Optimal planar immersions of prescribed winding number and Arnold invariants

TL;DR

This work develops a variational framework for optimal planar immersions with prescribed winding number and Arnold invariants by employing desingularized tangent-point energies. It establishes existence of energy-minimizing immersions within invariant-defined compartments, proves a Gamma-convergence from the truncated energy $\delta^{q-2}{\rm TP}_{q,\delta}$ to a renormalized energy $R_q$, and shows that minimizers converge to a $C^1$ limit curve $\Gamma^\eta$ whose self-intersections occur at right angles and which minimizes $R_q$. The limit curve serves as an optimal representative in its topological class, being almost-minimal for the original energies when $\delta$ is small and locally optimal among curves with right-angle intersections. These results provide a rigorous angle-energy correspondence for planar immersions and establish a principled method to select canonical representatives in compartments $\mathcal{C}(j_\pm,s,\omega)$.

Abstract

Vladimir Arnold defined three invariants for generic planar immersions, i.e. planar curves whose self-intersections are all transverse double points. We use a variational approach to study these invariants by investigating a suitably truncated knot energy, the tangent-point energy. We prove existence of energy minimizers for each truncation parameter $δ > 0$ in a class of immersions with prescribed winding number and Arnold invariants, and establish Gamma convergence of the truncated tangent-point energies to a limiting renormalized tangent-point energy as ${δ\to 0}$. Moreover, we show that any sequence of minimizers subconverges in ${C^1}$, and the corresponding limit curve has the same topological invariants, self-intersects exclusively at right angles, and minimizes the renormalized tangent-point energy among all curves with right self-intersection angles. In addition, the limit curve is an almost-minimizer for all of the original truncated tangent-point energies as long as the truncation parameter $δ$ is sufficiently small. Therefore, this limit curve serves as an "optimal" curve in the class of generic planar immersions with prescribed winding number and Arnold invariants.

Figures (7)

• Figure 1: Examples of numerically computedminimizers of the energy ${\rm{TP}}_{q,\delta}$ in admissibility classes $\mathcal{F}(\eta,j_\pm,s,\omega)$ of curves with different prescribed Arnold invariants $J^+=j_+,$$J^-=j_-$, $\mathop{\mathrm{\textit{St}}}\nolimits=s$, and winding numbers $W=\omega$. All these minimizers seem to self-intersect exclusively in right-angles. The competing curves in $\mathcal{F}(\eta,j_\pm,s,\omega)$ are affine linear within arclength $\eta$ around every self-intersection; see Definition \ref{['admissibile_curves']}. The ratio of the truncation parameter $\delta$ (green or red) to $\eta$ (blue or yellow) is approximately $1/2$ in our computations.
• Figure 2: Homotopy through a positive direct self-tangency.
• Figure 3: Arnold's representative curves $\gamma_R^\omega$ for various winding numbers $\omega \in \mathbb{Z}$ (Arnold).
• Figure 4: The space of planar $C^1$-immersions with fixed winding number equal to $\omega$. The three colored lines without their mutual intersections correspond to the good part $\Delta^d\cup\Delta^i\cup\Delta^t$ of the discriminant $\Delta$. $\gamma^\omega_R$ is the representative curve with winding number $\omega$, and the dotted line represents a generic path connecting $\gamma^\omega_R$ with a generic immersion $\gamma$ of equal winding number.
• Figure 5: Two curves with Arnold invariants $J^+=0$, $J^-=-2$, $\mathop{\mathrm{\textit{St}}}\nolimits=0$ and winding number $W=1$

Theorems & Definitions (36)

• Definition 1.1
• Theorem 1.2: Existence of admissible curves
• Theorem 1.3: Existence of minimizers
• Theorem 1.4: Gamma convergence as $\delta\to 0$
• Theorem 1.5: Limit immersion is almost-minimizer
• Corollary 1.6: Optimal immersion minimizes among curves with intersection angles $\frac{\pi}{2}$
• Lemma 3.1: Global individual bilipschitz estimate
• proof
• Definition 3.2
• Lemma 3.3