Discrete and Continuous Green Energy on Compact Manifolds

The Projective Ensemble and Distribution of Points in Odd-Dimensional Spheres

“…For example, one can study the natural analogues of Riesz's energy as in Theorem 3.3 below. A very natural measure is given by the value of Green's energy of [3]: let G : ( d+1 ) × ( d+1 ) → [0, ∞] be the Green function of ( d+1 ), that is, G(x, •) is zero-mean for all x, G is symmetric and ∆ y G(x, y) = δ x ( y)− vol( ( d+1 )) −1 , with δ x the Dirac's delta function, in the distributional sense. The Green energy of a collection of r points ω r = (x 1 , .…”

“…Minimizers of Green's energy are assymptotically well-distributed (see [3,Main Theorem]). Our second main result will follow from the computation of the expected value of Green's energy for the projective ensemble.…”

“…Proof of Theorem 1.3. From[3], the Green function of( d+1 ) is given G(x, y) = φ(r) where r = d ( d+1 ) (x, y) and φ (r) = − 1 Vol( ( d+1 )) π/sin 2d−1 t cos t d t sin 2d−1 r cos r = − 1 2d Vol( ( d+1 )) 1 − sin 2d r sin 2d−1 r cos r .Integrating the formula above (see for example [10, 2.517-1] we have:φ(r) = 1 2d Vol( ( d+1 )) d − k) (sin r) 2d−2k − log (sin r) + C.…”