Why Mean Curvature Flow On Geometry in General
Why Mean Curvature Flow 5th May 2023 The Mean curvature flow is an extrinsic flow on immersed hypersurfaces in Euclidean space or arbitrary Riemannian manifold. It is the negative gradient of the volume functional so not a completely unmotivated object. The main thing which makes it work in some natural way was the codimension one condition; however generalizations to higher codimension hypersurfaces exist.
Why do we want to study this flow and if we do what results can we achieve? To start with it is a very natural flow to consider on hypersurfaces in Euclidean space. It bends the higher curved parts with more speed than the lower curved parts to uniformize the curvature across the hypersurface. The one dimensional case which is called the curve shortening flow has been a very successful test of this flow. Also, the parabolic nature of the equation directly gives short time existence and uniqueness; so we know that given a hypersurface we have one way to evolve to possibly study its geometry. For its twin ''Ricci flow'' as Hamilton calls it the motivation was uniformizing Riemannian manifolds with an eye towards Poincaré conjecture. This is of course with benefit of hindsight after Perelman's seminal resolution using surgery methods. For Yamabe flow the plan was also clear - solve the PDE and get a conformal metric with uniform scalar curvature.
So the question begs itself : What do we want to do with the Mean curvature flow on hypersurfaces? A generic answer is to study the geometry of hypersurfaces and attempt a classification. This is severely restricted by the assumption of mean convexity (\(H >0\) everywhere) which makes the maximum principle work in several cases. Huisken's result on the convergence of convex hypersurfaces into a round sphere is the first step towards it, but it doesn't achieve much topologically. A uniformly convex hypersurface is diffeomorphic to a unit sphere by Gauss map, to begin with. For non-convex hypersurface singularities might develop which prohibit a direct analysis. To overcome this we blow up the manifold near singularity and this limiting process gives an ancient solution. So we shift our attention to the classification of ancient solutions which is still a difficult problem.
Another direction in which the mean curvature flow is being explored is the Lagrangian Mean curvature flow in order to find special Lagrangians inside symplectic manifolds. This is related to a problem in symplectic geometry(?) and in the case of Calabi-Yau manifolds the condition of being Lagrangian is preserved under Mean curvature flow.
Mean curvature flow can also be potentially used to find nice codimension one hypersurface inside Riemannian manifolds. Lawson conjecture is one example of it. It states that the Clifford torus is the only minimally embedded torus in \(S^3\). While Brendle's proof of Lawson conjecture didn't directly involve the flow however it did use some techniques coming from his sharp estimate analysis of the inscribed radius in noncollapsing. From what I have read on the internet Yau gave a big push for the question of finding minimal surfaces inside Riemannian manifolds. I don't know why this is an interesting question to consider and what we can get out of it.
On Geometry in General 2nd May 2023 Disclaimer: I am still largely inexperienced in the ways of geometry. These are just some thoughts I have collected over the years and need not be accurate, but I still wanted to share.
It has been a long pressing question for me to find out what is the goal of doing differential geometry in the large and what is the future of the field. A typical question to throw is the classification problem - if you're doing algebra classify all groups, rings, modules, fields; if you're doing analysis classify all Hilbert spaces, functions between spaces; if you're doing topology classify all spaces up-to homeomorphism and it is a great question to ask in geometry as well. What are all the Riemannian/complex/symplectic manifolds? Like other subfields this seems to be an incredibly difficult and almost unsolvable problem.
So we do what is most common and add more conditions/structures on manifolds. The question of what good manifolds are is a tricky one I feel. One condition to consider is possession of a large isometry group, i.e. the best manifold should be very symmetric. A popular definition in this regard is the definition of homogenous spaces. A space \(X\) is said to be homogenous if it is equipped with a transitive action of a group \(G\). This is pretty close to the spirit of Erlangen program.
At Sorbonne my advisors Prof. Alix Deruelle and Prof. Gilles Courtois introduced me to a question by Heinz Hopf of finding the best metric on a given smooth manifold. Best is subjective and we can have various definitions. In fact there is an entire book A Perspective on Canonical Riemannian Metrics by Giovanni Catino and Paolo Mistrolia which explores some meanings one can associate with the best metric. This is a also a chapter in Berger's book A Panaromic View of Riemannian Geometry . For two dimensions the problem is easy to consider and has been solved for compact manifolds. Here the best metric is the metric with constant curvature and Uniformization theorem gives a complete answer. For three dimension there is Thurston's conjecture that one can break a three manifold into pieces with canonical geometry but I am not really familiar with this theory (there is a lot happening on the topological side as well).
Another approach considered for best metric is to think of metrics coming from critical values of a functional. One example is the scalar curvature functional which gives Einstein metrics as critical metrics. On the complex side there is a subfield based out of study of critical points of the Calabi functional which I am completely unaware of.
There is a very interesting paper by Chern called 'Differential Geometry: Its past and its future' which gives an overview. Another source of a broad perspective of geometry (or more specifically geometric analysis) I recently got to know about is Yau's big survey paper which will be an ambitious read if I am ever able to complete it.