International Congress of Mathematicians

740 S. Zelditch[Pi,Pj] = 0 and whose symbols define a moment map V := (pi,... ,p n ) satisfyingdpi A dp2 A • • • A dp n ^Oona dense open set QcT*M-0. Since {Pi,Pj} = 0, thefunctions pi,---,p n generate a homogeneous Hamiltonian R"-action whose orbitsfoliate T*M — 0. We refer to this foliation as the Liouville foliation.We consider the IP norms of the L 2 -normalized joint eigenfunctions Pj*fix =Xjipx- The spectrum of A often has bounded multiplicity, so the behaviour of jointeigenfunctions has implications for all eigenfunctions.Theorem 5 [22, 23] Suppose that the Laplacian A g of(M,g) is quantum completelyintegrable and that the joint eigenfunctions have uniformly bounded L°° norms.Then (AT, g) is a flat torus.This is a kind of quantum analogue of the 'Hopf conjecture' (proved by Burago-Ivanov) that metrics on tori without conjugate points are flat. In [23], a quantitativeimprovement is given under a further non-degeneracy assumption. Unless (M,g) isa flat torus, the Liouville foliation must possess a singular leaf of dimension < n.Yet £ denote the minimum dimension of the leaves. We then construct a sequenceof eigenfunctions satisfying:n-l , (n-»(p-2) ,\\Vk\\L~>C(e)X k * ', \\v>k\\L P > C(e)X k 4 * , (2 < p)for any e > 0. It is easy to construct examples were £ = n — l, but it seems plausiblethat in 'many' cases £ = 1. To investigate this, one would study the boundary facesof the image V(T*M — 0) of T * M — 0 under a homogeneous moment map. For arelated study in the case of torus actions, see Lerman-Shirokova [12].3.3. Quantum ergodicityQuantum ergodicity is concerned with the sums (A £ \P 0 (Af)):S P (X)= J2 \(A