Some remarks on spherical harmonics

Some remarks on spherical harm onics V.M. Gic hev ∗ Abstract The article contai ns severa l observ ati ons on spherical harmonics and their no dal sets: a construction for harmonics with prescrib ed zeroes; a natural representa tion of this t yp e f or harmonics on S 2 ; upp er and lo w er b ounds for no d al length and inner radius (th e upp er b ounds are sharp ) ; the precise upp er bound for the num ber of common zeroes of tw o spherical harmonics on S 2 ; the mean Hausdorff measure on th e in tersection of k nod al sets for h armonics of different degrees on S m , where k ≤ m (in particular, the mean number of common zero es of m harmonics). In tro duction This a rticle contains several observ ations on spheric a l harmo nics a nd their no dal sets; the emphasis is on the ca se of S 2 . Let M be a compact connected ho mogeneous Riemannian manifold, G be a compac t Lie gro up acting o n M tr ansitively b y isometries, and E b e a G - inv ariant subspace of the (real) eigenspace for some non-zero eigenv alue o f the Laplace–Be ltr ami opera to r. W e show that eac h function in E can b e realized as the determinant o f a matrix, whose entries are v alues of the repro ducing kernel for E . There is a similar w ell-known construction for the orthogonal poly no- mials. How ever, the metho d do e s not work for an arbitrar y finite dimensional G -inv ariant subspace of C ( M ) (see Remark 2). Ther e is a natural unique up to scaling factor s rea lization of this type for spherical ha rmonics on S 2 . It can be obtained by complexifica tion and restr ic tion to the n ull-cone x 2 + y 2 + z 2 = 0 in C 3 . There is a tw o-sheeted eq uiv a riant cov ering of this cone by C 2 , whic h ident ifies the spa ce H n of harmonic homog e ne o us complex-v alued p olynomials of degree n on R 3 with the space P 2 2 n of homogeneo us holomorphic polynomials on C 2 of degree 2 n . 1 ∗ Pa rtially supported by RFBR gran ts 06-08-01403, 06-07-89051 and SB RAS pro j ect No. 117 1 In 1876, Sylve ster used an equiv alen t const ruction to refine Maxw ell’s method for repre- sen tation of spherical harmonics. According to i t, one has to differentiate the function 1 /r , where r is the distance to origin, i n suitable dir ections i n R 3 to get a real harm onic. The directions are uniquely defined; the corresp onding p oints in S 2 are called p oles (see [15, Ch. 9] or [3, 11.5.2]; [7, Ch. 7, s ection 5] and [1, App endix A] contain extended exp ositions and further information). 1 The se t of all zero es of a real s pherical har monic u is ca lled a no da l set . W e say that u a nd its no dal set N u are r e g ular if zero is not a critical v alue of u . Then each comp onent of N u is a Jordan con tour. Accor ding to [1 1], a pair of the no da l sets N u , N v , where u , v ∈ H R n and n > 0, hav e a non-void int ersection; moreover, if u is r e gular, then each compo nent of N u contains a t least t wo po int s of N v . The set N u ∩ N v may b e infinite but the family of such pair s ( u, v ) is closed and nowhere dense in H R n × H R n . If N u ∩ N v is finite, then car d N u ∩ N v ≤ 2 n 2 . The estimate follows from the Bezout theorem a nd is precise. This gives a n upp er b ound for the num b er of cr itical p oints of a generic spherical har mo nic, which pro bably is not sha r p. The configura tion of critical p oints is alwa ys deg e ne r ate in some s ense (see Remar k 5). The problem of finding low er bo unds seems to be more difficult. Accor ding to pa rtial r esults and computer exp eriments, 2 n may be the shar p low er b ound. The inv es tigation of metr ic and topo logical pro pe r ties of the no dal s ets has a long and ric h history; we only g ive a few remarks on the sub ject o f this pap er. Let ∆ b e the Laplace–Beltra mi op era tor and λ b e an eig env alue of − ∆. In 1978, Br ¨ uning ([5]) found the low er bo und c √ λ for the leng th o f a no dal set on a Riemann surface. Y au conjectured ([2 2, Pro blem 74]) that the Hausdo rff measure of the no dal set of a λ -eigenfunction on a co mpact Riemannian manifold admits upp er and low e r bo unds of the type c √ λ . This conjecture was prov ed by Donnelly and F efferman for real analytic manifolds in [8]. In ([18]), Sav o prov ed that 1 11 Area( M ) √ λ is the low er b ound for the length o f a no da l set in a surface M for all sufficient ly la rge λ in any surface a nd for all λ if the curv ature is nonnegative. The upper and lower estimates o f the inner ra dius w e re found by Mangoubi ([13], [14]); in the c ase of surfaces , they are of order λ − 1 2 ([13]). One ca n find the 1-dimensio nal Ha usdorff measure of a set in S 2 int egrating ov er SO(3) the co un ting function for the num b er of its common po in ts with translates of a suitable subset o f S 2 (see Theo rem 4). Using estimates o f the nu m be r of common zero es, we give upp er and lower b ounds for the leng th of a no dal set and for the inner r a dius of a no dal domain in S 2 . The upp er bo unds are precise. Let H m +1 n be the spa ce