Fruits of procrastination

Tag: Geometric Analysis

Day 3- Conference on Geometric Analysis

Notes from the third day of the NSF conference on Geometric Analysis at Princeton University are given below.

Relation of Alain-Cohn equations with minimal surfaces– The first talk of the day was given by Marco Guaraco. The theme of the talk was finding a function u that satisfies a particular PDE, and then making that u converge to a minimal surface in a nice way. Remember that yesterday we determined that every function that satisfies the minimization of energy condition need not be a minimal surface. Hence, it is not obvious that u itself would be a minimal surface. However, we can make it converge to the minimal surface.

The speaker wrote a nice set of notes on his talk, which pretty much contain all that he talked about and more. Hence, I am not going to write notes for this talk, although there seem to be a couple of misprints that I could have elaborated about.

Harmonic maps to metric spaces– The second talk of the day was given by Christine Breiner.

Let u:(M,g)\to (N,h) between Riemannian manifolds, and define E^M=\int_M |du|^2 dvol_g (this could perhaps be thought of as a form of energy). The critical points for E are harmonic maps (which means that as we vary u, the functions that are stationary points are harmonic maps). This is clearly a variational problem. Some examples are geodesics, harmonic forms, and totally geodesic maps.

There is a theorem by Ahlfors-Bera ’60 and Morrey ’38 which states that if g is a bounded, measurable Riemannian metric on S^2, then \exists an almost conformal homeomorphism u:(S^2,g_0)\to (S^2,g). Here I suppose g_0 is the metric induced on the sphere from Euclidean space. A question one can then ask is, if (X,d) is a geodesic space that is homeomorphic to S^2, is there a quasi-symmetric or quasi-conformal homeomorphism u:(S^2,g_0)\to (X,d)? Note that we have weakened conformal to quasi-conformal. We have put in a homeomorphism, but taken away boundedness of d.

There are some partial results in this direction. If (S^2,d) is a compact, locally CAT(1) space and u:(S^2,g_0)\to (S^2,d) has finite energy, then \exists an almost conformal harmonic homeomorphism. Note that we don’t have boundedness of d here. However, X being a CAT(1) space suffices. But what is a CAT(1) space? It is a complete geodesic space if \forall geodesics with perimeter<2\pi, the comparison triangles on (S^2,g_0) are “fatter”. One way to think of this is that CAT(1) spaces are “less curved” than S^2 under the usual metric.

The speaker then goes on to talk about other things that I could not fully understand, including the following definition: for u\in L^2(M,X), e^n_u(X)=\int_{S^{n-1}} \frac{d^2(u(x),u(y))}{\epsilon^2}\frac{d\epsilon}{\epsilon^{n-1}}. The point that I did understand is the following: a map is harmonic if it’s locally minimizing. Moreover, CAT(1) spaces are hugely useful in this area as they crop up everywhere where we do not have a bounded metric.

K-stability– The third talk of the day was given by Sean Paul from the University of Wisconsin Madison. Let (X^n, \omega) be a compact Kahler manifold, where \omega is \frac{\sqrt{i}}{2\pi}\sum\limits_{i,j} g_{ij}dz_id\overline{z}_j. Clearly, such a form can only be defined on an even dimensional manifold. X^n just denotes that it is n-dimensional, and not an n-product of X.

Let us define \mu=\int_X scal(\omega) \frac{w^n}{V}. The reason why we have \omega raised to the nth power is that we want to create a volume form, as that is the only way that we can integrate over the whole manifold. V perhaps is the volume of X. Hence, we want some sort of a normalized integral of Scal(\omega).

Let us now define U_{\omega}=\{q\in C^{\infty}(X)|\frac{\sqrt{i}}{2\pi}\partial\overline{\partial}q>0\}. One also refers to this as the set of Kahler metrics on [\omega]. Let us define \omega_{\phi}=\frac{\sqrt{i}}{2\pi}\partial\overline{\partial}\phi. An important open question is: does there exist an function \phi\in U_{\omega} such that scal(\omega_{\phi})=\mu? We are integrating on the right, and hence we’re sort of taking an average (the division by the volume of X is but a trivial calculation, and let us assume we do that here). On the left, we are just finding the scalar curvature of a particular function. So does there exist a function whose scalar curvature models the average curvature of the manifold X?

