Some Uniformization Problems for a Fourth order Conformal Curvature

2023/05/14 by Sanghoon Lee, Lee, Sanghoon
Mathematics · #53C44 (Primary) #58J35 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2305.08027

openalex publication_date 2023/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we establish the existence of conformal deformations that uniformize fourth order curvature on 4-dimensional Riemannian manifolds with positive conformal invariants. Specifically, we prove that any closed, compact Riemannian manifold with positive Yamabe invariant and total Q-curvature can be conformally deformed into a metric with positive scalar curvature and constant Q-curvature. For a Riemannian manifold with umbilic boundary, positive first Yamabe invariant and total (Q, T)-curvature, it is possible to deform it into two types of Riemannian manifolds with totally geodesic boundary and positive scalar curvature. The first type satisfies Q≡ constant, T ≡ 0 while the second type satisfies Q≡ 0, T ≡ constant.

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