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

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.