Submanifolds with nonpositive extrinsic curvature

TL;DR

This paper demonstrates that under certain conditions, complete submanifolds with low codimension and bounded by small cylinders must contain points with large positive extrinsic curvature, unifying previous results.

Contribution

It generalizes and unifies earlier findings on submanifolds with nonpositive extrinsic curvature by establishing new curvature point richness results under specific geometric constraints.

Findings

Submanifolds with low codimension have many points with large positive extrinsic curvature.

Smaller bounding cylinders lead to larger positive extrinsic curvatures at certain points.

The results extend previous work on submanifolds with nonpositive extrinsic curvature.

Abstract

We prove that complete submanifolds, on which the Omori-Yau weak maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower the codimension is, the richer such points are. The smaller the radius is, the larger such curvatures are. This work unifies and generalizes several previous results on submanifolds with nonpositive extrinsic curvature.