Self-repulsiveness of energies for closed submanifolds

We show that the regularized Riesz $\text{a}$-energy for closed submanifolds $M$ in ${\text{RR}}^n$ blows up as $M$ degenerates to have double points if $\text{a}\le -2\dim M$. This gives theoretical foundation of numerical experiments to evolve surfaces to decrease the energy which have been carried out since 90’s.