Differential geometry, including: 

• Riemannian geometry;
• Relationship between local curvature conditions and global geometry/topology;

Partial differential equations, including:

• Prescribed curvature problems;
• Curvature driven parabolic equations, including Ricci flow, mean curvature flow and flows of curves, surfaces and hypersurfaces by other functions of curvature;
• Application of such flows to global differential geometry and general relativity.

Curve shortening flow 

Curve shortening flow is the gradient flow of length for regular curves. It drives an immersed curve on a Riemannian surface (e.g. the plane) with velocity equal to its curvature vector. Appropriately interpreted, this (differentio-geometric) process may be studied from the point of view of parabolic partial differential equations.

Curve shortening flow models the evolution of grain boundaries and the shapes of worn stones in two dimensions, and has been exploited in a multitude of further applications.

The curve shortening flow has much better convergence properties than other geometric flows. Indeed, Grayson (1987), building on work of Gage–Hamilton (1986), proved that it "untangles" every closed embedded planar curve, eventually causing the curve to contract a single point, with limiting shape that of a round circle. In fact, this behaviour holds in more general ambient spaces (e.g. compact surfaces), leading to numerous geometric applications (e.g. simple proofs of the Lusternik–Schnirelmann theorem on the existence of at least three closed geodesics on any Riemannian two-sphere and Smale’s theorem on the retractibility of Diff(S²) to SO(3)). The curve shortening flow also improves the isoperimetric ratio of an embedded curve, leading to further applications.

The natural Neumann boundary value problem for curve shortening flow, called the free boundary problem, asks for a family of curves whose endpoints lie on (but are free to move on) a fixed barrier curve which is met by the solution curve orthogonally. In this setting, it is known that convex curves with free boundary on a smooth, convex, locally uniformly convex barrier remain convex and shrink to a point on the barrier curve with limiting shape that of a round semicircle.

Despite its relative simplicity and the intensity with which it has been studied since the 1980's, many interesting questions about curve shortening flow (and other closely related flows) remain open. 

Applications of geometric flows in geometry

Geometric flows have proved to be an indispensable tool in the analysis of a number of important problems in differential geometry and related fields. Particularly spectacular examples are the proofs of the Poincaré and geometrization conjectures using Ricci flow and the proof of the Riemannian Penrose inequality in general relativity using inverse mean curvature flow. 

Further geometric applications include proofs of the Lusternik–Schnirelmann theorem and the Smale theorem (on Diff(S²)) using curve shortening flow, "local to global" results involving curvature using various intrinsic and extrinsic flows, and "pinched implies compact" theorems using Ricci flow.

There remain many interesting settings in which geometric flows may yield powerful new results.

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.