Riesz and Green energy on projective spaces

Abstract

In this paper we study Riesz, Green and logarithmic energy on two-point homogeneous spaces. More precisely we consider the real, the complex, the quaternionic and the Cayley projective spaces. For each of these spaces we provide upper estimates for the mentioned energies using determinantal point processes. Moreover, we determine lower bounds for these energies of the same order of magnitude.

A lower bound for the logarithmic energy on $\mathbb{S}^2$ and for the Green energy on $\mathbb{S}^n$

“…Our main new result is the following lower bound for the Green energy on S n . Note that similar results are known for the Green energy in projective spaces [1]. Theorem 4.2 (Main result).…”

“…where G(M; •, •) is the Green function in M associated to the Laplace-Beltrami operator, is a more natural choice since it does not depend on extrinsic quantities, and is attracting more attention in the last few years, see [13], [20], [1]. It turns out that the Green function in S 2 is…”

“…The Green function of a general Riemannian manifold can be very hard to compute. In [4], a method is given to compute it in compact harmonic manifolds, that is to say, in spheres and projective spaces (see [1] for an alternative, equivalent method). We have used it to get the explicit expression of G(S n ; p, q) in Proposition 3.1.…”

A Lower Bound for the Logarithmic Energy on $$\mathbb {S}^2$$ and for the Green Energy on $$\mathbb {S}^n$$

“…where G(M; •, •) is the Green function in M associated to the Laplace-Beltrami operator, is a more natural choice since it does not depend on extrinsic quantities, and is attracting more attention in the last few years, see [1,14,23]. It turns out that the Green function in S 2 is…”

On Gegenbauer Point Processes on the Unit Interval

“…The second property implies that the random points exhibit some repulsion, and has been used to give upper bounds on the minimum value of the energy E log (K, n) (and other energies) for different sets: in [2] for the 2-sphere S 2 , in [7] for the d-sphere S d , in [5] (see also [8]) for the complex projective space, in [3] for 2-point homogeneous spaces, in [18] for the flat torus and in [6] for the rotation group SO (3).…”