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 $c\in R$ m 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.