On the rigidity of manifolds with respect to Gagliardo-Nirenberg inequalities

In this paper, we investigate local rigidity properties related to Gagliardo-Nirenberg constants and unweighted Yamabe-type constants. Let $V$ be an open bounded subset of an $n$-dimensional Riemannian manifold $(M,g)$ whose Gagliardo-Nirenberg constant satisfies $${\mathbb{G}}_{\alpha }^{\pm }(V,g)\ge {\mathbb{G}}_{\alpha }^{\pm }({\mathbb{R}}^n,g_{{\mathbb{R}}^n}),$$ where $({\mathbb{R}}^n,g_{{\mathbb{R}}^n})$ denotes the $n$-dimensional Euclidean space with its standard metric. We show that for $\alpha \in (0,1)\cup (1,\frac{n+6}{n+2})$ when $n\le 6$ or $\alpha \in (0,1)\cup (1,\frac{n}{n-2}]$ when $n\ge 7$, if the first eigenvalue of the Ricci tensor satisfies $${\int }_V{\lambda }_1(\mathrm{Rc})d{\mu }_g\ge 0,$$ then $V$ must be flat. When $\alpha$ belongs to a specific subinterval around $1$ within the above range, ${\mathbb{G}}_{\alpha }^{\pm }(V,g)\ge {\mathbb{G}}_{\alpha }^{\pm }({\mathbb{R}}^n,g_{{\mathbb{R}}^n})$ and the weaker curvature condition of the scalar curvature $${\int }_V\mathrm{Sc}d{\mu }_g\ge 0$$ already imply that $V$ is flat. Moreover, we prove that for $\alpha$ sufficiently close to 1, the condition $${\mathbb{Y}}_{\alpha }^{\pm }(V,g)\ge {\mathbb{G}}_{\alpha }^{\pm }({\mathbb{R}}^n,g_{{\mathbb{R}}^n})$$ on the unweighted Yamabe-type constants guarantees the flatness of $V$.