Shape Gradient: a Driving Force Induced by Space Curvatures

摘要

On the basis of the second gradient operator defined on biomembranes, the concept of the shape gradient is introduced. The universal existence of the shape gradient is confirmed: As a driving force induced by the curvatures of spaces, the shape gradient occurs frequently in various scientific problems: In the mechanics of biomembranes and liquid crystals with smectic-A phases, the shape gradient is the force maintaining equilibriums. In the mechanics of elastic thin shells, the shape gradient is either the force keeping equilibriums and deformation compatibilities or the force preserving stability or stimulating instability. In diffusion dynamics on curved surfaces, the shape gradient is the force driving surface diffusions and morphology evolutions. In short, the shape gradient is both the driving force induced by curved spaces and the force driving the redistributions of matters on curved spaces.