Topology and Morse Theory of Allen-Cahn Moduli Spaces

Definitions and mathematical statement

The supplied problem concerns the planar Allen--Cahn equation \[ \Delta u+u-u^3=0\qquad\text{on }{\mathbb{R}}^2. \] Under the conventional classical, real-valued interpretation, a solution is a function $u\in C^2({\mathbb{R}}^2;{\mathbb{R}})$ satisfying this equation pointwise. The source does not explicitly specify regularity or the target space.

The notation $M_{2m}$ denotes the source's proposed moduli space of solutions having $2m$ ends. However, the supplied record does not define an end, prescribe asymptotic conditions, specify admissible values of $m$, or state whether solutions are identified under translations, rotations, reflections, or the transformation $u\mapsto -u$. It also does not specify a topology or differentiable structure on $M_{2m}$. These choices remain unresolved and are necessary for a precise moduli-space formulation.

With those qualifications, the requested problem is: for each admissible $m$, determine whether $M_{2m}$ is connected, determine its dimension, and classify the Morse indices of its solutions.

For clarity, one conventional meaning of the Morse index of an individual classical real-valued solution is obtained from the quadratic form \[ Q_u(\phi)=\int_{{\mathbb{R}}^2}\left(|\nabla\phi|^2+(3u^2-1)\phi^2\right)\,{\,\mathrm{d}} x, \qquad \phi\in C_c^\infty({\mathbb{R}}^2;{\mathbb{R}}). \] Under this convention, \[ \operatorname{ind}(u)=\sup\left\{\dim V:\ V\subset C_c^\infty({\mathbb{R}}^2;{\mathbb{R}})\text{ is a finite-dimensional real vector subspace and }Q_u(\phi)<0\text{ for every }0\ne\phi\in V\right\}, \] with value $+\infty$ if these dimensions are unbounded. This convention is stated explicitly for interpretation; the supplied record does not select a variation space or an index convention. Nor does it assert that all solutions with the same number of ends have the same index. A complete formulation must specify whether dimension refers to a smooth manifold, local dimensions, or dimensions of strata.

Short English statement

The problem asks whether the space of planar Allen--Cahn solutions with $2m$ ends is connected, what its dimension is, and which Morse indices its solutions have. The supplied record leaves the meaning of ends, asymptotic conditions, symmetry identifications, topology, and dimension unspecified, so the full formulation requires clarification and human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30004484$, source problem code catalogue entry, title Topology and Morse Theory of Allen-Cahn Moduli Spaces, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2020.
• Supplied source citation: Calculus of Variations (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/22, pp. 1139--1196. https://doi.org/10.4171/owr/2020/22. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: OWR Calculus of Variations report. https://ems.press/content/serial-article-files/46860.
• Supplied literature source, with attribution and title retained as given: Pacard--Kowalczyk--Liu, Classification of four-end solutions. https://arxiv.org/abs/1109.5604.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that four-ended solutions are classified, while connectivity, dimension, and Morse-index classification for the general $2m$-ended moduli space remain incomplete. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming self-containment and preservation of the mathematical target. These are supplied editorial assertions; this draft does not independently verify them or resolve the missing moduli-space specifications.