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A general nonuniqueness result for Yamabe-type problems for conformally variational Riemannian invariants

João Henrique Andrade, Jeffrey S. Case, Paolo Piccione, Juncheng Wei

TL;DR

This work develops an axiomatic framework for conformally variational Riemannian invariants (CVIs) to study Yamabe-type problems and nonuniqueness phenomena. By formalizing geometric Aubin sets and associated Yamabe constants, it provides sufficient conditions under which compact manifolds admit many nonhomothetic conformal rescalings with constant $I$-curvature on finite covers, and infinitely many periodic conformal representatives on universal covers. The approach unifies and extends known nonuniqueness results for $Q$-curvatures of orders $2$, $4$, and $6$, and yields new nonuniqueness results for higher-order $Q$-curvatures and renormalized volume coefficients, including new geometric Aubin sets for higher-order cases. The results are applicable to a broad class of CVIs, with key contributions including a general tower-covering argument, explicit geometric Aubin-set constructions (e.g., via $\Gamma_k^+$ ellipticity and SEP structures), and corollaries covering both singular and complete Yamabe-type problems. Overall, the paper broadens the landscape of nonuniqueness phenomena in conformal geometry by providing a versatile, axiomatic toolkit grounded in the theory of CVIs.

Abstract

Given a conformally variational scalar Riemannian invariant $I$, we identify a sufficient condition for a compact Riemannian manifold to admit finite regular coverings with many nonhomothetic conformal rescalings with $I$ constant. We also identify a sufficient condition for the universal cover to admit infinitely many geometrically distinct periodic conformal rescalings with $I$ constant. Using these conditions, we improve known nonuniqueness results for the $Q$-curvatures of orders two, four, and six, and establish nonuniqueness results for higher-order $Q$-curvatures and renormalized volume coefficients.

A general nonuniqueness result for Yamabe-type problems for conformally variational Riemannian invariants

TL;DR

This work develops an axiomatic framework for conformally variational Riemannian invariants (CVIs) to study Yamabe-type problems and nonuniqueness phenomena. By formalizing geometric Aubin sets and associated Yamabe constants, it provides sufficient conditions under which compact manifolds admit many nonhomothetic conformal rescalings with constant -curvature on finite covers, and infinitely many periodic conformal representatives on universal covers. The approach unifies and extends known nonuniqueness results for -curvatures of orders , , and , and yields new nonuniqueness results for higher-order -curvatures and renormalized volume coefficients, including new geometric Aubin sets for higher-order cases. The results are applicable to a broad class of CVIs, with key contributions including a general tower-covering argument, explicit geometric Aubin-set constructions (e.g., via ellipticity and SEP structures), and corollaries covering both singular and complete Yamabe-type problems. Overall, the paper broadens the landscape of nonuniqueness phenomena in conformal geometry by providing a versatile, axiomatic toolkit grounded in the theory of CVIs.

Abstract

Given a conformally variational scalar Riemannian invariant , we identify a sufficient condition for a compact Riemannian manifold to admit finite regular coverings with many nonhomothetic conformal rescalings with constant. We also identify a sufficient condition for the universal cover to admit infinitely many geometrically distinct periodic conformal rescalings with constant. Using these conditions, we improve known nonuniqueness results for the -curvatures of orders two, four, and six, and establish nonuniqueness results for higher-order -curvatures and renormalized volume coefficients.
Paper Structure (9 sections, 13 theorems, 57 equations)

This paper contains 9 sections, 13 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Conformally variational Riemannian invariants
  3. Examples of geometric Aubin sets
  4. Second-order Q-curvature
  5. Renormalized volume coefficients
  6. Fourth-order Q-curvature
  7. Higher-order Q-curvatures
  8. Proofs of our main results
  9. The Aubin Lemma for the Q-curvature

Key Result

Theorem 1.1

Fix $k,m \in \mathbb{N}$. There is an $N = N(k,m) \in \mathbb{N}$ such that if then for each $\ell \in \mathbb{N}$, there is a finite regular covering $\pi \colon \widetilde{X}^n \to M_1^m \times M_2^{n-m}$ for which there exist pairwise nonhomothetic metrics $\{ \sigma_j \}_{j=1}^\ell \subset [\pi^\ast (g_1 \oplus g_2)]$ of constant $Q_{2k}$-curvature. Moreover, the conformal

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Remark 3.3
  • ...and 17 more