2] Here we consider it only in the relation with the pre

mentioned above

Definition 1 Identify R?" = C" with real orthonormal basis ¢1, Jey, ény Jén- Any n-fo

is called a torus form on R?" if it belongs to @ A‘ span (ex, Jex) C A" R™

k=

Note that the Lagrangian forms Ree’ dz; A - Adz, are torus forms The torus forms belong to the class of V-torus forms defined below:

Definition 2 Let V be a 2-dimensional subspace of R" (n > 2) Any form @ € A!V @! (€> 0) is called a V-torus form

We note that if e,, ¢2 is an orthonormal basis of V then each V-torus form can be expr

Ø=t1A0i+t2A09a, where @¡ and @¿ are forms on VÌ,

Obviously, for any p-form f1 on #*, ||0||* = ener ||» 1N||* Moreover, for V-torus :

we have the following

Conversely, take € € G(f) and put € in canonical form with respect to V (see [7, Le

7.5}), that is

€ = (cos 6; €, + sin 9, f,) A (cos 2 e2 + sin bz fo) AfaA -A fy,

where ¢1, ¢2 is and orthonormal basis of V, fi, f2,- , fp are orthonormal vectors in V+ and a 6, 05 6; < %, fort =1, 2 Then

A(E) = cos 4; sindz M(e A foA+-'A fy) + sin, cosb2 (fi Nez A fy) =acos@, sinf2 + b sin 8, cos 62 < max{|al, |b|} (cos 4, sin 62 + sin 1 cos 62)

= max(|a|, |ð|} sin(# + 62) < max{|al, ||} < |||",

where a=2(e,A foA-:-A fp), b= A(firesd- A fp)

Hence the inequalities become equality, in particular, max{|a|, |b|} = ||ô||*

Therefore, |a| = ||M||* or |b] = |||"

But |a| = |M(e, A faA -A fy)| = lex IN (fo A -A fp)I S [lex 1M"

and |b] = |O(fr AezA A fy)| =|—e2 IN (frA -A fp) < || — 2 IOI", therefore |If|* < ||e; -1f||* or |||Í* < || - «;-19|l*

Thus, we have ||f|*®= max |lu-10|" 9€V,|u|=1

The lemma is proved

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Suppose v, -, Yp+y € R"+™ and vy =v We have

0.1(ø A 0)(05,s tu+a) = ÍØ A 9) 9, 9i s3 9+)

= >> index(Ø) Ø(9z(1)i «ý 93p] ) Ó{Uztp+ll› sen Uơp+al) ý

ø€Shlp.4]

Sh(p, 4) consists of all permutations ø of {1, , p + q)} such that ø increases on the set of

p} and the set of {p + 1, , p + g}

Fince vj 1p =0 the above sum equals to

32 index(c) PY, Val2yseoy 9z(p)) 9{9ztp+)› oy Vaip+q)) Z€Shlp~ 1.4) = YS index(Ø)(9-1Ø)(9s(al, - Yai) {9z(p+l)› => 92(p+a)} øcSh[p—1,q) = (vi Tp) A (v2, s005 Ups ess Up4y)) > tơ is a permutation of {2, 3, , p+ q} 4 Donsequently v -1(~ A #) = (vi 1p) Ay The lemma is proved

vrem 1 Let be g a torus form on R?" = C" and y be a p-form on R™ Consider N= pA

a (n+ p)-form on R" ® R™, Then ||Q\|* = ||pl|*.||¥||" and G(M) > G(p) A G(y)

ark The second conclusion holds for every case when the first one holds

n

\ Let e1, Jes, én) Jén is an orthonormal basis of R?" and p € @ A'span(ex, Jey) C

k=1

'" Then # and @A9 are V-torus forms, where V = span(e¡, Je,), We note that v_Iy where is a torus form on R2("~!) = V+_ Using Lemma 1, Lemma 2 by induction on n we have:

le^ 9Ì” = „ppt, 91A9) ) = max, Mute) Avil

= carole IW" = lll Iv

take n € G(£) and À € G(ý), we have

(0d) = e(n).¥(A) = Hell? = HA?

