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- W2953445534 abstract "Let $(M,g)$ be an $n-$dimensional complete open Riemannian manifold with nonnegative Ricci curvature verifying $rhoDelta_g rho geq n- 5geq 0$, where $Delta_g$ is the Laplace-Beltrami operator on $(M,g)$ and $rho$ is the distance function from a given point. If $(M,g)$ supports a second-order Sobolev inequality with a constant $C>0$ close to the optimal constant $K_0$ in the second-order Sobolev inequality in $mathbb R^n$, we show that a global volume non-collapsing property holds on $(M,g)$. The latter property together with a Perelman-type construction established by Munn (J. Geom. Anal., 2010) provide several rigidity results in terms of the higher-order homotopy groups of $(M,g)$. Furthermore, it turns out that $(M,g)$ supports the second-order Sobolev inequality with the constant $C=K_0$ if and only if $(M,g)$ is isometric to the Euclidean space $mathbb R^n$." @default.
- W2953445534 created "2019-07-12" @default.
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- W2953445534 date "2016-08-14" @default.
- W2953445534 modified "2023-09-27" @default.
- W2953445534 title "Second-order Sobolev inequalities on a class of Riemannian manifolds with nonnegative Ricci curvature" @default.
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