Ansatz

For flows with arbitrary symmetric functions of curvature I need to understand symmetric polynomials of eigenvalues of a matrix. Let \(S(n)\) denote the set of \(n \times n\) symmetric matrices. Let \(M \in S(n)\) be a symmetric matrix with eigenvalues \(\lambda_1, \ldots, \lambda_n\). Also denote the elementary \(k\)-th symmetric polynomial by \(S_k(\vec{\lambda}) = \sum_{1 \le i_1 < \ldots < i_k\le n}\lambda_{i_1}\cdots \lambda_{i_k}\). Let \(f : \mathbb{R}^n \to \mathbb{R}\) be a smooth symmetric function (invariant under permutaion of coordinates). Then there exists a function \(G\) such that \(f (\vec{\lambda}) = G(S_1, ... , S_n )\) because eigenvalues satisfy the polynomial equation \(\det(\lambda I - M) = 0\) which is completely determined by the symmetric sums.

A convenient way to get \(S_i\)'s as a function of \(M\) is the following. Recall that the coeffiecient of \(\lambda^{n-k}\) in \(\det(\lambda I - M) \) is \((-1)^k S_k\). Now the expansion for eigenvalue polynomial reads \[ \begin{aligned}\det(\lambda I - M) & = \sum_{\sigma \in \text{Perm}(n)}(-1)^{|\sigma|}(\lambda \delta_{1 \sigma(1)} - M_{1 \sigma(1)})\cdots (\lambda \delta_{n \sigma(n)} - M_{n \sigma(n)}) \\ & = \sum_{k=0}^{n}\lambda^{n-k} \sum_{\substack{1\le i_1 < \cdots < i_k \le n \\ \{j_1, ..., j_{n-k}\} = \{1,...,n\} \backslash \{i_1, ..., i_k\}}}(-1)^{k}\sum_{\sigma \in \text{Perm}(n)}(-1)^{|\sigma|}M_{i_1 \sigma(i_1)} \cdots M_{i_{k} \sigma(j_{k})}\delta_{j_1\sigma(j_1)} ... \delta_{j_k \sigma(j_{n-k})} \\ & = \sum_{k=0}^{n}\lambda^{n-k} \sum_{1\le i_1 < \cdots < i_k \le n}(-1)^{k}\sum_{\sigma \in \text{Perm}(k)}(-1)^{|\sigma| }M_{i_1 \sigma(i_1)} \cdots M_{i_{k} \sigma(i_{k})}. \end{aligned} \] Thus, \[ S_{k}(\lambda) = \sum_{1\le i_1 < \cdots < i_k \le n}\sum_{\sigma \in \text{Perm}(k)}(-1)^{|\sigma| }M_{i_1 \sigma(i_1)} \cdots M_{i_{k} \sigma(i_{k})}. \]

Another way to do this which looks much more cooler is to use exterior algebra. The map \(M : \mathbb{R}^n \to \mathbb{R}^n \) induces a map \(M^k : \bigwedge^k \mathbb{R}^n \to \bigwedge^k \mathbb{R}^n \) in the \(k\)-th exterior algebra . For a basis element \(e_{i_1} \wedge ... \wedge e_{i_k}\) with \(1\le i_1 < \cdots < i_k \le n \), the induced map is \[ \begin{aligned} M^k(e_{i_1} \wedge \cdots \wedge e_{i_k}) & = M e_{i_1} \wedge \cdots \wedge M e_{i_k} \\ & = M_{i_1}^{j_1}e_{j_1} \wedge \cdots \wedge M_{i_k}^{j_k}e_{j_k} \\ & = \sum_{\sigma \in \text{Perm}(k)} (-1)^{|\sigma|} M_{i_1}^{j_1} \cdots M_{i_k}^{j_k} e_{\sigma(j_k)}\wedge \cdots \wedge e_{\sigma(j_1)} \end{aligned} \] where \(\sigma\) in the last line is used to order \(\sigma(j_1) < \cdots < \sigma(j_k)\). It is easy to see that we will hit every permutation while doing so. Notice that \(\text{tr}(M^k) = S_k(\lambda)\) and its not a coincidence; the reason is that trace is preserved after diagonalization and trace of the diagonal matrix \(\text{diag}[\lambda_1 , ..., \lambda_n] \) in the exterior algebra is the symmetric sum \(S_k(\lambda)\). Also notice that \(S_k\) is the sum of determinants of all \(k \times k\) minors. This is a very useful property for calculations.