8A G(N), this implies that G(A) 5 G(e) A G(0)

Ivery parallel differential form having comass one is a calibration (the notion of calibration

e given later in section 4) The following corollary follows directly from Theorem 1

Mary 1, Let Redz = Reda, A »A dan be the special Lagrangian calibration on R2" % Ơ"

f?]) and @ be a cahbration on R™ Then w = Redz Ag is a calibration on R?°+™ and

2 6(Lag) A G(@) uhere S(Lag) consists of all special Lagrangian subspaces of R?" = C

3ecause for an arbitrary 3-form on R° there is a convenient basis so that this 3-form is a torus

(see [11] , from Theorem 5.1 in {11] and Theorem 1 we have the following

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Corollary 2 Let p be a calibration on R® and p be a calibration on R™ Then p

calibration on R°+™ and G(p A 9) 5 G(e) A G(9)

3 THE COMASS OF A PRODUCT WITH A FACTOR BEING AN AVERAGED F Let § c O(n) be a compact Lie group, each k-form p = Sg g°w dg for any w € A* R”

an averaged form by group § Some known averaged forms are the normalized powers o nionics Kahler forms, the Euler forms and their “adjusted powers” Using them as cali one showed certain submanifolds are homologically minimal in quaternionic Kahler manif

in Grassmannian manifold (see (6, 12]

In this section we prove that the equality on the comass of a product holds when oi is a certain averaged form

Let G be a compact Lie group Consider the Haar measure on G such that the measu whole group G equals to 1 We have the following

Theorem 2 Let G C O(n) be a compact Lie subgroup and œ € A* R", suppose that € € C

spané ts G-invariant Then

* = Jolt

| [ det (glepane) 9° w dol

and € is a mazimal direction of the form on the left side

Proof Note that since spané is G-invariant and § C O(n), det(g|apang) = +1 or —1 we h | [devas soda < ol

Indeed, take n € G(k, RE"), then

| 2eke e)Ze)4|< [ loloe mda < fel do = hol

for any n € G(k, R"), therefore | seo z4 < lt Conversely, we have g„ € = det(0|spane)-€ Therefore |(se6t-.oz+ da|

Py [ det(glspane) ø°ø(£) đợ = [ det? (glepane) w(€) da

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is a maximal direction of the form on the left side The theorem is proved

vem 8 Letp EAE R", PEAR", DAVE AFR t™ ouch that oA Yl’ = loll’ Ilvll*-

icf xo g edg) Awl| = | f xlore eda) tor Le ie gu Miciant terproverthe inequality

I( [ x(a) 9" eda) Av < | [ x(o)9" ed" i

ve

l(ƒ x6)924) ^vl[ = I foo g eA y) dg| `" LỆ x)ge^9)4| , (+) here G is considered as a subgroup of O(n + m) such that g|pm =idzm, for any g € G

rom (*) it follows that

ICf xo sted) nv" s fila sen vila

= ƒ le" dạ =lleAw|* (+e)

lle 4 villt = llell* wll" from (e+) we have

i J, x(a) 9°e do) AvI" < livlt vil = | ƒ x(o) 9° dal" v

eorem is proved

ow we apply the above results for the powers of quaternionic Kahler forms, the Euler forms

1eir “adjusted powers”

ine showed that submanifolds G, R**? is homologically minimal in Grassmannian manifold

+” for k-even and they are calibrated by forms A, given below (see [6])

et us consider an orthogonal complex structure J on R* then J defines a complex structure

same name on R*@ R" by J(u@v) = J(u)@v Let wy denote the corresponding Kahler form

P

® R" Consider kp-form 2 = es (k = 2r) A twisted average of these powers of Kahler

by the space of all possible complex structure J on R*, equivalently, by the group O(k) as s Fix the complex structure J on R* Then for each g © O(k) consider the corresponding ex structure g~!J.g on RF and by our convention, also on R* @ R", we have g*wy = Wy-1 Jo

der the form

Àp= [ (det)? g* Ndg

O(k)

Ap is a SO(k) x SO(n)-invariant form, hence it induces a SO(k + n)-invariant differential

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power” of the Euler form (the term of “adjusted power” was used by Le Hong Van in [10], b

the above construction, see [6]

