Abstract:
The classical minimisation problems in differential geometry lead to variational equations of second order. Minimising curvature integrals leads instead to fourth-order equations whose analysis is entirely different in tone from that of their second-order counterparts.
In this work, we focus on the gradient flows for curvature integrals defined on curves and surfaces. The natural questions to ask are: can the gradient flow equations be solved for any period of time? If so, for how long? And if there are solutions which exist for all time, what is their asymptotic behaviour? We provide clean answers on all three counts.
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