A third way to do this is to polarize the determinant and let mixed discriminant come into the picture. Let \(M^1, ..., M^n\) be fixed matrices, the mixed discriminant of the set is denoted by \(\det(M^1,...,M^m)\) and is the coefficient of \(t_1 \cdots t_n\) in \(\frac{1}{n!}\det(t_1M^1 + ... + t_nM^n)\). As was hinted in the last post, this can be evaluated as \[\det (M^1,...,M^n) = \frac{1}{n!}\sum_{\sigma, \tau \in \text{Perm}(n)}(-1)^{|\sigma| + |\tau|}M_{\sigma(1)\tau(1)}^1 \cdots M_{\sigma(n)\tau(n)}^n.\] Now the coefficient of \(\lambda^{n-k}\) in the expansion \(\det(\lambda I + M)\) is \(\binom{n}{k}\det(\underbrace{M,...,M}_{k-\text{times}},I,...,I)\).

The main purpose of the above exercise was to find out how \(S_k\)'s perturb under a perturbation of \(M\) in \(S(n)\). Consider a basis of \(S(n)\) given by elementary matrices \(e_{ii}\) and \(e_{ij} + e_{ji}\) (\( j \neq i\)). We will only be concerned with perturbation around a diagonal matrix \(M = \text{diag}[\lambda_1, ..., \lambda_n]\). Let \(M_{\epsilon} = M + \epsilon e_{ii}\), then \[\frac{d S_k}{d \epsilon}\bigg|_{\epsilon = 0} = S_{k-1}((\lambda_1, ...,\hat{\lambda}_i,..., \lambda_n))\] and for \(M_\eta = M+ \eta (e_{ij} + e_{ji})\) the expansion of \(S_k\) will have no linear term implying \[\frac{d S_k}{d \eta}\bigg|_{\eta = 0} = 0,\] so for a symmetric functions \(f(\lambda) = G(S_1,...,S_n)\), only the diagonal directions have non-trivial perturbation.

The notion of convexity is central to many areas of mathematics ranging from PDEs to combinatorics. There is a book by Lars Hörmander titled Notions of convexity exploring convexity in various branches. The first definition of convexity comes from geometry; a set is said to be convex if for any two points in the set, the line segment joining them is in the set too. I will be concentrating on the Brunn-Minkowski theory in convex geometry which is concerned with mixed volume and various measures coming from Minkowski addition of convex bodies. Given two convex bodies \(A, B \subset \mathbb{R}^{n}\) we can simply add all possible combinations of points in these and define \(A+B = \{a+b | a \in A, b \in B\}\) and it turns out to be a convex set. Moreover a convex set scaled by a positive numbers is also a convex set. These two operations allow us to do some kind of calculus in set of convex bodies which is even rich enough to get some Sobolev inequalites. A.D. Aleksandrov exploited this calculus really well leading to a number of far reaching results in convex geometry.

One of the cornerstones in Brunn-Minkowski theory is the Brunn-Minkowski inequality. It states that under Minkowski addition volume power function \(V(\cdot)^{\frac{1}{n}}\) is concave. Specifically, \[V(\lambda A + (1-\lambda)B)^{\frac{1}{n}} \ge \lambda V(A)^{\frac{1}{n}} + (1-\lambda)V(B)^{\frac{1}{n}} \qquad \forall \, \lambda \in [0,1].\] Because \(V(\cdot)^{\frac{1}{n}}\) is degree one homogenous, the ineuqality is equivalent to a multiplicative form \(V(\lambda A + (1-\lambda)B) \ge V(A)^{\lambda}V(B)^{1-\lambda}\) which can be proved using Prékopa-Leindler inequality. Brunn-Minkwoski theorem doesn't require \(A\) and \(B\) to be convex but I guess it was proved that way first. The ineuqality is valid as long as the Minkowski added sets are measurable. A really great exposition which every source recommends about Brunn-Minkowski inequality is Gardner's article The Brunn-Minkowski inequality.

Steiner obtained a polynomial formula for \(V(K+tB)\) in dimension \(2\) and \(3\) where \(K\) is a bounded convex set, \(t\) is a positive real number and \(B\) denotes the ball of unit radius. He proved that this function is a polynomial in \(t\) and the linear cofficient in this expansion is area of the convex set. Formally, \[A(K) = \lim_{t \to 0^+}\frac{V(K+tB)-V(K)}{t}\] essentially saying that area is a differential of volume. This settles the question I had in school wondering why the derivative of volume of sphere \(V(r) = \frac{4}{3}\pi r^3\) gives the surface area of the sphere \(A(r) = 4 \pi r^2\). Steiner formula along with Brunn-Minkowski theorem easily establishes isoperimetric inequality for convex sets.

