Ancient solutions to geometric flows
A very important (hard but tractable) open problem concerns the meaningful classification of convex ancient solutions to mean curvature flow (and related flows).
Ancient solutions to geometric flows (such as the mean curvature, Ricci, or Yamabe flows) are solutions which have existed for an infinite amount of time in the past. Such solutions arise naturally (through "blow-up" procedures) in the study of singularities of the flow, and a deep understanding of them will have profound implications for the continuation of the flow through singularities (a prerequisite for many important applications). In the context of extrinsic geometric flows such as the mean curvature flow, the study of convex ancient solutions (i.e. solutions whose timeslices bound convex regions in space) is particularly pertinent, since blow-ups are guaranteed to be "codimension-one" and convex in many settings.
Ancient solutions also model the ultra-violet regime in certain quantum field theories, and early research on ancient solutions was undertaken by physicists in this context.
Ancient solutions are known to exhibit rigidity phenomena resembling those of their elliptic counterparts — minimal hypersurfaces. For example, there is a "Bernstein-type" theorem which guarantees that shrinking round spheres are the only ancient solutions satisfying certain geometric conditions.
Moreover, in the one-space-dimension case, there is a (rather satisfying) classification: the stationary halfspaces and strips, the shrinking circles, the Angenent ovals and the Grim Reapers are the only convex ancient curve shortening flows.
Another important theorem implies that the only regions "swept-out" by convex ancient mean curvature flows are slab regions ("slab case") or the whole ambient space ("entire case"). The shrinking sphere is an example which sweeps out all of space. The "ancient pancake" is an example that sweeps-out a nontrivial slab region. The slab-entire dichotomy is closely related to "noncollapsing" phenomena.
Both the slab and entire settings have been studied (separately) in recent years, with some success. For example, in the entire case, it is known that the shrinking spheres, the (admissible) shrinking cylinders, the (admissible) ancient ovaloids, and the bowl solitons are the only convex ancient mean curvature flows which are uniformly two-convex. In the slab case, it is conjectured (and partly proven) that every circumscribed convex body arises as the "squash-down" to a (non-entire) convex ancient mean curvature flow.
The problem of classifying ancient solutions has also been studied in more general ambient spaces. It is known, for example, that the only geodesically convex ancient mean curvature flows in spheres are the shrinking hyperparallels.
In the free boundary setting, there is exactly one convex ancient mean curvature flow in the unit ball in each dimension.
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