ExpandersEE91DF9C2CB0

Expansion from a singular initial shape

A self-expander is a submanifold whose mean curvature flow evolves by dilation away from a center. With $\vec H=\Delta_\Sigma X$, the normalized equation and flow are

The profile is the time-one slice. For a hypersurface with $\vec H=-H\nu$, the scalar equation is $H=-\langle X,\nu\rangle/2$. This sign is opposite to the shrinker equation. A smooth compact boundaryless expander of positive dimension cannot exist in Euclidean space: the identity $\Delta_\Sigma|X|^2=2n+|X^\perp|^2$ contradicts the maximum principle. Planes through the origin are stationary solutions of the expander equation, and nontrivial complete examples are noncompact.

The introductions of Bernstein–Wang place expanders at the intersection of two flow problems: emergence from a conical singularity and possible large-time behavior after rescaling. Both concern a scale growing like $\sqrt t$. Unlike reversing a shrinking flow, the expander equation still generates forward mean curvature flow, with area decreasing locally according to the evolution equation. Growth of the whole noncompact image is compatible with this local area decrease.

A particularly important class consists of properly embedded hypersurfaces $\Sigma^n\subset\mathbb R^{n+1}$ asymptotic to a regular cone $\mathcal C$. Here regular means smooth away from the vertex, and, in a typical $C^k$ formulation,

Consequently $\sqrt t\,\Sigma$ approaches $\mathcal C$ away from the vertex as $t\downarrow0$. Prescribing the cone is an analogue of prescribing boundary data at infinity. It does not automatically prescribe the topology of the smoothing or select a unique solution. Asymptotic conicality is an assumption in the results discussed here, not a property asserted for every complete expander.

Examples and the meaning of uniqueness

The simplest models smooth a planar corner or a graphical round cone. For an expanding graph $\sqrt t\,(x,u(x))$, its time-one height satisfies

Reflection or rotational symmetry reduces this to an ODE. The profiles in the catalogue are defined by those equations and their asymptotic conditions, rather than elementary closed formulas. Ding's constructions also address one-sided smoothings of mean-convex cones, with geometric and minimizing hypotheses that are recorded in the relevant entry (Ding20, Sections 4–6).

A rotational double cone makes nonuniqueness visible. A disconnected smoothing consists of two graphical sheets, while a connected smoothing can join the two ends through a neck. Their existence ranges depend on the cone angle. Angenent–Ilmanen–Chopp prove a critical-angle picture and also provide numerical investigations; the computed angles should not be mistaken for exact constants (AIC95, Lemma 2 and Theorem 4). Distinct expanders from one cone give distinct smooth evolutions for positive time with the same conical initial trace. A claim about fattening additionally concerns a specified weak flow: for level-set flow, fattening means that the evolving zero set has nonempty interior.

Networks introduce another distinction. A regular planar expanding network solves the curve expander equation on its branches and has three branches meeting at $120$ degrees at each junction. It is not a smooth immersed curve at those junctions. Ilmanen–Neves–Schulze establish uniqueness within a prescribed topological class for regular expanders with the same asymptotic rays; this does not identify different combinatorial types (INS19, Section 4). The network and hypersurface theories therefore have related equations but different admissible objects.

Weighted area and relative expander entropy

The expander equation is the stationarity condition, under compactly supported variations, for

The growing weight makes $E$ infinite on an asymptotically conical expander. Compactly supported first and second variations remain meaningful, but a global comparison requires cancellation at infinity. For a two-sided hypersurface the Jacobi operator is

Its quadratic form is $-\int fL_Ef\,e^{|X|^2/4}\,d\mu$ for compactly supported $f$. Stability refers to nonnegativity of this form; in deformation problems at a fixed cone, the permitted behavior at infinity also matters. These conventions should not be confused with the Gaussian weight of a shrinker.

Bernstein–Wang develop the relative quantity

The cancellation is a theorem under suitable hypotheses, not a formal subtraction of two infinities. In their two-sided obstacle setting, the reference expanders have the same cone and bound nested domains, and the competing hypersurface bounds a domain between them. The relative limit exists, possibly with value $+\infty$; under the additional $C^2$ asymptotically conical hypothesis in Remark 1.2, it is finite. This framework supports minimization and a forward monotonicity formula (BWRelative2022, Theorem 1.1 and Remark 1.2). Relative expander entropy is not the translation-and-dilation invariant Gaussian entropy $\lambda$.

Deformation, compactness, and degree

In BWSpace2021, Theorem 1.1, Bernstein–Wang describe the space of asymptotically conical self-expander embeddings of a fixed model hypersurface, modulo reparametrizations fixing the data at infinity. For $n,k\geq2$ and $0<\alpha<1$, their $C^{k,\alpha}_*$ framework gives a smooth Banach manifold. The map taking an expander to its parametrized cone is Fredholm of index zero. Its derivative has kernel given by normal Jacobi fields that fix the cone. Generic cone data are therefore nondegenerate in the precise sense of that theorem. Nondegeneracy controls local deformations; it is not global uniqueness.

Global arguments also need compactness. For surfaces in $\mathbb R^3$, Bernstein–Wang obtain local smooth subsequential convergence when genus and number of ends are fixed and the cone links converge in $C^{k,\alpha}$ to a regular embedded link. They prove corresponding results in the uniformly mean-convex setting, and in low-entropy classes subject to the stated exclusion of low-entropy singular minimal cones (BWCompactness2021, Theorem 1.1). These hypotheses prevent degeneration; fixing an arbitrary cone alone is not the general compactness theorem.

When the cone projection is proper on the domain under consideration, Bernstein–Wang define an integer degree. At a regular cone, it counts preimages with signs $(-1)^{\operatorname{index}}$, rather than simply counting all surfaces. Invariance of this signed count can force additional solutions, including unstable annular expanders for an open set of cones in $\mathbb R^3$ (BWDegree2023, Introduction and Theorem 1.1). Properness and the chosen domain are indispensable parts of this conclusion.

Mountain passes and topological uniqueness

The relative variational theory can produce an expander between two others. For $2\leq n\leq6$, let two distinct strictly stable, properly embedded $C^2$-asymptotically conical expanders in $\mathbb R^{n+1}$ have the same $C^3$-regular cone and bound nested domains. Bernstein–Wang's mountain-pass theorem produces a third smooth expander between them, with a point in the open intervening region. For generic cone data, it can be chosen unstable. In higher dimensions, their general existence theorem permits a singular set of codimension at least seven (BWMountain2022, Theorem 1.1 and Corollary 1.2). Thus the dimension range determines whether the variational output is guaranteed to be smooth.

Nonuniqueness of surfaces can coexist with uniqueness of topology. For $2\leq n\leq6$ and $k\geq2$, if a $C^{k+1}$-regular cone in $\mathbb R^{n+1}$ has entropy strictly below that of $S^{n-1}\times\mathbb R$, all $C^{k+1}$-asymptotically conical self-expanders with that cone are $C^k$ asymptotically conical isotopic with the cone fixed. In particular, they are diffeomorphic; the theorem does not say that they coincide (BWTopology2022, Theorem 1.1). For these $C^2$-asymptotically conical expanders, Gaussian entropy equals the entropy of the cone, as recalled in that paper's introduction. Neither value is generally the centered Gaussian area of the time-one expander.

The resulting questions are geometric as well as analytic: how many smoothings a cone admits, which topologies occur, how solutions change when the cone changes, and which expanders are selected by a given flow. The results above answer substantial parts of these questions under specific hypotheses. The examples below provide concrete entry points into this theory, with dimensions, asymptotic conditions, and numerical limitations stated separately for each family.