Self-Intersecting Closed Evolutes in the $T_\infty$ Problem

Determine whether a regular immersed, self-intersecting closed evolute of the form $C=\alpha+\rho\dot\alpha$, with $\rho>0$ and $\operatorname{rank}\dot\alpha=1$, can lie entirely in the constitutive set of a strictly hyperbolic, genuinely nonlinear $2\times2$ system with a strictly convex entropy.

Background

The paper proves nonexistence for embedded closed evolutes of the Iqbal type. Its global contradiction uses embeddedness to infer that a one-signed-curvature curve is a strictly convex oval and then derives incompatible integral identities.

The authors explicitly limit the theorem because one-signed curvature does not force a self-intersecting closed immersion to bound a convex domain. Thus the genuinely immersed, self-intersecting case remains unresolved by the argument presented.

References

Thus our nonexistence statement should be understood with the embeddedness assumption; the genuinely immersed, self-intersecting case is not settled by this particular global argument.

— The computational ansatz for convex integration of hyperbolic systems and a resolution of the Strong Trace Conjecture  (2609.09353 - Krupa, 8 Sep 2026) in Section “Nonexistence of $T_\infty$ configurations,” final paragraph of the proof of Theorem \ref{thm:main_t_infty}