Other special flows52A98BA8F5C1

Time domains and special behavior

The preceding chapters organize flows by a symmetry of their evolution. This chapter also includes distinguished solutions characterized by their time interval, asymptotic behavior, or singularity formation. For a smooth immersion $X:M^n\times I\to N^N$, the equation is $(\partial_tX)^\perp=\vec H$, with mean curvature computed in the stated ambient metric. We call a flow ancient if it is defined for every $t<T$ for some finite $T$, immortal if it is defined for all future time, and eternal if it is defined for all real time.

These terms describe time domains rather than shapes. Shrinking solitons are ancient, translators are eternal, and expanding solitons exist for every positive time. Ancient and eternal solutions need not be solitons. The grouping here avoids repeating those earlier families and instead emphasizes examples whose shape changes, stationary minimal submanifolds, higher-codimension phenomena, and flows with special finite-time or infinite-time limits.

For a smooth closed flow, area satisfies

This dissipation identity explains both the appeal and the difficulty of using mean curvature flow to find minimal submanifolds. Area decreases, but curvature can concentrate, topology can change, and several sheets can converge to the same limiting surface. A stationary limit, when one is obtained with sufficient compactness and vanishing mean curvature, is minimal. Area decrease alone does not provide those hypotheses or guarantee smooth multiplicity-one convergence.

Ancient solutions as singularity models

Parabolic rescaling about a developing singularity stretches the preceding interval of existence. When the rescaled geometry has the necessary local curvature and mass bounds, a subsequence can converge to a flow defined arbitrarily far backward in time. This produces an ancient model. Fixed-spacetime tangent flows are self-similar by monotonicity; limits obtained by moving the centers and curvature scales can retain non-self-similar ancient dynamics. Distinguishing these limiting procedures prevents the mistaken conclusion that every ancient solution is a shrinker.

Convexity provides important rigidity results. Huisken–Sinestrari show, for closed convex ancient hypersurface flows of dimension $n\geq2$ in Euclidean space, that a uniform pinching $h_{ij}\geq\varepsilon H g_{ij}$ with $\varepsilon>0$ forces shrinking spheres. Their equivalent criteria include suitable diameter growth and Type I behavior backward in time (HuiskenSinestrari2015, Theorem 1.1). Convexity at each time without uniform pinching is a weaker condition and admits richer ancient behavior.

The planar Angenent oval makes that behavior explicit. In the normalization used here, it is the closed curve

Every slice is embedded, compact, and strictly convex. Its changing aspect ratio rules out evolution by a single dilation. At extinction its rescaled shape approaches a round circle; far backward in time, its upper and lower tips approach oppositely directed grim reapers after translation. Thus one ancient flow connects shrinking and translating models at different scales. The entry derives these limits from the implicit equation and distinguishes it from a graph parametrization (BV11, the paperclip solution).

For noncompact ancient surfaces, assumptions remain decisive. Brendle–Choi's bowl classification requires strict convexity and uniform two-sided noncollapsing in $\mathbb R^3$ (BrendleChoi19, Theorem 1.1). It does not classify arbitrary immersed ancient curves, higher-codimension ancient submanifolds, or ancient flows with no curvature restrictions.

Higher codimension and changing effective dimension

Stryker–Sun's introduction asks how the ambient dimension of an ancient flow is reflected in its backward limit and entropy. Their torus curves give compact ancient curve-shortening flows that span arbitrarily large even-dimensional Euclidean spaces. They explicitly acknowledge the earlier construction of Altschuler–Altschuler–Angenent–Wu (SS19, Introduction and Remark 1.3; AAAW13, Section 3). These are one-dimensional curves lying on products of circles, rather than surfaces of torus topology.

For example, for integers $1=k_1<\cdots<k_m$, their coordinate pairs have the form

The parameter is $\theta\in\mathbb R/2\pi\mathbb Z$, the ambient space is $\mathbb R^{2m}$, and the codimension is $2m-1$. The first frequency makes this subfamily embedded. Different coordinate radii dominate at different times: the backward rescaled limit is a circle with multiplicity $k_m$, whereas the extinction limit is a multiplicity-one circle. The full flow nevertheless spans $\mathbb R^{2m}$. This shows why a low-dimensional extinction profile need not capture the ambient dimension of an ancient flow (SS19, Section 2.1).

