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

Abstract

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 \[ \mathbb{G}_{\alpha}^{\pm}(V,g) \geq \mathbb{G}_{\alpha}^{\pm}(\mathbb{R}^n,g_{\mathbb{R}^n}), \] where $(\mathbb{R}^n,g_{\mathbb{R}^n})$ denotes the $n$-dimensional Euclidean space with its standard metric. We show that for $\alpha \in (0,1) \cup \left(1,\frac{n+6}{n+2}\right)$ when $n \leq 6$ or $\alpha \in (0,1) \cup \left(1,\frac{n}{n-2}\right]$ when $n \geq 7$, if the first eigenvalue of the Ricci tensor satisfies \[ \int_V \lambda_1(\operatorname{Rc}) \, d\mu_g \geq 0, \] then $V$ must be flat. When $\alpha$ belongs to a specific subinterval around $1$ within the above range, $\mathbb{G}_{\alpha}^{\pm}(V,g) \geq \mathbb{G}_{\alpha}^{\pm}(\mathbb{R}^n,g_{\mathbb{R}^n})$ and the weaker curvature condition of the scalar curvature \[ \int_{V} \operatorname{Sc} \, d\mu_g \geq 0 \] already imply that $V$ is flat. Moreover, we prove that for $\alpha$ sufficiently close to 1, the condition \[ \mathbb{Y}_{\alpha}^{\pm}(V,g) \geq \mathbb{G}_{\alpha}^{\pm}(\mathbb{R}^n,g_{\mathbb{R}^n}) \] on the unweighted Yamabe-type constants guarantees the flatness of $V$.