Let ¢:, , ¢e is an oriented orthonormal basis in R* such that this orientation agrees

canonical orientation of complex structure J Let f,, , fy is an orthonormal basis of R” '

E=(a Sh)A Alea @h)A -A (1B fp) A> A (ee @ fp), forpSn

is a canonically oriented complex rp-plane for complex structure J in Rk @ R", ie EE and spané is O(k)-invariant Each g € O(k) is extended on R* @ R" by g(u®v) = g(u) @v

det(g|spang = (detg)? (on the right side, we consider the determinant of transformation g or

Applying Theorem 2, Theorem 3 and Theorem 6.1 in [8] for powers of the Kahler forms we ‹

the following

Corollary 8 Let Ap = fio, (detg)? 9° M.dg be the form mentioned above Then

làÀs|l" = Jl@l|* = 1

and lA; ^ 9|" = lÀ;|:IIwll,

tuhere ý 3s any form on a space orthogonal to R* @ R"

In fact, the first conclusion of Corollary 3 had been proved in (6)

Now we consider the quaternionic Kahler form on a quaternionic Kahler manifold it is d

as follows: on a quaternionic Kahler manifold M and parallel differential form determin Sp(n) x Sp(1)-invariant form = ‡(Qƒ + 03 +%) on H” is called the quaternionic F

form and it is also denoted by M, where w;, ws, wx are Kahler form corresponding to co:

structures J, J, K on H” and where H” is identified with a tangent space T, M for some +

(see [12])

Tasaki [12] has proved that

(Sp(1) = {z € H, |z| = 1} and Sp(1) acts on H” by the left handed multiplication)

We note that €=uAutAu 7 AvKA \A Um AUnt A Und A Um ky Wher€ UỊ, , 9,

system of H-linearly independent orthonormal vectors in H”, satisfies the conditions in Th:

nn (2m)!

=1, 2, , m with the determinants of transformations equal to 1, therefore det(g|spane) = any g € Sp(1) Because of the same reason as for corollary 3 we obtain the following

2 forw = and G = Sp(1) Further since Sp(1) acts on span vp, v, i, vp Jy vp k) for

Corollary 4 Let 1 be the form on H" defined as above Then =1 am = | (Em) |

and a” A pl = JIaII.Iwll"

where wp is any form on a space orthogonal to H"

In fact, the first conclusion of Corollary 4 had been proved in [12]

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4, PRODUCT OF MINIMAL CURRENTS

We consider the product M x N of Riemannian manifolds M and N Let S and T be minimal mts on M and N respectively In general, one does not know whether the Cartesian product

*is also minimal on M x N Applying the calibration method with using results on the comass

oducts, we can give some new examples of minimal currents as Cartesian products of minimal

int in the class of normal currents

First we recall some necessary notions and facts (for details see [4])

Let y and S be a differential p-form and a p-current in a Riemannian manifold M respectively comass of y is defined by ||¢||* = sup{{jyz||", 2 € M} and the set of maximal directions of 9 fined by

G() = LJ(G(ez), liezl" = llel"}-

HÍ @ is closed and has comass one then it is called a calibration The mass of 8 is deRned by

M(S) = sup{S(), lel)’ = 1}

is a surface in M then M(S) = volume(S) A current S in a Riemannian manifold is called

logically minimal with respect to the mass if M(S) < M(S') for any current S’ homologous and the S is called homologically mass-minimizing current, or simply, homologically minimal

ont A fundamental theorem of the calibration method [4, Theorem 3.6] says that a current S Riemannian manifold M is homologically minimal if and only if there exists a closed form 2 that the tangent S, of S belongs to G(M) almost every where (in the sence of the measure | In this case, we say that S is calibrated by 2

