One shrinking solution stays stable in four, five and six dimensions

An equation whose whole job is to smooth things out has, in each of four dimensions, infinitely many ways to fail. That is the starting point for a preprint posted to arXiv on 29 July 2026, in which Johannes Angerer, Sarah Kistner and Birgit Schörkhuber ask which of those failures is the one that counts.

The equation is the harmonic map heat flow. Think of a map that assigns, to every point of flat d-dimensional space, a point on a d-dimensional sphere. Some of those maps are taut and economical, others are twisted and full of stored stretching, and the heat flow is a rule that lets the twisted ones relax, much as heat spreads through a metal bar until the temperature evens out. The hope is that the relaxing map stays smooth forever.

The dimensions in this paper do not cooperate. For maps from flat space into the sphere of the same dimension, with that dimension equal to 3, 4, 5 or 6, the problem is supercritical: concentration at small scales can outrun the smoothing, and a solution that begins perfectly smooth can lose regularity in finite time. The map does not drift gently toward a tidy final state. It breaks, at a specific moment, at a point.

Mathematicians already know a great deal about how that break can look. Each of these dimensions admits infinitely many self-similar solutions, shapes that reproduce themselves at ever smaller scales as the singular time approaches. These are the shrinkers of the paper's title, and each one is a worked example of a smooth start turning singular on a schedule.

An infinite supply of examples is less useful than it sounds. If infinitely many shapes each describe a legitimate way for the flow to tear itself, the interesting question becomes which of them you would actually run into. Angerer, Kistner and Schörkhuber approach that by asking about stability. A shrinker that shrugs off small disturbances is a shape solutions can settle into. One that any small nudge destroys is a curiosity.

What they proved

In dimensions 4, 5 and 6, the authors establish that there is a self-similar profile, which they call f₀, that increases monotonically and is asymptotically stable under small corotational perturbations. Corotational is a symmetry condition, and the authors work inside that class throughout, so the profile and the disturbances it has to survive carry the same rotational structure. The three-dimensional case was handled in earlier work the paper cites, from 2017 and 2018. What is new is that the same picture holds one, two and three dimensions further up.

Getting there took a computer. The profile is not something you write down in closed form, and the stability question reduces to a spectral problem, an eigenvalue calculation for the flow linearized around the profile. The team resolved both with what they describe as rigorous computer assistance. The word rigorous carries the weight there: the claim is not that a numerical experiment looked convincing, but that the computation comes with certified bounds and stands as a step in the proof. Arguments built this way let a proof reach objects that exist but refuse to be written down.

The analysis also says something about all the other shrinkers. Within the corotational class, the authors obtain finite-codimension stability for arbitrary self-similar profiles. For any one of them, the directions in which a perturbation can knock a solution off course are finite in number, and in everything else the profile holds. Even the unstable members of that infinite family are unstable in only finitely many ways.