of all real s pherical har monics of degre e n on the unit sphere S m in R m +1 . Cor resp onding to a p oint of S m the ev aluation functional at it on H m +1 n , we get an equiv a riant immersion of S m to the unit spher e in H m +1 n , which is lo ca lly a metr ic homothety with the co efficient q λ n m , where λ n = n ( n + m − 1 ) is the eig env alue of − ∆ in H m +1 n . This makes it p o ssible to calculate the mean Hausdorff mea sure of the intersection of k harmonics of degrees n 1 , . . . , n k : it is equal to c p λ n 1 . . . λ n k , where c dep ends only on m and k and k ≤ m (Theore m 6). In particular, if k = m , then w e get the mea n nu m be r of common zero es of m ha rmonics: it is e q ual to 2 m − m 2 p λ n 1 . . . λ n m ; if m = 2 , then p λ n 1 λ n 2 . In ar ticle [8], Donelly and F efferman wro te: “A main theme of this pap er is that a solution of ∆ F = − λF , on a rea l analytic manifold, behaves like a po lynomial of deg ree c √ λ ”. F ollo wing this idea, L. Poltero v ich conjectured that the mean num b er of common zero es is sub ject to the Bezout theorem, i.e ., that it is a s above. Th us, the result in the case k = m confirms 2 this conjecture up to m ultiplicatio n by a cons tant , and may b e treated as “ the Bezout theorem in the mea n” for the spherical har monics. F or k = 1, the mean Hausdorff measure, b y different but similar metho ds, was found by Bera r d in [4] and Neuheisel in [16]. The cas e of a flat tor us was inv es tigated by Rudnick and Wigman ([17]). 1 Construction of eigenfunctions whic h v anish on prescrib ed finite sets In this sec tion, M is a compact connected oriented homogeneo us Riemannian manifold of a compact Lie group G a cting by iso metries on M , ∆ is the Laplace– Beltrami op era tor o n M , λ > 0 (1) is a n eigenv alue of − ∆, E λ is the corr esp onding real eigenspace (i.e., E λ consists of rea l v alued eigenfunctions), and E is its G -inv aria n t linear subspace. Thu s, E is a finite sum of G -inv ariant irreducible subspaces of C ∞ ( M ). The inv aria nt measure with the total ma ss 1 on M is denoted by σ , L 2 ( M ) = L 2 ( M , σ ). F or any a ∈ M , there exists the unique φ a ∈ E that realiz es the ev alua tion functiona l at a : h u, φ a i = u ( a ) for all u ∈ E . Set φ ( a, b ) = φ a ( b ) , a, b ∈ M . It follows that φ ( a, b ) = φ a ( b ) = h φ a , φ b i = h φ b , φ a i = φ b ( a ) = φ ( b, a ) , (2) u ( x ) = h u, φ x i = R φ ( x, y ) u ( y ) dσ ( y ) for all u ∈ E , (3) x ∈ N u ⇐ ⇒ φ x ⊥ u , (4) φ x 6 = 0 for all x ∈ M . (5 ) The latter holds due to the homog eneity o f M . Acco r ding to (3), φ ( x, y ) is the r epr o ducing kernel for E (i.e., the mapping u ( x ) → R φ ( x, y ) u ( y ) dσ ( y ) is the orthogo nal pro jectio n o nt o E in L 2 ( M )). Let a 1 , . . . , a k , x, y ∈ M . Set a = ( a 1 , . . . , a k ) ∈ M k and let a also denote the corr esp onding k -s ubset of M : a = { a 1 , . . . , a k } . Set Φ a k ( x, y ) = Φ a k,y ( x ) = det      φ ( a 1 , a 1 ) . . . φ ( a 1 , a k ) φ ( a 1 , y ) . . . . . . . . . . . . φ ( a k , a 1 ) . . . φ ( a k , a k ) φ ( a k , y ) φ ( x, a 1 ) . . . φ ( x, a k ) φ ( x, y )      . (6) 3 Obviously , Φ a k ( x, y ) = Φ a k ( y , x ). Let us fix y and set v = Φ a k,y . Then, b y (6), v ∈ E and a 1 , . . . a k ∈ N v . (7) W e say that a 1 , . . . a k are indep endent if the vectors φ a 1 , . . . , φ a k ∈ E are linearly independent. F or a subset X ⊆ M , put N X = s pan { φ x : x ∈ X } . (8) If X = N u , where u ∈ E , then we abbrevia te the notation: N N u = N u . Set n = dim E − 1 . It follows from (1) that n ≥ 1 (note that E is real and G -inv a riant). Lemma 1. L et a ∈ M k , wher e k ≤ n . Then a 1 , . . . a k ar e indep endent if and only if Φ a k,y 6 = 0 for some y ∈ M . Pr o of. It follows from (4) tha t E = N M ; since k ≤ n , N a 6 = E . Ther efore, if a 1 , . . . a k are indep endent, then we ge t an indepe nden t set adding y to a , for some y ∈ M . Then Φ a k,y 6 = 0 since Φ a k,y ( y ) > 0 (by (2) a nd (6), Φ a k,y ( y ) is the determinant of the Gr am matrix for the vectors φ a 1 , . . . , φ a k , φ y ). Clearly , Φ a k,y = 0 for a ll y ∈ M if a 1 , . . . a k are dep endent. The following prop osition implies that ea ch function in E can b e r e alized in the form (6). Prop ositio n 1. F or any u ∈ E , N u = u ⊥ ∩ E . Lemma 2. If u , v ∈ E and N v ⊇ N u , then v = c u for some c ∈ R . Pr o of. This immedia tely follows from the inclusion N v ⊇ N u and Lemma 1 o f [11], which states that v = cu for so me c ∈ R if there exist no dal domains U, V for u, v , r esp e ctively , such that V ⊆ U . Here is a sketc h of the pro of of the mentioned lemma; it is based on the same idea as Courant’s No dal Domain Theore m. Since u doe s not change its sign in U , λ is the first Dirichlet eigenv alue for U . Hence, it has multiplicit y 1 and D ( w ) ≥ λ k w k L 2 ( U ) for all w ∈ C 2 ( M ) that v anish on ∂ U , where D is the Dirichlet form o n U . Mo reov er, the equa lit y holds if and o nly if w = cu for some c ∈ R . On the other hand, if w v a nishes outside V and co incides with v in V , then the equality is fulfilled. Pr o of of Pr op osition 1. If v ∈ E and v ⊥ N u , then N v ⊇ N u by (4). Thus, v ∈ R u b y Lemma 2. Therefore, N u ⊇ u ⊥ ∩ E . The reverse inclusion is evident. Let Φ : M n +1 → E b e the mapping ( a, y ) → Φ a n,y and set U = Φ( M n +1 ). 