The Log-Sobolev inequality for a submanifold in manifolds with asymptotic non-negative intermediate Ricci curvature

We prove a sharp Log-Sobolev inequality for submanifolds of a complete non-compact Riemannian manifold with asymptotic non-negative intermediate Ricci curvature and Euclidean volume growth. Our work extends a result of Dong-Lin-Lu which already generalizes Yi-Zheng: arXiv:2104.05045 and Brendle: arXiv:1908.10360v3.