Almost non-negative scalar curvature on Riemannian manifolds conformal to tori

Abstract

In this article we reduce the geometric stability conjecture for the scalar torus rigidity theorem to the conformal case via the Yamabe problem. Then we are able to prove the case where a sequence of Riemannian manifolds is conformal to a uniformly controlled sequence of flat tori and satisfies the geometric stability conjecture. We are also able to handle the case where a sequence of Riemannian manifolds is conformal to a sequence of constant negative scalar curvature Riemannian manifolds which converge to a flat torus in $C^1$. The full conjecture from the conformal perspective is also discussed as a possible approach to resolving the conjecture.

22 pages, comments welcome. v2: Adjustments made to the introduction and background. To appear in the Journal of Geometric Analysis

References in corpus (3)

• Stability of the positive mass theorem for graphical hypersurfaces of Euclidean space
• $d_p$ convergence and $\epsilon$-regularity theorems for entropy and scalar curvature lower bounds
• Volume Above Distance Below

Cited by in corpus (2)

• $d_p$ convergence and $\epsilon$-regularity theorems for entropy and scalar curvature lower bounds
• Sobolev Inequalities and Convergence For Riemannian Metrics and Distance Functions