4 Theorem 1. (i) L et u ∈ E , u 6 = 0 . F or ( a, y ) ∈ N n u × M , Φ( a, y ) = c ( a, y ) u, (9) wher e c is a c ontinuous nontr ivial function on N n u × M . (ii) U is a c omp act s ymmetric neighb ourho o d of zer o in E . (iii) F or every a ∈ M n , ther e exists a nontrivial no dal set which c ontains a ; for a generic a , this set is unique. Pr o of. Let a ∈ N n u . If a 1 , . . . , a n are independent, then co dim N a = 1 ; since u ⊥ N u by (4), w e get (9), where c ( a, y ) 6 = 0 for some y ∈ M by L e mma 1. If a 1 , . . . , a n are dep endent, then Φ( a, y ) = 0 for all y ∈ M by the sa me lemma. The function c is cont in uous b y (6 ); it is nonzero s ince the set N u contains independent p oints a 1 , . . . , a n by Prop os ition 1. This proves (i). According to (6), Φ is co n tin uous. Hence, U is co mpact. Since M is con- nected, for any u ∈ U , we may g et the segment [0 , u ] mo ving y ; hence, U is starlike. Since transp osition of every tw o p oints in a changes the sign of c ( a, y ), U is symmetric if n > 1; for n = 1, U is a disc in E b ecause it is G -inv ariant a nd starlike. Thus, U is compa ct, symmetric, s tarlike, and ∪ t> 0 t U = E . Hence U is a neighbourho o d of z ero, i.e., (ii) is true. Let a ∈ M n and a ′ ⊆ a b e a maximal indep endent subset of a . Then Φ a ′ k,y 6 = 0 for some y ∈ M by Lemma 1, where k = card a ′ . Set v = Φ a ′ k,y . According to (7), a ′ ⊂ N v . By (4), N v contains any po in t x ∈ M such that φ x ∈ N a ′ . Hence N v includes a . The set N v is unique if a 1 , . . . , a n are independent b ecause co dim N v = 1 in this case. Since M is homo geneous and E is finite dimensional, the functions φ x , x ∈ M , a r e re al ana lytic. Ther e fo re, either Φ a n,y = 0 for all ( a, y ) ∈ M n +1 or Φ a n,y 6 = 0 fo r g eneric ( a, y ) (note that M is connec ted). Finally , Φ a n,y 6 = 0 for s ome ( a, y ) ∈ M n +1 since N M = E due to (4) and (5). A closed subset X ⊆ M is called an interp olation set for a function sp ac e F ⊆ C ( M ) if F | X = C ( X ). Corollary 1. L et k ≤ dim E . F or generic a 1 , . . . , a k ∈ M , a = { a 1 , . . . , a k } is an interp olation set for E . Remark 1. The function c may v anish on some comp onents of the s et N n u × M . F or example, let M b e the unit sphere S 2 ⊂ R 3 and E be the restriction to it of the space o f harmo nic homogeneous p olynomials of de g ree k ; then dim E = 2 k + 1, n = 2 k . If k > 1, then any big circ le S 1 in S 2 is contained in several no dal sets (for example, nodal sets of the functions x 1 f ( x 2 , x 3 ), where f is harmonic, contain the big circ le { x 1 = 0 } ∩ S 2 ); moreover, if k is o dd, then S 1 may be a comp onent of N u . Hence, c o dim N S 1 > 1 and Φ( a, y ) = 0 for all ( a, y ) ∈  S 1  n × S 2 . Remark 2. Theorem 1 fails for a generic finite dimensional G -inv ariant sub- space E ⊆ C ( M ). Indee d, if dim E > 1 and E co n tains cons ta n t functions, then it 5 includes an o pe n s ubset consisting of functions without zero es, which ev ide ntly cannot b e realiz e d in the form (6). F urthermore, it follows from the theorem that the pro ducts φ a 1 ∧ · · · ∧ φ a n fill a neighbour ho o d of z ero in the n th exterio r power of E , whic h may b e identified with E . This prop erty evidently imply the int erp olation prop erty o f Corollar y 1 but the conv ers e is not tr ue; a n example is the spac e of all homogeneo us po lynomials of degree m > 1 on R 3 , restricted to S 2 (or the s pace o f a ll po lynomials of deg ree les s than n o n [0 , 1] ⊂ R , where n > 2). 2 Spherical harmonics on S 2 Let P m n denote the spa ce o f a l homo g eneous holomor phic p olynomials of degr ee n on C m or/and the space of all complex v alued homogeneous p olynomials of deg ree n on R m ; clea rly , there is one-to-o ne corre spo ndence betw een these spaces. Its subspace of p o ly nomials whic h ar e har monic o n R m is denoted by H m n ; we omit the index m in H m n if m = 3. Then dim H n = 2 n + 1. The po lynomials in H m n , a s w ell a s their tr aces o n the unit sphere S m − 1 ⊂ R m , are called spheric al harmonics . They ar e eigenfunctions of the Lapla ce–Beltrami op erator; if m = 3, then the eigenv alue is − n ( n + 1). F or a pr o of of these facts, see, for example, [19]. W e say that u ∈ P m n is r e al if it takes rea l v alues on R m . The standard inner pro duct in R m and its bilinear extension to C m will b e denoted by h , i , r ( v ) = | v | = p h v , v i , v ∈ R m , r 2 is a ho lomorphic quadratic for m o n C m . F or a ∈ C m , set l a ( v ) = h a, v i . The functions Φ a k ( x, y ) admit holo mo rphic extensio ns o n a ll v ar iables (except for k ). If M = S 2 ⊂ R 3 , then the extension to C 3 and subse q uent restr iction to the null-cone S 0 = { z ∈ C 3 : r 2 ( z ) = 0 } makes it p os s ible to cons truct a kind of a natural r epresentation in the form (6), which is unique up to multiplication by a complex num b er, for a ny c omplex v a lued spher ical harmonic. The pro jection of S 0 to CP 2 is Riemann sphere CP 1 . The cone S 0 admits a na tural parametr ization: κ ( ζ 1 , ζ 2 ) = ( z 1 , z 2 , z 3 ) = (2 ζ 1 ζ 2 , ζ 2 1 − ζ 2 2 , i ( ζ 2 1 + ζ 2 2 )) , ζ 1 , ζ 2 ∈ C . (10) Lemma 3. The m apping R : H n → P 2 2 n define d by Rp = p ◦ κ is one-to-one and intertwines the n atur al re pr esentations of SO(3) and SU(2) in H n and P 2 2 n , r esp e ctively. 