We consider the problem of minimizing the energy of $N$ points repelling each other on curves in $\mathbb{R}^d$ with the potential $|x-y|^{-s}$, $s\geq 1$, where $|\, \cdot\, |$ is the Euclidean norm. For a sufficiently smooth, simple, closed, regular curve, we find the next order term in the asymptotics of the minimal $s$-energy. On our way, we also prove that at least for $s\geq 2$, the minimal pairwise distance in optimal configurations asymptotically equals $L/N$, $N\to\infty$, where $L$ is the length of the curve.

We define sets with finitely ramified cell structure, which are generalizations of post-crit8cally finite self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local self-similarity, and allow countably many cells connected at each junction point. In particular, we consider post-critically infinite fractals. We prove that if Kigami's resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates.

We consider the $s$-energy $$ E(\ZZ_n;s)=\sum_{i \neq j} K(\|z_{i,n}-z_{j,n}\|;s) $$ for point sets $\ZZ_n=\{ z_{k,n}:k=0,\dots,n\}$ on certain compact sets $\Ga$ in $\R^d$ having finite one-dimensional Hausdorff measure, where $$ K(t;s)= \begin{cases} t^{-s} ,& \mbox{if } s>0, \\ -\ln t, & \mbox{if } s=0, \end{cases} $$ is the Riesz kernel. Asymptotics for the minimum $s$-energy and the distribution of minimizing sequences of points is studied. In particular, we prove that, for $s\geq 1$, the minimizing nodes for a rectifiable Jordan curve $\Ga$ distribute asymptotically uniformly with respect to arclength as $n\to\infty$.