Lectures on Immersions with Controlled Curvatures

TL;DR

This work presents a unified program for constructing and constraining immersions with small normal curvature $curv^\perp$ in Euclidean balls, tubes, and related domains. It develops a toolkit of geometric and analytic methods—focal radius, encircling, bowl inequalities, and optimal-control perspectives—to bound geometric size, understand rigidity, and connect to scalar curvature via the Gauss formula and Petrunin-type estimates. Key contributions include extremal examples (spheres and Veronese varieties), Petrunin’s $\sqrt{3}$-inequalities for tori, and width-type bounds for Riemannian bands, revealing deep links between immersion geometry and global curvature/topology. The results yield both explicit constructions of low-curvature immersions (including Clifford tori and products of spheres) and robust obstructions grounded in scalar curvature, while outlining open problems about sharp constants, rigidity, and the topology of immersion spaces under curvature constraints.

Abstract

Construction of immersions with "small" curvatures between Riemannian manifolds and indicating obstructions to such immersions