Homothetic Expanders & Shrinkers: Geometric Solitons

Updated 31 January 2026

Homothetic expanders and shrinkers are self-similar solutions defined by rescaling in geometric flows, clarifying singularity formation and evolution.
They are classified across various flows such as Ricci flow, mean curvature flow, and higher-order planar flows using rigorous analytical methods.
Their study informs practical understanding of blow-up behavior, rigidity phenomena, and the long-time behavior of nonlinear geometric evolution.

Homothetic expanders and shrinkers are self-similar solutions to geometric evolution equations that evolve purely by scaling about a fixed point. These solitons appear in a wide range of contexts: geometric flows on manifolds, curvature-driven dynamics in planar curves, Lagrangian mean curvature flow, higher-codimension submanifold evolution, harmonic map heat flow, and even flows driven by higher-order elliptic operators. Their structure, classification, and analytic properties are central to understanding singularity formation, long-time behavior, and rigidity in nonlinear evolution.

1. Definitions and Self-Similarity Equations

A homothetic soliton is a time-independent solution to a geometric flow whose evolution in time is given by rescaling. For a general family of embeddings or metrics $X(\cdot,t)$ , this reads

$X(p,t) = \psi(t)(X(p,0)-p_*) + p_*$

where $\psi(t)$ is a scaling function, $p_*$ is a fixed center, and $X(p,0)$ is the profile at $t=0$ .

Ricci Flow

A homothetic (gradient) Ricci shrinker is a triple $(M^m,g,f)$ with $Rc+\nabla^2 f = \frac{1}{2}g$ and normalization $R+|\nabla f|^2 - f=0$ , where $R$ is scalar curvature and $f$ is a potential (Li et al., 2018).

Mean Curvature Flow (MCF)

A self-shrinker satisfies $H + \frac{1}{2}X^\perp = 0$ ; a self-expander satisfies $H - \frac{1}{2}X^\perp = 0$ , where $H$ is the mean curvature and $X^\perp$ the normal component of the position vector (Khan, 2020). The corresponding time evolution is $X_t(p) = \sqrt{t} X(p)$ for $t>0$ .

Lagrangian Mean Curvature Flow

On graphs $X(x,t) = (x, Du(x,t))$ , the self-similar equation in potential variables reads

$\sum_{i=1}^n \arctan \lambda_i = s_1 + s_2(x \cdot Du - 2u)$

with $\lambda_i$ the eigenvalues of $D^2u(x)$ ; $s_2 > 0$ defines a shrinker, $s_2 < 0$ an expander (Bhattacharya et al., 2024).

Planar Curve Flows

In the area-preserving curve-shortening flow (APCSF), homothetic shrinkers solve

$k - \bar{k} = \delta \langle x, N \rangle,$

with $k$ curvature, $\bar{k}$ its average, $N$ inward normal, and $\delta>0$ (Cernomazov, 1 Aug 2025).

In higher-order flows such as elastic flow and curve diffusion, expanders and shrinkers are defined via scaling-invariant profile equations involving higher derivatives of curvature (Andrews et al., 29 Jan 2026).

Laplacian Flow on $G_2$ -Structures

Self-similar “Laplacian” solitons satisfy $d\varphi=0$ and $\Delta_{\varphi}\varphi=\lambda\varphi+\mathcal{L}_X\varphi$ . Homothetic expanders have $\lambda > 0$ , shrinkers $\lambda < 0$ (Haskins et al., 9 Jan 2025).

2. Analytical Properties and Regularity

Homothetic solitons often admit strong regularity and compactness properties, controlled by scaling invariants and normalization functionals.

Ricci shrinkers (with $\mu(g) \ge -A$ ) possess weak compactness: any sequence converges (pointed Gromov-Hausdorff) to a limit space $M_\infty$ with codimension- $\geq4$ singular set $\mathcal{S}$ , and full $C^\infty$ Cheeger-Gromov convergence on the regular part $\mathcal{R}$ (Li et al., 2018).
Harmonic radius and curvature integral bounds in Ricci shrinkers follow from density bounds and conformal covering arguments, independent of pointwise curvature assumptions.
For Lagrangian mean curvature shrinkers/expanders, quantitative interior $|D^2u(0)|$ estimates are obtained under hypercritical phase ( $|\Theta|\ge(n-1)\pi/2$ ), leveraging a Jacobi-type inequality and local Sobolev/maximum principle via Lewy-Yuan rotation. Entire smooth solitons with such phase constraints and bounded oscillation are rigid: only quadratic solutions exist (Bhattacharya et al., 2024).
In planar curve flows (APCSF), homothetic shrinkers are critical points of the length functional under area constraints, exhibiting a unique negative direction in the second-variation quadratic form; this is the “saddle-point property” (Cernomazov, 1 Aug 2025).

3. Classification, Existence, and Uniqueness

Homothetic expanders and shrinkers have rich moduli and classification schemes across flows.