Let us now perhaps place some extra conditions on X to make it more amenable- let us assume it is Fano, which means it has positive curvature. Define the function V_{\omega}=\frac{1}{V}\int_X \log (\frac{\omega^n_{\phi}}{\omega^n})\omega^n_{\phi}-(I_{\omega}(\phi)-J_{\omega}(\phi))+O(1). We want to prove that it is bounded below, as we vary the function \phi. Although it may be negative, it cannot go to -\infty. Note that I_{\omega}(\phi)-J_{\omega}(\phi) is always positive. This is just an aside, and perhaps only increases the chances of V_{\omega} being unbounded below.

It is a theorem of Tian’s from ’97 that V_{\omega} is positive on H_{\omega} iff \exists constant A,B such that V_{\omega}(\phi)\geq AJ_{\omega}(\phi)-B. Here the speaker notes that J_{\omega}(\phi)\sim\int |\nabla\phi|^2\frac{\omega^n}{V}, in other words, a rescaled average norm of first derivative.

If V_\omega is not bounded below, then \exists \{\phi_i\} such that V_\omega(\phi_i)\to -\infty. We have to somehow find a way to contradict this.

Remember that X is a Fano variety, which means a variety with positive scalar curvature. Let G=SL(n+1,\Bbb{C})=Aut(\Bbb{P}^n). It turns out that G\hookrightarrow H_\omega. How can a matrix be embedded into a set of functions? You make each elemeent of G act on \Bbb{P}^n. That makes is a function. As X itself is also a subset of \Bbb{P}^n ($X$ is a variety, and hence a subset of \Bbb{P}^n, G\hookrightarrow H_\omega makes sense.

It turns out that proving V_\omega is bounded below in G implies that it is bounded below in H_{\omega} also. This is because for k>>0, there exist B_k=\{\frac{\phi_{\sigma}}{k}|\sigma\in SL(N_{k}+1,\Bbb{C})\} such that \cup_{k>>0} B_k=H_\omega. I don’t understand the notation \phi_\sigma.

The speaker then goes on to discuss related results, like d^2V_{\omega}(\phi_{\lambda(t)})=\#(\lambda)\log|t|^2+O(1). As far as my understanding of the talk goes, the speaker did not state that the above theorem had been proven, but only talked about possible approaches that one could take to prove it.

Mean Curvature Flow– The last talk of the day was given by Lu Wang. Given an arbitrary curve in R^2, let its mean curvature at a point be denoted be the vector \overrightarrow{H_\Sigma}(t). Then \frac{d}{dt}vol(\Sigma_t)=-\int_{\Sigma_t}|\overrightarrow{H}_{\Sigma_t}|. The rate of change of volume in gradient flow is proportional to the integral of the mean curvature on manifold. Sounds intuitive enough.

It turns out that “maximal surfaces are stable solutions”. What I think this means (although I cannot be sure) is that during gradient flow, the manifold ultimately becomes this maximal surface.

The speaker then goes on to give examples of various kinds of gradient flows- that of a sphere contracting to a point, a cylinder contracting to its axis, etc. As one can see, contraction further increases curvature, that only accelerates the rate of contraction. Hence, contraction to the final state only takes finite time. One may also think of the example of a two dimensional dumbbell, which can be thought of as two spheres connected by a long narrow rod (with 0 curvature). The two spheres soon separate. In fact, it is a fact that closed surfaces form singularities in finite time. Whilst one may think that the rod does not contract, and hence the two spheres dissociate from the rod, that is false, to maintain continuity, the rod does in fact contract with the spheres (continuity is maintained until the formation of the singularity).

We shall now discuss the Avoidance Principle, which states that if we have two hypersurfaces \Gamma_0 and \Sigma_0 (one may imagine them as two shells in a space that is of one higher dimension), and one is contained within the other (one may think of two concentric circles), and \Gamma_0\cap \Sigma_0=\emptyset, then \Gamma_t\cap \Sigma_t=\emptyset for all t. This is because the inner hypersurface has higher curvature than the outer one, and hence the gradient flow for it is faster. Although pointwise mean curvature may not always be larger for the inner hypersurface, the rate of change of volume depends on the total integral of the mean curvature (hence the average mean curvature in some sense). And the average mean curvature of the inner hypersurface is definitely higher.