6 Pr o of. Clea rly , p ◦ κ is a homogene o us poly nomial on C 2 of degr ee 2 n for any p ∈ P 3 n . F urther, κ is equiv a riant with r esp ect to the natural actions of SU(2) in C 2 and SO(3) in C 3 : an easy calcula tio n with (10) shows tha t the c hange of v ariables ζ 1 → aζ 1 + bζ 2 , ζ 2 → − bζ 1 + aζ 2 , where | a | 2 + | b | 2 = 1, induces a linear tra nsformation in C 3 which preser ves r 2 and leaves R 3 inv ariant (in o ther words, the transformatio n of P 2 2 , induced by this c hange of v a riables, in the base 2 ζ 1 ζ 2 , ζ 2 1 − ζ 2 2 , i ( ζ 2 1 + ζ 2 2 ) corres p onds to a matrix in SO (3)). Hence R is an intert wining op er ator. It is well known that P 3 n = H n ⊕ r 2 P 3 n − 2 (see, for example, [1 9]). Since R 6 = 0 a nd Rr 2 = 0 , we get R H n 6 = 0. It remains to note that the natural r epresentations of these groups in H n , P 2 n are irreducible. Corollary 2. F or any p ∈ H n \ { 0 } , the set p − 1 (0) ∩ S 0 is the u nion of 2 n c omplex lines; s ome of them may c oincide. If the lines ar e distinct, q ∈ H n , and p − 1 (0) ∩ S 0 = q − 1 (0) ∩ S 0 , then q = cp for some c ∈ C . Pr o of. Clea rly , κ ma ps lines o nt o lines and induces an em b edding of CP 1 int o CP 2 . The functions φ a of the pr evious section can be written explicitly: φ a ( x ) = c n P n ( h a, x i ) , where a, x ∈ S 2 , c n is a no rmalizing constant, and P n is the n th Leg e ndre p olynomia l: P n ( t ) = 1 2 n n ! d n dt n ( t 2 − 1) 2 n . Ther e is the unique extension of φ ( a, x ) = φ a ( x ) int o R 3 which is homoge neous of degr ee n and harmonic on bo th v aria ble s (it is als o s y mmetric and extends int o C 3 holomorphica lly). F or example, if n = 3, then 2 P 3 ( t ) = 5 t 3 − 3 t a nd φ ( a, x ) is prop ortio na l to 5 h a, x i 3 − 3 h a, a i h a, x i h x, x i (if a = (1 , 0 , 0), then to 2 x 3 1 − 3 x 1 x 2 2 − 3 x 1 x 2 3 ). Of course, the r epresentation of p ∈ H n in the fo rm (6) holds for M = S 2 but ther e is a more natura l version in this case. F or ζ = ( ζ 1 , ζ 2 ) ∈ C 2 , set j ζ = ( − ζ 2 , ζ 1 ) . Theorem 2. L et p ∈ H n . Supp ose t hat p − 1 (0) ∩ S 0 is the union of distinct lines C a 1 , . . . , C a 2 n . Then ther e exists a c onstant c 6 = 0 such that p ( x ) p ( y ) = c det      h a 1 , a 1 i n . . . h a 1 , a 2 n i n h a 1 , y i n . . . . . . . . . . . . h a 2 n , a 1 i n . . . h a 2 n , a 2 n i n h a 2 n , y i n h x, a 1 i n . . . h x, a 2 n i n h x, y i n      (11) 7 for al l y ∈ S 0 , x ∈ C 3 . Mor e over, re placing h x, y i n with φ ( x, y ) in the matrix, we get s u ch a r epr esent ation of p ( x ) p ( y ) for al l x, y ∈ C 3 (with another c in gener al). Pr o of. A calcula tio n shows tha t h a, x i n is harmonic on x fo r all n if a ∈ S 0 . Hence, the function Φ a y ( x ) = Φ a ( x, y ) in the right-hand s ide b e lo ngs to H n for each y ∈ S 0 . Clearly , Φ a y ( a k ) = 0 for all k = 1 , . . . , 2 n . By Corolla ry 2, Φ a y is prop or tional to p . Since Φ a ( x, y ) = Φ a ( y , x ), we get (11) if the r ight-hand side is no n trivial. Th us, we have to prove tha t c 6 = 0. Let x ∈ S 0 . There exist α 1 , . . . , α 2 n , ξ , η ∈ C 2 such that a k = κ ( α k ) fo r all k , x = κ ( ξ ), and y = κ ( η ). By a straig ht forward calculation, for an y a, b ∈ C 2 we get h κ ( a ) , κ ( b ) i = − 2 h a, j b i 2 . (12) Hence, the r ight-hand side of (11) is equal to − 2 (2 n +1) n c det      h α 1 , j α 1 i 2 n . . . h α 1 , j α 2 n i 2 n h α 1 , j η i 2 n . . . . . . . . . . . . h α 2 n , j α 1 i 2 n . . . h α 2 n , j α 2 n i 2 n h α 2 n , j η i 2 n h ξ , j α 1 i 2 n . . . h ξ , j α 2 n i 2 n h ξ , j η i 2 n      . (1 3) The determina nt can b e ca lc ulated explicitly . More gener ally , if C = ( c r s ) k +1 r,s =1 , where c r s = h a r , b s i k , a r , b s ∈ C 2 , then det C = k Y r =1  k r  Y s 0 is co ns tant and h is a p olynomia l. Therefore, we can get a function f 6 = 0 on C 3 , which co incides with p ( x ) on S 0 up to a cons tant factor , replacing h x, y i n with φ ( x, y ) in (11) and fixing generic y ∈ C 3 . By Cor o llary 2 , the same is true on C 3 since f ∈ H n according to (11) (all functions in the last row ar e harmonic on x ). Since φ ( x, y ) = φ ( y , x ), this proves the s econd asser tion. Remark 3. The set p − 1 (0) ∩ S 0 , wher e p ∈ H n , is also distinguished by the orthogo nality condition Z S 2 p ( x ) h x, w i n dσ ( x ) = 0 , where σ is the inv aria nt measure on S 2 and w ∈ S 0 . This is a consequence of (15) since R p ( x ) φ ( x, y ) dσ ( x ) = p ( y ) for all y ∈ S 2 , hence for a ll y ∈ R 3 ( p ( y ) and φ x ( y ) ar e ho mo geneous of deg ree n ), mo reov er , for all y ∈ C 3 (bo th sides are holomorphic on y ). In particular, this is true for y ∈ S 0 but φ ( x, y ) = s n h x, y i n in this case. If p − 1 (0) ∩ S 0 is the union of distinct lines C a k , k = 1 , . . . , 2 n , then the functions h x, a k i n , k = 1 , . . . , 2 n , form a linea r base fo r the space o f functions in H n which a r e o rthogonal to p with resp ect to the bilinear form R f g dσ . This is a cons equence o f (12): it is easy to c heck that the functions h ζ , b s i k on C 2 , where s = 1 , . . . , k , are linearly indep endent if the lines C b s are distinct (the V andermonde determinant). W e conclude this s e ction with rema r ks on num b er of zero es in S 2 of functions in H n . Let f ∈ H n , u = Re f , v = Im f . A zero of f is a common zero of u and v . The following prop osition, in a slightly more gener al form, w as proved in [11]. W e say that u is r e gu lar if zero is no t a critical v alue for u . Prop ositio n 2 ([11]) . L et n > 0 , u ∈ H n . If u is r e gular, then for any v ∈ H n e ach c onne cte d c omp onent of N u c ontains at le ast two p oints of N v . The ass ertion fo llows from the Green formula which implies that Z C v ∂ u ∂ n ds = 0 , (16) where C is a comp onent of N u , which is a Jo rdan contour, ds is the length measure on C , and ∂ u ∂ n is the nor mal deriv a tive; note that ∂ u ∂ n keeps its sign on C . F or the standard sphere S 2 , (16 ) follows from the classic al Gree n formula for the do main D ε = (1 − ε, 1 + ε ) × S 2 , where ε ∈ (0 , 1 ), and the homog eneous of degree 0 e x tensions of u, v in to D ε . Let u, v ∈ H n be real and r egular. Set ν ( u, v ) = card N u ∩ N v . F or singular u, v , zero es must b e counted with multiplicities; if u , v ∈ H n , then the multiplicit y o f a zer o ca n b e defined as the num ber o f s mo oth no dal lines which meet at it; if u, v hav e m ultiplicities k , l at their co mmo n zero , then o ne 9 hav e to c o unt them k l times (the gr eatest nu mber o f common zer o es whic h app ear under small p ertur bations). If u = φ a , where a ∈ S 2 , then N u is the union of n pa rallel c ir cles h x, a i = t k , x ∈ S 2 , whe r e k = 1 , . . . , n and t 1 , . . . , t n are the zero es of P n ( t ). Since they are distinct, P ′ n ( t k ) 6 = 0 for all k . It follows from Pro p os ition 2 tha t for any r e al v ∈ H n ν ( φ a , v ) ≥ 2 n , where a ∈ S 2 . If b ∈ S 2 is sufficie ntly clo s e to a , then the e q uality holds fo r v = φ b . In the inequality ab ov e, φ a and n may b e replaced with any reg ula r u and the num b er of comp onents of N u , resp ectively . The latter ca n b e less than n (according to [12], it can b e equal to one or t wo if n is o dd or even, r esp ectively 2 ). How ever, computer exp e r iments supp or t the following conjectur e : for all real u, v ∈ H n , ν ( u, v ) ≥ 2 n. The common zero es m us t b e counted with multip licities. Otherwis e , there is a simple exa mple of tw o harmo nic s which hav e only tw o c o mmon zero es: Re( x 1 + ix 2 ) n and Im( x 1 + ix 2 ) n . On the other hand, for g e neric re al u, v ∈ H n there is a trivial sharp upp er bo und for ν ( u , v ). W e prov e a version tha t is s tr onger a bit. Prop ositio n 3. L et u, v ∈ H n b e r e al. If ν ( u , v ) is finite, then ν ( u, v ) ≤ 2 n 2 . (17) By the Bezout theorem, if u, v ∈ P 3 n hav e no prop er common divisor , then the set { z ∈ C 3 : u ( z ) = v ( z ) = 0 } is the union of n 2 (with multiplicities) complex lines. Then ν ( u, v ) ≤ 2 n 2 since each line has at most t wo common po int s with S 2 . The prop os itio n is not an immediate consequence of this fac t since u , v may hav e a nontrivial common divis or which has a finite num b er of zero es in S 2 . This cannot happ en for u, v ∈ H n by the following lemma. Lemma 4. L et u ∈ H n b e r e al, x ∈ S 2 , and u ( x ) = 0 . Supp ose t hat u = v w , wher e v ∈ P 3 m , w ∈ P 3 n − m ar e r e al. If w ( y ) 6 = 0 for al l y ∈ S 2 \ { x } that ar e sufficiently close to x , then w ( x ) 6 = 0 . Pr o of. W e may ass ume x = (0 , 0 , 1). If u has a zer o o f multiplicit y k at x , then u ( x 1 , x 2 , x 3 ) = p k ( x 1 , x 2 ) x n − k 3 + p k +1 ( x 1 , x 2 ) x n − k − 1 3 + · · · + p n ( x 1 , x 2 ) , where p j ∈ P 2 j , p k 6 = 0. Since ∆ u = 0 , we hav e ∆ p k = 0. Hence, p k ( x 1 , x 2 ) = Re( λ ( x 1 + ix 2 ) k ) 2 The corresp onding harmonic is a small p erturbation of the function Re( x 1 + ix 2 ) n . 