Weyl Law Improvement for Products of Spheres

The classical Weyl Law says that if NM(?) denotes the number of eigenvalues Laplace operator on a d-dimensional compact manifold M without boundary are less than or equal to ?, then $${N_M}(\lambda ) = c{\lambda ^d} + O({\lambda ^{d - 1}}).$$ This paper explores prospects improvements remainders products manifolds. In particular we obtain polynomial improvement remainder for spheres, demonstrate how Duistermaat and Giullemin’s result implies little-o Riemannian manifolds boundary, conjecture hold these more general products.