On the restriction of various Laplace operators on submanifolds

TL;DR

This paper investigates the relationship between diffusion operators in ambient Euclidean space and on embedded surfaces, providing a precise characterization for general surfaces in three-dimensional space.

Contribution

It offers a detailed analysis of how Laplace operators restrict to submanifolds, specifically addressing the case of surfaces in three-dimensional space.

Findings

Characterization of Laplace operator restrictions on surfaces

Insights into diffusion operators in embedded manifolds

Framework applicable to Navier-Stokes equations on manifolds

Abstract

When considering Navier-Stokes equations on Riemannian manifolds one frequently encounters situations where the manifold is embedded in the ambient Euclidean space. In this context it is interesting to investigate what is the precise relationship of the diffusion operator in the ambient space to the diffusion operator on the manifold. The present paper gives a precise characterization of this situation for general surfaces in three dimensional space.