10 for s ome λ ∈ C \ { 0 } . Therefore, p k is the pr o duct o f k distinct linea r forms. Let w = q l ( x 1 , x 2 ) x n − m − l 3 + q l +1 ( x 1 , x 2 ) x n − m − l − 1 3 + · · · + q n − m ( x 1 , x 2 ) , v = r s ( x 1 , x 2 ) x m − s 3 + r s +1 ( x 1 , x 2 ) x m − s − 1 3 + · · · + r m ( x 1 , x 2 ) , where q j , r j ∈ P 2 j and q l , r s 6 = 0 . Since p k = q l r s , we hav e k = l + s ; moreover, either q l is constant or it is the pro duct o f distinct linea r forms. The latter implies that it change its sign near x ; then the same is true for w , co nt radictory to the ass umption. Hence l = 0 . Thus, q l 6 = 0 implies w ( x ) = q l ( x ) 6 = 0. Pr o of of Pr op osition 3. Let u, v ∈ H n be rea l and w be their grea tes t common divisor. Clearly , w is real. Since N u ∩ N v is finite, zero es of w in S 2 m ust be isolated; b y Lemma 4, w ha s no zero in S 2 . Applying the B ezout theorem to u/w and v /w , w e get the ass ertion. The equa lit y in (17) ho lds, for example, for the following pair s and for their small p erturbations: u = φ a , v = Re( x 2 + ix 3 ) n , where a = (1 , 0 , 0); (18) u = Re( ix 2 + x 3 ) n , v = Re( x 1 + ix 2 ) n . Corollary 3. If the nu mb er of critic al p oints for r e al u ∈ H n is fi nite, t hen it do es not exc e e d 2 n 2 ; in p articular, this is true for a generic r e al u ∈ H n . Pr o of. If x is a critical po int of u , then ξ u ( x ) = 0 for any vector field ξ ∈ so(3). It is p ossible to cho o se tw o fields ξ , η ∈ so (3) which do not annihilate u a nd are independent a t a ll critical p oints; then the critica l po int s of u ar e precisely the common zero es of ξ u, η u ∈ H n . Remark 4. This b ound is not s harp. At least, for n = 1 , 2 the num b er of critical p oints is eq ual to 2( n 2 − n + 1), if it is finite. Let u, v be as in (18). Then u + ε v , where ε is small, has 2( n 2 − n + 1 ) critical p oints. I know no example of a spherical ha rmonic with a greater (finite) num ber of c r itical p oints. Remark 5. The c o nsideration ab ov e prov es a bit more than Cor ollary 3 says. A nontrivial orbit of u under SO(3) is either 3-dimensional o r 2-dimensio nal, and the latter holds if a nd only if u = cφ a for some constant c and a ∈ S 2 . In the first case , the set C of critical po int s o f u is precise ly the s et of common zero es of three linea rly indep endent spherical harmo nics (a ba se for the tangent space to the orbit of u ). Hence, co dim N C ≥ 3 . Note that generic thre e harmonics have no c o mmon zer o. Th us, the configura tion of cr itica l p oints is a lwa ys degener ate. The pr o blem of estimation of the num b er of critical p o int s, comp onents o f no dal sets, no dal domains, etc., for spherica l harmonics on S 2 was stated in [2]. Prop ositio n 4. The set I of functions f = u + iv ∈ H n such that ν ( u, v ) = ∞ is close d and nowher e dense in H n . 11 Pr o of. If N u ∩ N v is infinite, then it contains a J ordan arc which extends to a contour since u and v are real a nalytic (by [6], a no da l s e t, o utside of its finite subset, is the finite union of smo oth a rcs). This contour ca nnot be included int o a disc D which is contained in some of no da l do mains: o therwise, its fir st Dirichlet eigenv alue would b e g r eater than n ( n + 1). Therefore, diameter o f the contour is bo unded from below. This implies that I is closed. If f ∈ I , then u and v hav e a no n trivial commo n divisor due to the Bezout theorem; hence, I is nowhere dense. In examples k nown to me, if f ∈ I , then N u ∩ N v is the union of c ir cles. 3 Estimates of no dal length and inner radius Let M b e a C ∞ -smo oth compac t connected Riemannian ma nifold, m = dim M , h k be the k -dimens ional Haus do rff measur e on M . Y au conjectured that ther e exists p ositive co ns tant c and C such that c √ λ ≤ h m − 1 ( N u ) ≤ C √ λ for the no dal s et N u of any eigenfunction u cor resp onding to the eig e n v a lue − λ . F or real a nalytic M , this conjecture was proved by Donnelly and F efferman in [8]. In the case of a sur face, lower b ounds were obta ined in pap ers [5] and [1 8]; in [18], c = 1 11 Area( M ). W e consider first the case M = S m ⊂ R m +1 , m ≥ 1. Set ψ ( x ) = Re( x 1 + ix 2 ) n . Clearly , ψ ∈ H m +1 n . Let φ de no te a zona l spher ical harmo nic; we o mit the index since the geo metric quantities that c haracterize N φ are indepe nden t o f it. Set ω k = h k ( S k ) = 2 π k +1 2 Γ  k +1 2  . Theorem 3. F or any nonzer o r e al u ∈ H m +1 n , h m − 1 ( N u ) ≤ h m − 1 ( N ψ ) = nω m − 1 . (19) The theorem is simply a n observ atio n mo dulo the following fact (a particula r case of Theorem 3 .2.48 in [10]). A s e t which can b e rea liz ed as the image of a b ounded subset of R k under a L ips chit z mapping is ca lled k - r e ctifiable (w e consider only the sets which can be realized as the coun table union o f compact sets). Since u ∈ H m +1 n is a p olyno mia l, the set N u is ( m − 1 )-rectifiable. Let µ m denote the in v a riant mea sure on O( m + 1) with the total mass 1. Theorem 4 ([10]) . L et A, B ⊆ S d b e c omp act, A b e k -r e ctifiable, and B b e l -r e ctifiable. Set r = k + l − d . Supp ose r ≥ 0 . Then Z O( d ) h r ( A ∩ g B ) dµ d ( g ) = K h k ( A ) h l ( B ) , (20) 12 wher e K = Γ  k +1 2  Γ  l +1 2  2Γ  1 2  d Γ  r +1 2  = ω r ω k ω l . If r = 0, then the left-hand side of (20) is a version o f the F av ar d mea sure for spheres (o n A or B ). Also, note that (20) can be proved directly in this setting since the left-ha nd side, for fixed A (o r B ), is additive o n finite