Now what about surfaces with singularities? How do they exhibit gradient flow? Imagine a cone (flat sides, and vertex with \infty curvature. One can show that such a cone also shows well understood gradient flow in which the vertex smoothens out, and the flat sides also become more rounded. The speaker goes on to explain this using the concept of expanders.

Consider the following integral \frac{d}{dt}(\int_{\Sigma_t} (4\pi t)^{-n/2} e^{\frac{|x|^2}{4t}}), which may be thought of as the rate of change of the rounded cone. This does not make sense, as the integral blows up at t=0. However, we can correct it by subtracting \int_{\Gamma_t} (4\pi t)^{-n/2} e^{\frac{|x|^2}{4t}}. Hence, the final expression that we have is \frac{d}{dt}(\int_{\Sigma_t} (4\pi t)^{-n/2} e^{\frac{|x|^2}{4t}}-\int_{\Gamma_t} (4\pi t)^{-n/2} e^{\frac{|x|^2}{4t}}). This is equal to \int_{\Sigma_t}|H_{\Sigma_t}-\frac{X}{2t}|^2(4\pi t)^{-n/2} e^{\frac{|x|^2}{4t}}.

As \to\infty, \frac{1}{\sqrt{t}}\Sigma_t\to\Gamma. Why do we need to divide by \sqrt{t}? I suppose that if we were to consider a gradient flow of a cone, it would keep expanding into something larger and larger. To somehow control the size of \Gamma_{\infty}, we divide by \sqrt{t}. Note that in this case, gradient flow does not speed up with time, as the curvature keeps on reducing. Hence, that phenomenon is only valid in closed curves in {R}^2.

The speaker then also talks about the fact that for generic cones \hat{C}, there will be a sequence of expanders \Gamma_0\leq \Gamma_1\leq \Gamma_2\leq\dots which are alternately stable and unstable. But isn’t gradient flow supposed to stop in finite time? No, for a cone, as we saw above, gradient flow continues for infinite time, although at a slowing rate. The \Gamma_n for n\in\Bbb{N} just denote the various phases that the cone passes through. What the the stability (or lack of it) of \Gamma_n mean? This I am not sure of.

I will try to record the talks that take place tomorrow too.

Day 2- Conference on Geometric Analysis

I am presenting my notes on today’s talks below. The talks were generally harder to follow than yesterday, not least because some of them were slide shows, and hence were prone to being thrown at the audience at brain-breaking speed.

Free Boundary Minimal Surfaces– The first talk today was by Lucas Ambrozio. Because it was a slide show talk, it sped by slightly fast, and I couldn’t take as many notes as I would have wanted. However, I will try to reproduce my impression of the talk.

Free Boundary Minimal Surfaces (we will call them FMBC’s from now on) are immersed hypersurfaces (M,g)\hookrightarrow (N^{n+1},\overline{g}) such that \partial M\subset \partial N, and M intersects N orthogonally. Moreover, image(M)\cap \partial N should not be contractible. As an example, one may imagine a circle along the boundary of a solid torus.

The first question of the day was whether out of all FMBC’s in a given manifold N, there exists one that minimizes the n-1 dimensional volume.

As simple examples, we will consider FMBC’s in the unit ball B_1^3(0)\subset R^3. One FMBC is any equatorial disc. Any other disc will not intersect the boundary orthogonally. Another example is the critical catenoid. Any catenoid is a minimal surface. However, only the critical catenoid intersects the boundary of the ball orthogonally. For any ball, there exists a unique critical catenoid.

The speaker then talked about the desingularization of intersecting minimal hypersurfaces. For instance, if we took the union of two intersecting equatorial discs in the unit ball, then we can smoothen out the area of the intersection by punching in some nice rounded holes (increasing the genus). We can do similar things for the intersection of the catenoid and an equatorial disc.

Let \gamma be the genus and r the number of boundary components of an FMBC. An open question is: Can we find an FMBC for any (\gamma, r)? This question seems like a hard, overarching question of the field.

We will talk briefly about Skeklov eigenvalues, which will play a role in the next section. Let \phi be a function on an embedded hypersurface \Sigma in N. Let \hat{\phi} be its harmonic extension beyond the hypersurface. Now consider the function \partial\Sigma\to \frac{\partial \hat{\phi}}{\partial \nu}. We now construct a functional involving \phi and \frac{\partial \hat{\phi}}{\partial \nu}. The minimal value of this functional across all possible functions \phi is the first Steklov eigenvalue.

