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

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.

The definition of the biharmonic tension field is not working in this case. Also I have been known by my colleagues that the introduced flow is not well-studied enough to being used. By all above, I have decided to withdrawn this manuscript and reconsider the issue to avoid any further misunderstanding