Equidistribution of expanding translates of smooth curves in homogeneous spaces under the action of a product of SO(n,1)'s

TL;DR

This paper proves that expanding translates of smooth curves in homogeneous spaces formed by products of SO(n,1) groups become uniformly distributed in the orbit closure, under certain generic conditions, using advanced ergodic and geometric methods.

Contribution

It establishes new equidistribution results for translates of smooth curves in complex homogeneous spaces involving product groups, extending prior understanding of orbit distributions.

Findings

Expanding translates of curves become equidistributed in the orbit closure.

Almost every point on the curve avoids algebraic obstructions for equidistribution.

The proof combines Ratner's measure classification, geometric invariant theory, and linearization techniques.

Abstract

We study the limiting distributions of expanding translates of a compact segment of a smooth curve under a diagonal subgroup of $G=\mathrm{SO}(n_1,1)\times\cdots\times\mathrm{SO}(n_k,1)$, where $G$ acts on a finite volume homogeneous space $L/\Gamma$ as a subgroup. We show that the expanding translates of the curve become equidistributed in the orbit closure of $G$, provided that Lebesgue almost every point on the curve avoids a certain countable collection of algebraic obstructions. The proof involves Ratner's measure classification theorem, Kempf's geometric invariant theory, and the linearization technique.