The logarithmic Sobolev inequality for a submanifold in manifolds with nonnegative sectional curvature (Q6564187)

Let \((M, g)\) be a complete non-compact Riemannian manifold of dimension \(k\) with non-negative Ricci curvature. The asymptotic volume ratio of \(M\) is\N\[\N\theta := \lim_{r\to +\infty} \frac{|B_r (p)|}{\omega_{k}r^{k}}\N\]\Nfor some (hence any) fixed point \(p \in M\), where \(B_r (p)\) denotes the geodesic ball in \(M\) , \(|B_r (p)|\) denotes its volume and \(\omega_k\) denotes the volume of the unit ball in \(\mathbb{R}^{k}\). By the Bishop-Gromov volume comparison theorem, the limit exists and \(0 \leq \theta \leq 1\). \N\NThe authors prove a sharp logarithmic Sobolev inequality which holds for compact submanifolds without boundary in \((M,g)\) with Euclidean volume growth (i.e, \(\theta>0)\) of arbitrary dimension and codimension. From this technical result, two important consequences follow:\N\begin{itemize}\N\item If \((M,g)\) is a complete noncompact Riemannian manifold of dimension \(k\geq 2\) with nonnegative sectional curvature and Euclidean volume growth, then there does not exist any closed minimal submanifold in \((M,g)\);\N\item If \((M,g)\) is a complete noncompact Riemannian manifold of dimension \(k\geq 2\) with nonnegative sectional curvature and there exists some closed minimal submanifold of some co-dimension in \((M,g)\), then \(M\) does not have maximum volume growth.\N\end{itemize}