Year: 2026
Type: preprint
Abstract: Let $V$ be a nonempty relatively compact open subset of a smooth Riemannian manifold $(M^n,g)$ without boundary. We prove that $C_{\mathrm N}(V,g)\ge C_{\mathrm N}(\mathbb R^n)$ and $\int_V \mathrm{Sc}_g dμ_g\ge0$ imply $\mathrm{Rm}_g=0$ on $V$, where $C_{\mathrm N}$ denotes the optimal constant in the Nash inequality and $\mathrm{Sc}_g$ denotes the scalar curvature. This improves an earlier rigidity result of Druet, Hebey, and Vaugon [Int. Math. Res. Not. (1999), no. 14, 735--779].
Source: arXiv (Cornell University)
Authors: Cheng, Liang, Lu, Zhenyu
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G8429268719
OpenAlex ID: W7223656947