Normal curvature bounds for immersions into Riemannian domains

Abstract

We study Gromov's problem on the minimal normal curvature of immersions. Our main result is a lower bound for the average normal curvature of a closed submanifold immersed in a Riemannian domain. The bound is expressed in terms of an invariant measuring the optimal $n$-trace convexity of the domain under a unit-gradient normalization. As applications, we recover and extend Petrunin's lower bound for closed submanifolds immersed in Euclidean balls to geodesic balls in Cartan-Hadamard manifolds and, more generally, to Riemannian domains satisfying suitable convexity conditions. In the Cartan-Hadamard setting, under a natural assumption on the average scalar curvature, we show that equality forces the submanifold to lie minimally in the boundary sphere and that the radial sectional curvature vanishes along it. We also obtain sharper estimates for immersions into hyperbolic balls and Euclidean tubes.