A quick road to Steiner formula at least for \(C^2_{+}\) bodies can be traversed as follows. I will be following notation from Extrinsic Geometric Flows which in turn is from Andrews's paper Entropy estimates for evolving hypersurfaces. For every compact strictly convex body \(K \subset \mathbb{R}^{n+1}\) with non-empty interior, the Gauss map \(G : \partial K \to S^n \) is a diffeomorphism from the hypersurface to sphere. Let \(\overline{\nabla}\) denote the standard connection on \(S^{n-1}\) with metric \(\overline{g}\). The inverse to Gauss map is given by \(X(z) = \sigma z + \overline{\nabla}\sigma\) where \(\sigma : S^n \to \mathbb{R}\) is the support function. This tells us that all information of the hypersurface is contained in the support function. This is exploited a lot in the book. The operator \(A[\sigma]_{ij} = \overline{\nabla}_i\overline{\nabla}_j \sigma + \overline{g}_{ij}\sigma \) is the inverse of second fundamental form. Its eigenvalues at \(z\) are the principal radii of the hypersurface at \(X(z)\). By divergence formula on the position vector \[V(K) = \frac{1}{n+1}\int_{\partial K} X \cdot \nu d \mu = \frac{1}{n+1}\int_{\partial K}\sigma d\mu = \frac{1}{n+1}\int_{S^n}\sigma \det A[\sigma] d \bar{\mu}\] where the last equality uses the fact that determinant of the Gauss map is the Gauss curvature. Also recall that support function is additive under Minkowski addition. This means that we have a Steiner formula staring at us right now (and a little more). It is easy to see that \(V(K+tL)\) is a polynomial of degree \(n+1\) in \(t\). Moreover we can polarize it! Formally define \(V(K_0, \cdots, K_{n})\) as the coefficient of \(t_0 \ldots t_{n+1}\) in \(V(t_0 K_0 + \cdots + t_{n}K_{n})\). This coefficient will called the mixed volume of \(K_{i}\)'s and is totally symmetric. Algebraically, this will be the intgeration of mixed discriminant of principal radii of \(K_1, \ldots, K_{n}\) with support function of \(K_0\) (or any other order) like volume was integration of support function and product of principal radii. In formula \[V(K_0, \cdots, K_{n}) = \frac{1}{n+1} \int_{S^n} \sigma_{K_0} Q[\sigma_{K_1}, \ldots, \sigma_{K_n}]d \bar{\mu}\] where \(Q[\sigma_{K_1}, \ldots, \sigma_{K_n}] = \frac{1}{n!}\sum_{\tau, \eta \in S_n}\text{sgn}(\tau)\text{sgn}(\eta)A[\sigma_{K_1}]_{\tau(1)}^{\eta(1)} \ldots A[\sigma_{K_n}]_{\tau(n)}^{\eta(n)} \).

A special case of this formula gives quermassintegrals. The \(j\)-th quermassintegral is defined as \(W_j(K) = V(\underbrace{K,\cdots, K}_{n-j \text{ times}}, \underbrace{B, \cdots, B}_{j \text{ times}})\) which make up the other coefficients in Steiner formula. A grand generalization of Brunn-Minkowski inequality is the Alexandrov-Fenchel inequality \[V(K_0,K_1, \cdots , K_{n})^2 \ge V(K_0,K_0,K_2, \cdots, K_{n}) V(K_1,K_1,K_2, \cdots, K_{n})\] which is like a reverse Cauchy-Schwartz. Minkowski inequality \(V(K,L, \cdots L) \ge V(K)^{\frac{1}{n+1}} V(L)^{\frac{n}{n+1}} \) also follows. Schneider in his book Convex Bodies: The Brunn-Minkowski Theory mentions that Alexandrov gave two proofs (one only for \(C^2_+\) bodies) while Fenchel independently stated and proved but his proof was sketchy (IMO it should be called Alexandrov-Alexandrov inequality for this reason). Alexandrov-Fenchel inequality also appears in algebraic geometry somehow as Hodge index theorem. Also, the fun doesn't stop here. Convex geometers have pushed the definition of mixed volumes for \(L^p\) additions of symmetric support functions (even for \( 0 < p < 1\)) and proved Minkowski inequalities in \(L^p\)-Brunn-Minkowski theory as well.

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 is 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.

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.