Nonhomothetic complete periodic metrics with constant scalar curvature

TL;DR

The paper proves that for $k<\frac{n-2}{2}$ there exist infinitely many pairwise nonhomothetic periodic conformally round metrics on $S^n\setminus S^k$ with constant scalar curvature, obtained by pulling back Yamabe metrics from products $S^{n-k-1}\times \Sigma^{k+1}$ under finite coverings. The approach blends residual finiteness and profinite techniques to build towers of finite regular coverings, a variational construction of Yamabe minimizers on each cover, and a Ferrand--Obata rigidity argument to prevent homotheties when lifted to the universal cover. A key novelty is the comparison between analytic and geometric moduli: although many Yamabe solutions exist on compact quotients, only infinitely many survive as nonhomothetic geometric representatives after lifting. This work strengthens the understanding of nonuniqueness phenomena in conformally invariant problems and highlights how group-theoretic finiteness properties interact with geometric analysis to produce rich moduli on noncompact singular spaces.

Abstract

We show that there are infinitely many pairwise nonhomothetic, complete, periodic metrics with constant scalar curvature that are conformal to the round metric on $S^n\setminus S^k$, where $k < \frac{n-2}{2}$. These metrics are obtained by pulling back Yamabe metrics defined on products of $S^{n-k-1}$ and compact hyperbolic $(k+1)$-manifolds. Our main result proves that these solutions are generically distinct up to homothety. The core of our argument relies on classical rigidity theorems due to Obata and Ferrand, which characterize the round sphere by its conformal group.

Theorems & Definitions (29)

• Theorem A: MR994021
• Theorem B: MR3504948
• Theorem C: MR3803113
• Definition A
• Definition B
• Theorem 1.1
• Proposition A: MR0125546MR0240748MR0448404MR788292
• proof
• Definition 2.1
• Proposition B: MR0303464MR1371767