families of disjoint compact s ets; thus, it is sufficient to chec k its a symptotic b ehavior on small pieces of submanifolds. Lemma 5. F or any r e al u ∈ H m +1 n and e ach big cir cle S 1 in S m , if S 1 ∩ N u is finite, then card( S 1 ∩ N u ) ≤ 2 n. (21) Pr o of. The restric tio n of u to the linear s pan of S 1 , which is 2-dimensio nal, is a homogeneous p olynomia l o f degree n of t wo v a r iables. Pr o of of The or em 3. Since S 1 int ersects in tw o p oints a ny hyper pla ne which do es not co n tain it, for almos t a ll g ∈ O ( m + 1 ) we have card( g S 1 ∩ N u ) ≤ 2 n = c a rd( g S 1 ∩ N ψ ) . Int egrating over O( m + 1) and a pplying (20) with k = 1, l = m − 1, A = S 1 , B = N u and B = N ψ , we get the inequality in (19 ). The equality is evident. A low er b ound can also b e obtained in this wa y . In what follows, we a ssume k = l = 1 and m = 2 ; then K = 1 2 π 2 , and (19) read as follows: h 1 ( N u ) ≤ 2 π n. (22) The noda l set N φ of a zonal s pher ical ha rmonic φ = φ a ∈ H n , where a ∈ S 2 , is the union of pa rallel circ le s of Euclidean r adii p 1 − t 2 k , where t k are zer o es of the Lege ndr e p olyno mia l P n . The smallest circle corr esp onds to the g reatest zero t n . Set r n = p 1 − t 2 n and let C n be a circle in S 2 of Euclidean radius r n . By Prop os ition 2, for any u ∈ H n , card( g C n ∩ N u ) ≥ 2 for a ll g ∈ O(3) . (23) Due to (20), h 1 ( N u ) ≥ 2 π r n . By [21, Theor em 6.3.4], t n = cos θ n , where 0 < θ n < j 0 n + 1 2 (24) 13 and j 0 ≈ 2 . 4048 is the leas t p ositive zero of Bessel function J 0 . This estimate, by [21, (6.3 .1 5)], is asymptotically shar p: lim n →∞ nθ n = j 0 . Thus, r n = s in θ n < sin j 0 n + 1 2 < j 0 n + 1 2 , and we get h 1 ( N u ) > 2 π j 0  n + 1 2  . (25) The b ound (25) is not the b est one but it is greater than 1 11 Area ( M ) √ λ , the bo und of pap er [18 ]: 4 π 11 p n ( n + 1) < 2 π j 0  n + 1 2  , since 4 π 11 ≈ 1 . 42 48, 2 π j 0 ≈ 2 . 6127; a c cording to [18], 1 11 Area ( M ) √ λ estimates from b e low the no dal length for all close d Riemannia n surfaces M (for suffi- ciently large λ in genera l and for all λ if the curv ature is nonneg ative). The length of the no dal set of a zona l har monic co uld b e the sha rp lower b ound. Ac- cording to [2 1, (6.21.5 )], k − 1 2 n + 1 2 π ≤ τ n − k ≤ k n + 1 2 π , where cos τ k , k = 0 , . . . , n − 1 , are the zero es of P n in the o rder of decrea sing (i.e., τ 1 = θ n ). Hence h 1 ( N φ ) = 2 π n X k =1 sin θ k ≈ 2 π n Z 1 0 sin π x dx = 4 n as n → ∞ . If this is true, then the upper b ound is r a ther clo se to the low er one since their r atio tends to π 2 as n → ∞ . It is also p os s ible to estimate the inner r adius of S 2 \ N u : inr( S 2 \ N u ) = sup  inf y ∈ N u ρ ( x, y ) : x ∈ S 2  , where ρ is the inner metric in S 2 : ρ ( x, y ) = ar ccos h x, y i . The least upp er b ound is evident: inr( S 2 \ N u ) ≤ inr( S 2 \ N φ ) = θ n by (24 ). Indeed, it is attaine d for u = φ a nd ca nnot be grea ter since the circle C n int ersects any no dal set by P rop osition 2. Let C ( θ ) b e the a circle of r adii θ in the inner metric of S 2 ; then Euclidean radius of C ( θ ) is r = sin θ . A num b er θ 0 > 0 is a low er b ound for the inner ra dius if and o nly if the following conditions hold: (i) θ 0 ≤ θ n , 14 (ii) for ea ch real u ∈ H n , there exis ts g ∈ O(3) such that g C ( θ 0 ) ∩ N u = ∅ . (note that the disc b ounded by C ( θ 0 ) c a nnot contain a comp onent of N u due to (i)). F urther, for almost a ll g ∈ O (3) the num ber card( g C ( θ 0 ) ∩ N u ) is even. Therefore, we may a ssume that card( g C ( θ 0 ) ∩ N u ) ≥ 2 if g C ( θ 0 ) ∩ N u 6 = ∅ . Set r 0 = s in θ 0 . If (ii) is false then 2 ≤ 1 2 π 2 h 1 ( C ( θ 0 )) h 1 ( N u ) = r 0 π h 1 ( N u ) ≤ 2 r 0 n by (20). Thus, if r 0 < 1 n , then θ 0 is a low er b ound for inr( S 2 \ N u ). Hence arcsin 1 n is a low er b ound for inr( S 2 \ N u ). The es timate seems to b e non-shar p; per haps, the lea st inner radius has the set S 2 \ N ψ (it is eq ual to π 2 n ). W e summarize the results on S 2 . Theorem 5. Le t M = S 2 . F or any nonzer o r e al u ∈ H n , 2 π j 0  n + 1 2  < h 1 ( N u ) ≤ 2 π n, (26) arcsin 1 n ≤ inr  S 2 \ N u  ≤ θ n < j 0 n + 1 2 . (27) In (26) , the upp er b ound is att aine d if u = ψ ; t he upp er b ound θ n in (27) is attaine d for u = φ . 4 Mean Hausdorff measure of in tersections of the no dal sets Let us fix m ≥ 2 and the unit s phere S m ⊂ R m +1 . W e shall find the mean v alue ov er u 1 , . . . , u k , k ≤ m , of the Hausdorff measure of s ets N u 1 ∩ · · · ∩ N u k ⊂ S m . If k = m , then this is the mean num b er of common z e ro es of u 1 , . . . , u m in S m . Set n = ( n 1 , . . . , n k ) , δ ( n ) = dim H m +1 n − 1 , where n, n j are p ositive integers. W e define the mean v a lue as follows: M n = Z S δ ( n 1 ) ×···× S δ ( n k ) h m − k ( N u 1 ∩ · · · ∩ N u k ) d ˜ σ δ ( n 1 ) ( u 1 ) . . . d ˜ σ δ ( n k ) ( u k ) , (28) where ˜ σ j denotes the inv a riant measur e on S j with the to ta l mass 1. Let λ n be the eigenv a lue of − ∆ in H m +1 n ; reca ll that λ n = n ( n + m − 1) . 