The related helical examples are noncompact eternal flows. Their radii decrease while the axial coordinate remains fixed, so the pitch-to-radius ratio changes and the evolution is not a homothetic soliton. Every finite-time slice is embedded, but the local limit as $t\to\infty$ is the axis. A three-coordinate projection of a higher-dimensional example can have crossings even when the original submanifold is embedded. The interactive figures therefore distinguish ambient geometry from its display.

Rotating curve solitons

The AAAW13 classification also supplies pure rotators, rotating–translating curves, and curves that expand or shrink while rotating. Their profiles solve an arclength ODE; exact ambient motions evolve those profiles. They belong here because nonzero normal rotation changes the pure soliton equation. Expanding rotators have two proper spiral ends, whereas genuinely rotating shrinking profiles have a bounded infinite-length tail. The irrational-frequency torus curves give another complete nonproper family, with a noncompact domain dense in a bounded torus. The entries distinguish these infinite-Gaussian-length immersions from the compact ancient torus curves above.

Stationary solutions and finite-time constructions

Every minimal surface or submanifold gives a stationary mean curvature flow. The entry discusses the general static equation, minimal graphs, affine planes, and curved ambient spaces. Its catenoid illustration is an exact complete properly embedded example in $\mathbb R^3$, with annular topology and two ends. Stationarity does not imply stability: its ordinary area second variation has a negative compactly supported direction, as computed in the entry. Minimal surfaces also serve as barriers and local models in constructions of evolving solutions. Their stability is measured by ordinary area, while the soliton problems in the earlier chapters use different weights.

Some special flows are recorded for the singularities they produce. The rotational degenerate neckpinches of Angenent–Velázquez are non-self-similar finite-time solutions with Type II curvature concentration. Their asymptotic analysis involves different spatial scales, including translating behavior near a tip (AV97, Introduction). Such a construction is an evolving solution of the parabolic equation; an asymptotic profile is not an exact formula for the whole flow.

The Whitney sphere gives a different example in higher codimension. It is a Lagrangian immersed sphere in $\mathbb C^n$, $n\geq2$, with a transverse double point. Its symmetric mean curvature evolution collapses in finite time and has a multiplicity-two Lagrangian plane as tangent flow, as established in the setting of Viana's theorem (Viana18, Introduction and Theorem 1.3). The initial immersion has an explicit formula, but the evolution is not self-similar and is not ordinary planar curve shortening of its profile. The ambient codimension and immersion hypothesis are essential to interpreting this example.

Immortal flows and multiplicity in the limit

Chen–Sun construct a connected embedded rotationally symmetric immortal surface flow in $\mathbb R^3$ whose limit as $t\to\infty$ is a plane with multiplicity two (CS26R3, Theorem 1.1). The finite-time surfaces are noncompact. Multiplicity two describes the limiting area measure: two sheets contribute mass to the same plane. It does not say that a finite-time slice is a double cover or is nonembedded. The construction concerns an unrescaled infinite-time limit, a different regime from a tangent flow at a finite-time singularity of a closed surface.

In a second construction, Chen–Sun obtain compact immortal embedded connected hypersurfaces in $S^n\times[-1,1]$ with the product metric, for every $n\geq2$. They converge to the minimal slice $S^n\times\{0\}$ with multiplicity two. The ambient product region can be included in a closed manifold (CS25Man, Theorem 1.1). This is mean curvature flow in a curved ambient space: it is not the Euclidean evolution of the same hypersurface viewed in a larger vector space. Their introduction uses the example to explain how long-time convergence can lose topology and retain additional mass.

These examples organize several recurring questions: what backward and forward limits exist, in which topology they converge, whether convergence is unique, and how dimension, entropy, stability, and embeddedness constrain them. The catalogue answers these questions only where the cited sources do. A closed formula, a prescribed initial immersion, a proven asymptotic model, and a numerical illustration convey different kinds of information, and the entries keep those distinctions visible.