Analyzing the Geometry of Immersions of Co-Dimension One via Shape Operator Dynamics

TL;DR

This paper investigates the extrinsic geometry of co-dimension one immersions using a novel fourth-order flow of the shape operator, inspired by bi-harmonic maps and Chen's conjecture.

Contribution

It introduces a new tensorial gradient flow that reduces curvature variation energy, advancing understanding of shape operator dynamics in geometric analysis.

Findings

Defined a moduli flow decreasing curvature variation energy.

Connected the flow to bi-harmonic map theory and Chen's conjecture.

Provided insights into the geometry of isometric immersions.

Abstract

We study the extrinsic geometry of isometric immersions into Riemannian manifolds of co-dimension one via a fourth-order geometric evolution of the shape operator. Motivated by bi-harmonic map theory and generalized Chen's conjecture, we introduce a moduli flow, a tensorial gradient flow that decreases a natural energy measuring curvature variation.