Some remarks on spherical harmonics

Let M be a compact connected homogeneous Riemannian manifold, G be a compact Lie group acting on M transitively by isometries, and E be a Ginvariant subspace of the (real) eigenspace for some non-zero eigenvalue of the Laplace-Beltrami operator. We show that each function in E can be realized as the determinant of a matrix, whose entries are values of the reproducing kernel for E. There is a similar well-known construction for the orthogonal polynomials. However, the method does not work for an arbitrary finite dimensional G-invariant subspace of C(M ) (see Remark 2). There is a natural unique up to scaling factors realization of this type for spherical harmonics on S 2 . It can be obtained by complexification and restriction to the null-cone x 2 + y 2 + z 2 = 0 in C 3 . There is a two-sheeted equivariant covering of this cone by C 2 , which identifies the space H n of harmonic homogeneous complex-valued polynomials of degree n on R 3 with the space P 2 2n of homogeneous holomorphic polynomials on C 2 of degree 2n. 1 The set of all zeroes of a real spherical harmonic u is called a nodal set. We say that u and its nodal set N u are regular if zero is not a critical value of u. Then each component of N u is a Jordan contour. According to [11], a pair of the nodal sets N u , N v , where u, v ∈ H R n and n > 0, have a non-void intersection; moreover, if u is regular, then each component of N u contains at least two points of N v . The set N u ∩ N v may be infinite but the family of such pairs (u, v) is closed and nowhere dense in 2 . The estimate follows from the Bezout theorem and is precise. This gives an upper bound for the number of critical points of a generic spherical harmonic, which probably is not sharp. The configuration of critical points is always degenerate in some sense (see Remark 5). The problem of finding lower bounds seems to be more difficult. According to partial results and computer experiments, 2n may be the sharp lower bound.

The investigation of metric and topological properties of the nodal sets has a long and rich history; we only give a few remarks on the subject of this paper. Let ∆ be the Laplace-Beltrami operator and λ be an eigenvalue of -∆.

In 1978, Brüning ([5]) found the lower bound c √ λ for the length of a nodal set on a Riemann surface. Yau conjectured ( [22,Problem 74]) that the Hausdorff measure of the nodal set of a λ-eigenfunction on a compact Riemannian manifold admits upper and lower bounds of the type c √ λ. This conjecture was proved by Donnelly and Fefferman for real analytic manifolds in [8]. In ( [18]), Savo proved that 1 11 Area(M ) √ λ is the lower bound for the length of a nodal set in a surface M for all sufficiently large λ in any surface and for all λ if the curvature is nonnegative. The upper and lower estimates of the inner radius were found by Mangoubi ([13], [14]); in the case of surfaces, they are of order λ -1 2 ( [13]). One can find the 1-dimensional Hausdorff measure of a set in S 2 integrating over SO(3) the counting function for the number of its common points with translates of a suitable subset of S 2 (see Theorem 4). Using estimates of the number of common zeroes, we give upper and lower bounds for the length of a nodal set and for the inner radius of a nodal domain in S 2 . The upper bounds are precise.

Let H m+1 n be the space of all real spherical harmonics of degree n on the unit sphere S m in R m+1 . Corresponding to a point of S m the evaluation functional at it on H m+1 n , we get an equivariant immersion of S m to the unit sphere in H m+1 n , which is locally a metric homothety with the coefficient λn m , where

. This makes it possible to calculate the mean Hausdorff measure of the intersection of k harmonics of degrees n 1 , . . . , n k : it is equal to c λ n1 . . . λ n k , where c depends only on m and k and k ≤ m (Theorem 6). In particular, if k = m, then we get the mean number of common zeroes of m harmonics: it is equal to 2m -m 2 λ n1 . . . λ nm ; if m = 2, then λ n1 λ n2 . In article [8], Donelly and Fefferman wrote: “A main theme of this paper is that a solution of ∆F = -λF , on a real analytic manifold, behaves like a polynomial of degree c √ λ “. Following this idea, L. Polterovich conjectured that the mean number of common zeroes is subject to the Bezout theorem, i.e., that it is as above. Thus, the result in the case k = m confirms this conjecture up to multiplication by a constant, and may be treated as “the Bezout theorem in the mean” for the spherical harmonics. For k = 1, the mean Hausdorff measure, by different but similar methods, was found by Berard in [4] and Neuheisel in [16]. The case of a flat torus was investigated by Rudnick and Wigman ([17]).

In this section, M is a compact connected oriented homogeneous Riemannian manifold of a compact Lie group G acting by isometries on M , ∆ is

…(Full text truncated)…