# 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 $n$th 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.