The speaker answered some related questions to the open question discussed above. We can distinguish between various FMBCs of the unit ball by placing further constraints on the properties of the desired FMBCs. For example, if we require that the FMBC be totally geodesic, homeomorphic to a disc, of least possible area and of least possible Morse index (1), then the FMBC has to be the equatorial disc. Similarly, if we require that the FMBC be homeomorphic to an annulus along with either being immersed by Steklov eigenfunctions, being symmetric with respect to reflection across coordinate planes, or have a Morse index of 4, then such an FMBC has to be the critical catenoid. These theorems are important because all the FMBCs in the unit ball have not yet been classified.

The author turns this into a variational problems, given certain constraints. He proved the following: consider M^2 to be an FMBC in the unit ball in R^3. If \phi(x)=|x^{\perp}|^2A^2(x)\leq 2 for all x\in M, then \phi can only take two values. Either \phi=0, in which case the FMBC is an equatorial disc, or \phi=2, in which case it is the critical catenoid. Note that here we’re talking about the nature of the immersion of M^2 inside B_0^3(0). M^2 might as well not be an FMBC at all. However, if \phi(x)=|x^{\perp}|^2A^2(x)\leq 2, then it has to be an FMBC.

Ambrozio then talks about a quadratic form Q(\phi,\phi), whose exact form is not important for our current elementary discussion of the topic. Let index(M) be the dimension largest subspace of C^{\infty} where Q(\phi,\phi) is negative definite (including 0 of course). He then goes on to state the following result: if \Omega\in R^{n+1} is smooth, and n=2, then the FMBC M satisfies the following condition: index(M)\geq \frac{1}{3}(2\gamma+r-1). If n>2, there is a similar result with a more complicated right hand side.

He then goes on to state a result by Tran from 2016: an FBMS in the three dimensional unit ball that has index four must be star shaped with respect to the origin. Clearly, the critical catenoid satisfies these conditions. However, there might be other FMBC’s with index four too, as homeomorphism with annulus has not been assumed. However, if index(M)\geq \frac{2}{3}(r-1)+3, then the critical catenoid are the only FMBC’s.

Ambrozio then goes on to state another theorem: let us take a three dimensional Riemannian manifold with non-negative Ricci curvature, and \Omega a strictly convex domain in it. Then the set of FMBC’s in \Omega with \gamma,r bounded above by the same bound C form a compact set. A corollary of this is the fact that the set of FMBC’s with Morse index bounded above by C also forms a compact set. Somehow, Morse index captures information from both \gamma and r. The quadratic form used to define the Morse index probably somehow contains information about the boundary too.

We now generalize this to a manifold of dimension higher than three, which inevitably brings a constraint with it. Let M^n be a Riemannian manifold with non-positive Ricci curvature, and 2\leq n\leq 6. Let \Sigma be a compact, strictly convex and smooth domain of M^n. Then the set of FMBC’s in \Omega with both Morse index and area bounded above by C, form a compact set.

Volume preserving stability of spherical space forms– The second lecture of the day was given by Celso Viana. The motivation for the talk was the isoperimetric problem: given the perimeter of a hypersurface (area (\partial \Sigma)), what is the maximum volume that it can enclose? This is a classical problem, and lots of advances in geometric analysis are made while attempting to solve this problem in various contexts. We shall assume that we are looking for solutions with constant mean curvature (or maybe the solution to the isoperimetric problem always happens to have constant mean curvature. I am not sure).

Let us consider an embedding [X, (0,\epsilon)]\to M (essentially again reducing this to a homotopy-type argument). Note the change in notation: here X is the hypersurface being immersed in M. Then \frac{d }{d t}|_{t=0} Area= -n\int Hfd_I, where H is the mean curvature and f=\langle \partial_t X,N\rangle. Note that H is not assumed to be constant. One may ask that even area seems not to be constant here. That is true. However, the area given to us is achieved at t=0. We are just looking for the rate of change of area at t=0 to study the properties of the volume enclosed by X. If this rate of change satisfies certain properties, we will know that the volume enclosed by X is maximal.

