9 problems

Square-root curvature conjecture. In fact,

Self-repulsiveness conjecture. The regularized $\alpha$-energy $E_\alpha$ for smooth closed surfaces is self-repulsive with respect to $C^0$-topology if and only if $\alpha\leq -4$…

The paper considers the energy … for embedded, arclength-parametrized curves $u:I\to\mathbb{R}^3$, where $\varkappa$ and $\varrho$ are positive and $\operatorname{TP}$ is a tangent…

Let $c$ be a sufficiently regular immersed closed curve, let $\tilde H^\alpha$ denote the first-variation expression for O'Hara's energy after the relevant decomposition, and let…

Let $\mathscr{M}_p$, $\mathscr{I}_p$, $\mathscr{E}_p$, and $\mathscr{E}_p^\textnormal{sym}$ denote the knot energies introduced in the paper. An energy is basic in the sense used b…

Let $\mathscr{E}_{\text{M\"o\"b}}$ denote the Möbius energy of a knot, and call an energy minimizable if every knot class contains a representative minimizing it within that class.…

Let $M$ be a Riemannian manifold, let $d_M(x,y)$ be the Riemannian distance, and for a knot $K\subset M$ define … For an isotopy class, an $E^{(\alpha)}_M$--minimizer is a knot in…

Let $(p,q)$ be integers, and let a $(p,q)$ torus knot mean the corresponding isotopy class of knots. A critical point of $E^{(2)}_{\circ}$ is unstable if the energy has a descendin…

Let $[K_1]$ and $[K_2]$ be isotopy classes of knots, and let $[K_1\sharp K_2]$ be their composite isotopy class. For an isotopy class $[K]$, write $E^{(2)}_{\circ}([K])$ for the in…