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

2025/07/16 by Kolaei, Mohammad Javad Habibi Vosta
#35K55 #53C42 #58J35 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.12068

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.

Citations

• Well-posedness of half-harmonic map heat flows for rough initial data
• A Monotonicity Formula for the Extrinsic Biharmonic Map Heat Flow
• Diffusive stability and self-similar decay for the harmonic map heat flow
• Rigidity and stability estimates for minimal submanifolds in the\n hyperbolic space
• A Note on Quasilinear Parabolic Equations on Manifolds
• The entropy formula for the Ricci flow and its geometric applications
• Minimal Varieties in Riemannian Manifolds

Related

• Extrinsic Characterisations of Immersions
• Isometric Immersions with Controlled Curvatures
• Lectures on Immersions with Controlled Curvatures
• Geometric flows and (some of) their physical applications
• The preservability of the curvature-adaptedness along the mean curvature flow