It turns out that V'(0)=\int f. Hence, if \int f=0, then the volume enclosed by X is extremal. I would imagine that f being identically 0 would be a condition for the volume being extremal. They have weakened that condition, to just the integral being 0, here.

An interesting fact here is that A''(0)\geq 0 whenever volume is extremized. Hence, whenever volume is maximal, area is minimal, which would imply that if volume is held constant, this is the smallest perimeter that X can have.

It is a famous theorem by Barbara Cormo(?)-Eschenburg that if \Sigma is a stable constant mean curvature hypersurface in R^n, S^n or H^n, then \Sigma has to be a geodesic sphere. By stability, I think they are referring to the stability of the area, as the second derivative being positive implies that the area there is a stable critical point.

In dimension 3, Rutone-Ros proved in 1992 that if \Sigma is a stable constant mean curvature torus in R^3/\Gamma, S^3/\Gamma or H^3/\Gamma, then \Sigma is flat. An example of this would be the Clifford torus embedded in S^3. This example tells us that the Clifford torus is of maximal volume given the perimeter.

The speaker then goes on to talk about holomorphic forms on manifolds, but I did not understand the relevance of these topics to solving the isoperimetric problem. I will try and read his paper to find out more.

Wave Equations and Conformal Geometry– The third talk of the day was given by Sagun Chanillo. He talks about the functions u on R^2 defined as u:R\times R^2\to R^3 satisfying the equation -\partial_t^2u+\Delta u=2u_x\wedge u_y. As the solution is basically defined on two-dimensional space (the third coordinate being time), it looks like a two-dimensional space being embedded in R^3.

Another function that we may consider is the following: u: R\times S^2\to R satisfying the equation \partial_t^2u-\Delta_g u=\alpha(\frac{e^{2u}}{\int_{S^2}r^{2u}}-1). Here, \alpha is a constant, and g is the standard metric on the sphere.

We may want to impose extra conditions on the shape of u, like making it the solution of an elliptic equation, and also a plateau, which would imply |u_x|=|u_y|=1 and u_x\wedge u_y=0. As solutions to elliptic equations are constant over time, the wave equation becomes \Delta u=u_x\wedge u_y. For the Liouville equation too, one might take the special case of \alpha=1 and \int_{S^2} e^{2u}=1, which would make the equation -\Delta u=e^{2u}. Interesting fact: If u is indeed a solution to such a specialized Liouville equation, then e^{2u}g is another metric on the sphere with constant curvature such that the area of the sphere is 4\pi.

Coming back to the general wave equation without plateau or elliptic assumptions on u, we define the conserved energy for “nice solutions” to be \int_{R^2} \frac{1}{2} (|\partial_t u|^2 + |\nabla u|^2)+\frac{2}{3}u.(u_x\wedge u_y). The first term on the right is some kind of energy term, and the second term is the volume term.

When does a global solution of the wave PDE fail to exist? Let u_0 be the part of u that depends only on time coordinates, and u_1 the part that depends only on spatial coordinates. Then if E(u_0,u_1)< E(W,0) and \|\nabla u_0\|_{L^2}\leq\|\nabla W\|_{L^2}, then u blows up in finite time. The existence of a global solution implies that u is finite at all spacial points for all times. It seems that W too only depends on the time coordinate, although the definition of W was not given before. Maybe W encodes some information about the energy bound for a system (beyond which the solution might blow up), and hence it makes sense that it only depends on time. This condition suggests that if the rate of increase of the energy of the solution is greater than the rate of increase of the upper bound of the energy, then despite being lower than the bound at first, within finite time, the solution will blow up. This makes intuitive sense. If E(u_0,u_1)>E(W,0), then the solution would not have existed even at t=0.

The speaker then goes on to talk about self-similar blow up: assume that there exists a solution u(x,y,t)=v(\frac{x}{t},\frac{y}{t}) for some v\in C^2(R^2). Then v is constant in the unit disc. This basically means that at any given point, there comes a time t_0 such that for t>t_0, u does not change. Moreover, every point comes to have the same value for u as t\to\infty. This is a much stronger situation than u not blowing up. From now on, we shall consider only those u for which there exists such a v. In other words, we shall only consider solutions that do not blow up in finite time.