We have the following

orem 4 Let S and T be two homologically minimal currents in Riemannian manifolds M and spectively If S 1s calibrated by p and T is calibrated by W such that |\pz Avyll* = llezll* lleyll*

ny z€ M,yEN Then Sx T is homologically minimal in M x N

tark @ and ý can be considered as differential forms on M x N by identifying with x] @ and respectively where ™ : M x N — M, m2: Mx N — N are canonical projections

if We have p Ap be closed and

ler vil" = sup{I(OAv) 2, ll’, (2, y) € Mx N} = sup{ii(ez ^ y)||”, ze M, € N} = sup{llezl\* llÚu||*, ze M, < W} = sup Íl@z|” sụp ||ý„ z€M vew | = lel'-ll#lˆ = 1: refore @ A 9 is calibration on M x N On the other hand, since 9 x Ï„,„) = Se AT, and g(Š,) = 1, 9(W) = 1 almost every where lave

s shown that S x T is calibrated by @A Ú

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Examples

Example 1 Let S be a special Lagrangian submanifold of R?" and T be a homologically mini current in a Riemannian manifold N Then S x T is homologically minimal current in R?” x

This follows form Corollary (3) and Theorem 4

Example 2 Let S be a homologically minimal current in a 6-dimensional Riemannian manifold and T a homologically minimal current and a Riemannian manifold N Then from Corollary

and Theorem 4 it follows that S x T is homologically minimal in M x N

Example 3 Let M be a 4n-dimensional quaternionic Kahler manifold and 3 be a quaternic

Kahler submanifold of M Let T be a homologically minimal current in a Riemannian manil

N Since S calibrated by differential form us (m < n) where M is quaternionic Kahler form |

{12]) from Corollary 4 and Theorem 4 it following that 8 x T is homologically minimal curren’ MxN

Example 4 For k even integer, by Proposition 3.1 in [6] the submanifold G, R**? of Gy R!

is homologically minimal and it calibrated by form A, mentioned in section 3 Let S be a hoi logically minimal current in a Riemannian manifold N, then from Corollary 3 and Theorem follows that G, R*+? x S is homologically minimal current in Gy R**" x N

Remark The above-results hold, in particular, when currents are replaced by surfaces

Acknowledgement The author expresses his gratitude to Prof Dao Trong Thi for his scien! advice

REFERENCES

1 M Berger Du Caté de chez Pu, Ann., Scient, Ec Norm Sup., 4 (1972), 1-44

2, J Dadok, R Harvey, and F Morgan Calibration on RŠ, Trans Amer Math Soc., : (1988), 1-40 3 Dao Trong Thi Minimal current on compact manifolds, Izv Akad Nauk USSR, Ser Ma 41 (1977), 853- 867 (in Russian) 4 Dao Trong Thi Globally minimal currents and surfaces in Riemannian manifolds, Acta Mi Vietnam., 10 (1985), 296- 333

5 H Federer Geometric Measure Theory, Berlin Springer, 1969

6 H Gluck, F Morgan, and W Ziller Calibrated geometry in Grassmannian manifolds, Cor Math Helv., 64 (1989), 256-268

7 R Harvey and H.B Lawson Calibrated geometries, Acta Math., 148 (1982), 47-157

8 Hoang Xuan Huan Separable calibrations and minimal surfaces, Acta Math Vietnam.,

(1994), 77-96

9 Le Hong Van Minimal surfaces on homogeneous spaces, Izv Akad, Nauk USSR, Ser Ma

52 (1988), 1-39

10 Le Hong Van Application of integral geometry to minimal surfaces, International Journ:

Mathematics, Vol 4, No.1 (1993), 89-111

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Morgan The exterior algebra A* R" and area minimization, Linear Algebra Appl., 66 15), 1-28

Tasaki Certain minimal or homologically volume minimizing submanifolds in compact

metric space, Tsukuba J Math., Vol.9, No.1, (1985), 117-131

KHOA HOC DHQGHN, KHTN, t.XII, n°1, 1996

VỀ DOI KHOI LUONG CUA TÍCH CÁC DẠNG

Nguyén Duy Binh

Khoa Todn - Dai hoc Sw pham Vink

leo phương pháp dạng cỡ, vấn đề xác định khối lượng và các hướng cực đại của tích các

lên các không gian trực giao có ý nghĩa quan trọng trong việc tìm các mặt cực tiểu thể

4o hàm tích các mặt cực tiểu thể tích Trong bài này chúng tôi chứng minh một đẳng thức

lượng của tích các dạng khi một nhân tử là một dạng xuyến hoặc là một dạng trung bình bởi một nhóm Áp dụng kết quả này, chúng tôi nhận thấy được một số ví dụ về các mặt

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