15 Theorem 6. Le t 1 ≤ k ≤ m . Then M n = ω m − k m − k 2 p λ n 1 . . . λ n k , (29) wher e M n is define d by (28) . If k = m , then we get the mea n v alue of car d ( N u 1 ∩ · · · ∩ N u m ); since ω 0 = 2 and h 0 = c ard, it is equal to 2 m − m 2 p λ n 1 . . . λ n m . There is a natural equiv aria nt immersio n ι n : S m → S δ ( n ) ⊂ H m +1 n : ι n ( a ) = φ a | φ a | . (30) If n is o dd, then ι n is one-to-o ne; for even n > 0, ι n is a tw o-sheeted cov ering, which iden tifies o pp os ite p oints. Clearly , the Riemannian metric in ι ( S m ) is O( m + 1 )- in v a riant a nd the stable subgroup of a acts transitively on spheres in T a S m . Hence, the mapping ι n is lo ca lly a metric homothety . Let s n be its co efficient. Clea rly , s n = | d a ι n ( v ) | | v | , (31) where the right-hand side is indep endent o f a ∈ S m and v ∈ T a S m \ { 0 } . F or any l -rectifia ble set X ⊆ S m such that X ∩ ( − X ) = ∅ , wher e l ≤ m , we hav e h l ( ι n ( X )) = s l n h l ( X ) . (32) Lemma 6. L et u ∈ H m +1 n and X ⊆ S m b e c omp act, symmetric, and ( r + 1 ) - r e ctifiable, wher e r ≤ m − 1 . Then Z S δ ( n ) h r ( N u ∩ X ) dσ δ ( n ) ( u ) = s n ω r ω r +1 h r +1 ( X ) . Pr o of. Since bo th sides are additive on X , we may assume X ∩ ( − X ) = ∅ . W e apply Theorem 4 to the sphere S δ ( n ) and its subs ets A = S δ ( n ) − 1 , B = ι n ( X ). In the notation of this theor em, d = δ ( n ), k = d − 1, l = r + 1; K ω k = ω r ω l . Replacing integration ov er S d by averaging ov er O( d + 1) and using (32 ), we g et Z S d h r ( N u ∩ X ) dσ d ( u ) = 1 s r n Z S d h r ( ι ( N u ∩ X )) dσ d ( u ) = 1 s r n Z S d h r ( u ⊥ ∩ ι ( X )) dσ d ( u ) = 1 s r n Z O( d +1) h r ( g S k ∩ ι ( X )) dµ d ( g ) = 1 s r n K h k  S k  h r +1 ( ι ( X )) = ω r s r n ω r +1 h r +1 ( ι ( X )) = s n ω r ω r +1 h r +1 ( X ) . 16 Corollary 4. The m e an value of h m − 1 ( N u ) over u ∈ H m +1 n is e qual to s n ω m − 1 . Pr o of. Set X = S m , r = m − 1. Corollary 5. L et M n , m , and k b e as in (28) . Then M n = ω m − k k Y j =1 s n j . (33) Pr o of. Set X = N u 1 ∩ · · · ∩ N u k − 1 . By Lemma 6, M n = s n k ω m − k ω m − k +1 M n ′ , where n ′ = ( n 1 , . . . , n k − 1 ). Applying this pr o cedure rep eatedly and using Cor o l- lary 4 in the final step, w e g et (33). It remains to find s n . Set d = dim O( m + 1) . Since the stable subgr oup O( m ) of the po int a = (0 , . . . , 0 , 1) acts transitively on spher es in T a S m , the inv aria n t Riemannian metric in S m can b e lifted up to a bi-in v ariant metric on O( m + 1) in s uch a w ay that the canonica l pro jection O( m + 1) → S m is a metr ic s ubmersion. Let ξ 1 , . . . , ξ m , . . . , ξ d be a n or thonormal linear bas e in the Lie algebr a so( m + 1 ). Realizing so( m + 1) by the left inv ariant vector fields on O( m + 1), we get the inv ariant Laplace – Beltrami op era tor on O( m + 1): ˜ ∆ = ξ 2 1 + · · · + ξ 2 d . The sum is indep endent of the choice of the base s inc e it is left inv ar iant and this prop er ty holds a t the identit y element e . Thus, we may a ssume that ξ m +1 , . . . , ξ d ∈ so( m ) . (34) F or f ∈ C 2 ( S m ), s et ˜ f ( g ) = f ( g a ). Then h ∆ f , φ a i = ˜ ∆ ˜ f ( e ). Since ι is equiv ar i- ant, we hav e d a ι ( ξ a ) = 1 | φ a | ξ φ a (35) for a ll ξ ∈ so( m + 1). It follows fro m (34) that ξ 1 a, . . . , ξ m a is a base for T a S m and ξ 1 φ a , . . . ξ m φ a is a base for T φ a ι ( S m ). Mor eov er , | ξ k a | = 1 , k = 1 , . . . , m, ξ k a = 0 , k = m + 1 , . . . , d, 17 where the first eq ua lit y holds since the pr o jection O( m + 1) → S m is a metric submersion. Due to these equa lities, (30), (31), and (3 5), we ge t ms 2 n = s 2 n d X k =1 | ξ k a | 2 = d X k =1 | d a ι ( ξ k a ) | 2 = 1 | φ a | 2 d X k =1 | ξ k φ a | 2 = − 1 | φ a | 2 d X k =1  ξ 2 k φ a , φ a  = − 1 | φ a | 2 h ∆ φ a , φ a i = λ n . Pr o of of The or em 6. Due to the ca lculation ab ov e, s n = r λ n m . Thu s, Corolla ry 5 implies (29). In the case n 1 = · · · = n k = n , there is ano ther natural ex pla nation of the equalities (29), (33 ): M n = ω m − k  λ n m  k 2 = ω m − k s k n . The mean v alue can b e defined as the average ov er the ac tio n of the group O( m + 1) on the set of subspac e s of co dimension k in H m +1 n , which can b e realized as N u 1 ∩ · · · ∩ N u k = u ⊥ 1 ∩ · · · ∩ u ⊥ k : M n = R O( m +1) h m − k ( ι − 1 n ( g S δ ( n ) − k ∩ ι n ( S m ))) dµ m ( g ) = s k − m n R O( m +1) h m − k ( g S δ ( n ) − k ∩ ι n ( S m )) dµ m ( g ) = s k − m n ω m − k ω m h m ( ι ( S m )) = ω m − k s k n . The metho d of calculation of the mean Haus do rff measur e e a sily can b e extended to families o f inv aria nt (may b e, r educible) finite dimensio nal function spaces on a ho mogeneous space whose isotr opy gro up acts tra ns itively on spheres in the tangent spa ce. Ac knowledgemen ts. I a m gr a teful to D. Jakobson for useful references and comments and to L. Polterovich for his question/conjecture on “ the Bezo ut theorem in the mean”. 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