We shall now discuss Bourgain spaces H^{s,b}. All functions that satisfy a particular integrability condition of their Fourier transforms are part of H^{s,b}. A function u belongs to \mathcal{H}^{s,b} if u\in H^{a,b} and \partial_t u\in H^{s-1,b}. The reason we want to use \mathcal{H}^{s,b} is that estimating energy of u becomes very easy using the \mathcal{H}^{s,b} norm.

The speaker then goes on to discuss the cases in which u does exist, and proof of existence involves using the concept of Bourgain spaces, and also using randomization to overcome the difficulties faced whilst using traditional methods to prove the existence of u.


Min-max construction for Constant Mean Curvature hypersurfaces– Much like yesterday, the last talk of the day provided the most intuitive and easy to understand results. The talk was given by Xin Zhou.

By now, we have had multiple talks on the fact that finding a hypersurface enclosing the largest volume, given a perimeter, can be phrased as a variational problem. The speaker here actually phrased the question of finding a hypersurface (we are going to refer to such a hypersurface as \Sigma from now on) with constant mean curvature (CMC) as a variational problem. One’s intuition might suggest that a surface with CMC would extremize some property, like area of boundary, or volume enclosed inside. We are going to test this intuition and see how far it takes us.

The speaker then gets into the specifics. If a disc D^2\subset R^2 is embedded into a hypersurface in R^3 by the map \phi, then its image has CMC if \Delta \phi=C\phi_x\wedge \phi_y and |\phi_x^2|-|\phi_y|^2=0=\phi_x.\phi_y. This is the same as the plateau condition in the previous lecture, aisde form the fact that |\phi_k| do not necessarily have to be 1. Moreover, the following term is also minimized: E(\phi)=\frac{1}{2}\int_D |\nabla\phi|^2 (Energy term) + \frac{2}{3}c\int\phi.(\phi_x\wedge \phi_y) (volume term). Hence, although the volume or energy might not be individually minimized (our intuition would suggest that both are minimized at the same time), their sum is indeed minimized for a CMC. Intuition not being exactly right, but being right in a weak form, is a feature of much of variational calculus.

Methods to find CMC’s are:

1. Solving the isoperimetric problem- it would appear that \Sigma that encloses the maximum volume given a perimeter has CMC. This agrees with the simple fact that a ball encloses the largest volume of air when it is fully inflated (of course we are assuming that the surface of the ball cannot be stretched to increase the area).

2. Perturbation method- Start with a hypersurface, and keep on perturbing it at various points until you get something with CMC.

3. Glueing method- one may also glue together CMCs in a “nice way” to create larger CMCs.

An earth-shattering result of Zhou’s, from 2017, says the following: for 3\leq n+1\leq 7, for any given c\in Rm there exists a smooth, closed, almost embedded hypersurface \Sigma\subset (M,g) such that H_{\Sigma}\equiv c. An almost embedded hypersurface is one that may not be completely embedded, and may hence have self intersections. However, at the self intersection points, the hypersurface decomposes into two parts. Two spheres touching is an example, and the graph of y^2=x^2(x+1) is not an example. Another condition is that the hypersurface is the boundary of an open set, and also has multiplicity 1.

Then the speaker talks about the method of getting such a hypersurface with CMC. \Sigma has H\equiv c\iff \Sigma is a critical point of A^c(\Omega)= Area(\partial\Omega)- c vol(\Omega). Here again, our intuition is only correct in a weak form. We wold have expected both the area and volume to have been extremized at the same time for the existence of a CMC. However, it is only their difference that has to be extremized. Hang on. That’s not completely correct either.

A^c doesn’t have to be extremized: the speaker said that A^c can be varied with respect to two independent parameters (I don’t remember which). The saddle point of this variation is the one corresponding to the hypersurface with CMC; in other words we determine L^c= \inf\max A^c. It is the hypersurface corresponding to this L^c that gives us the CMC hypersurface.

Zhou then goes on to state an even more earth shattering result: given ANY function h:M^n\to R, he can create a hypersurface \Sigma^n in M^{n+1} with the mean curvature being modeled by the function h. This is a vast generalization from just the constant mean curvature case. He proves it by proving the following: \Sigma^n\subset M^{n+1} has a prescribed mean curvature \iff  \Sigma is a critical point of A^n(\Omega)= Area(\Omega)-\int_{\Omega} h dvol. He then probably proved the existence of this critical point by min-max considerations.

I will try to write notes for the talks tomorrow as well.