In APCSF, all shrinkers are classified: either the circle, or for each coprime $(m,n)$ with $1/2 < m/n < 1$, a unique $n$ -fold symmetric non-circular shrinker, parametrized by average curvature $\lambda$ or energy $E$ (Cernomazov, 1 Aug 2025).
In MCF, parabolicity (vanishing capacity at infinity) implies that only shrinkers ( $\lambda > 0$ ) are parabolic for $n>2$ ; expanders are necessarily non-parabolic (Gimeno et al., 2018).
For higher codimension MCF, uniqueness theorems state that any two self-shrinkers smoothly asymptotic to the same cone coincide up to reparametrization; for expanders, uniqueness holds under rapid Hausdorff-separation decay, precluding nontrivial multi-sheeted asymptotics (Khan, 2020).
In Laplacian flow, for Sp(2)-invariant expanders on the bundle $\Lambda^2_-S^4$ , there is a 1-parameter family of smooth complete expanders, each uniquely determined (up to scale) by its asymptotic cone ( $L: q\mapsto\ell$ bijection). Shrinkers in this context have distinct asymptotic behavior, and conjectures extend the uniqueness structure to SU(3)-invariant solitons on $\Lambda^2_-CP^2$ (Haskins et al., 9 Jan 2025).
For planar flows governed by higher-order operators (elastic flow, curve diffusion, ideal flow), infinite families of geometrically distinct expanders and shrinkers exist, with orders of dihedral symmetry parametrized by integers or rational rotation indices. This is established by boundary-value shooting and gluing arguments (Andrews et al., 29 Jan 2026).

4. Singularities, Blow-up, and Continuation

Homothetic shrinkers and expanders model singularity formation and post-blow-up continuation in geometric flows.

In harmonic map heat flow between spheres, Type I blow-up is governed by a unique, linearly stable homothetic shrinker; the blow-up profile matches a specific solution $f_1(y)$ of the self-similar ODE. Continuation beyond blow-up is achieved by gluing a homothetic expander, with matching determined by shooting for the limit $B(A) = -b_1$ ; the process changes topological degree by one at each blow-up time (Biernat et al., 2011).
In Ricci flow and mean curvature flow, the structure of singularities (codimension, regular set, tangent cones) is governed by the limit geometry of homothetic shrinkers. No compact Ricci shrinker can be approximated by a metric cone (gap theorem) (Li et al., 2018).

5. Geometric, Variational, and Symmetry Properties

Homothetic solitons exhibit variational extremality, symmetry, and geometric invariants.

APCSF shrinkers are saddle points of length under area constraint; perturbation in normal directions bifurcates the evolution into global existence or finite-time cusp-type singularity depending on area constraint sign (Cernomazov, 1 Aug 2025).
In higher-order planar flows, dihedral symmetry and rotation index invariant under similarity transformations distinguish infinite families of expanders and shrinkers. Dihedral gluing closes fundamental arcs into smooth closed curves, and no hidden symmetries arise outside this construction (Andrews et al., 29 Jan 2026).
In Laplacian flow, asymptotic conicality precisely determines the expander, and trichotomy of end behaviors (asymptotically conical, super-Euclidean volume growth, finite extinction) characterize the phase portrait for Sp(2)-invariant, and conjecturally SU(3)-invariant, solitons (Haskins et al., 9 Jan 2025).

6. Extensions, Applications, and Implications

The classification and analytic control of homothetic expanders and shrinkers have far-reaching consequences.

Rigidity results and interior Hessian estimates exclude high-frequency singularities and guide blow-up profile classification in Lagrangian mean curvature flow (Bhattacharya et al., 2024).
No properly immersed MCF self-shrinker with bounded second fundamental form ( $|A|^2 < 5n/(3n-4)$ ) other than spheres or cylinders exists; no noncompact parabolic self-shrinker is confined strictly in a Euclidean ball (Gimeno et al., 2018).
Unique continuation from infinity for asymptotic eigenfunctions on conical ends, and associated weighted Sobolev inequalities, preclude the existence of exotic higher-codimension self-shrinkers/expanders (Khan, 2020).
In Laplacian flow, the conjectured “flow-through surgery” may allow CP $^2$ to shrink to a cone and re-emerge as an expander with altered topology, a scenario not possible in Sp(2)-invariant cases but plausible in SU(3)-invariant families (Haskins et al., 9 Jan 2025).

7. Examples, Sharpness, and Model Solutions

Canonical examples provide equality cases and testbeds for sharp estimates and classifications.

The Gaussian soliton $(\mathbb{R}^m, g_{\mathrm{E}}, f=|x|^2/4)$ for Ricci flow, saturates density and curvature integral bounds (Li et al., 2018).
In planar curve flows, circles and Abresch-Langer curves realize the full set of allowed turning ratios and symmetry orders (Cernomazov, 1 Aug 2025).
For harmonic map heat flow, explicit parameter tables and matched asymptotics describe the full sequence of shrinkers $f_n(y)$ and their expansion rates (Biernat et al., 2011).
Concrete diagrams illustrate “jellyfish” expanders for elastic flow, epicyclic shrinkers for curve diffusion, and associated symmetry orders (Andrews et al., 29 Jan 2026).

Homothetic expanders and shrinkers thus constitute a central class in geometric analysis, essential for understanding exact solutions, singularities, and rigidity phenomena in nonlinear geometric flows.