我們愛老師 老師愛我們

M: O, you may start to ask questions.

O: But this is my first time in a committee.

M: That's fine.

O: Can I start with Riemannian geometry? (Unusual. Professors always begin with the major, PDE in my case.)

O: Can you state the first and second variation of length?

I did the variation of energy and said the one for length is the same as long as the curve is a geodesic.

O: Can you use that to prove Cartan-Hadamard theorem?

(OS: Shit. This is the easiest theorem in comparison geometry but I know I forgot to review it in my final preparation.)

I have to prove there is no Jacobi fields vanishing at both endpoints. I remembered Petersen calculate

$\frac{1}{2}\frac{d}{dt} \langle J, J \rangle = \langle J, J' \rangle$ and $\frac{1}{2} \frac{d}{dt^2} \langle J, J \rangle = \langle J', J'\rangle + \langle J, J'' \rangle.$ Then I'm stuck. Now the guardians come in.

M: You haven't defined the Jacobi field.

Using the Jacobi equation and nonpositive sectional curvature, $\langle J, J \rangle$ is increasing. It's the proof on Petersen. But I totally forgot it.

M: What can you say if you have a variation along a Jacobi field?

If $J(0) = J(l) =0,$ then $ E''=0.$ (The integrand of E'' is $\langle J, J'' \rangle- \langle J, R(J, \gamma')\gamma' \rangle$)

O: You can also write the integrad as $ \langle J', J' \rangle - \langle J, R(J, \gamma')\gamma' \rangle$

Hence $E''>0$, a contradiction.

M: Can you state the Hessian and Laplacian comparison theorem?

I gave the statement when $r$ is smooth.

M: What can you say when $r$ is not smooth?

I remember this well. The inequality holds in the sense of distribution.

M: Now show the proof of Laplacian comparison.

Using the Riccati equation, it's easy to derive a second order differential inequalityfor $\triangle r$ and it is an equality for $\triangle r_k$, the distance function of the space form. Then compare the ODE. Actually, I made a wrong calculation at that time. Then I say it's easy to show it's an equality for $\triangle r_k$. They didn't notice that. Nobody wants to see the comparison of ODE. (In fact it takes some work to show $\triangle r$ and $\triangle r_k$ has "the same" initial value.)

M: Prove the distributional inequality.

I'm glad that I have prepared it although I thought no one would ask. But

M: Why $\langle \nabla r, \vec{n} \rangle  \geq 0$? ($\vec{n}$ is the outward normal of a star-shaped domain.)

It looks trivial, but I'm stuck again. (I still don't know how to prove it.)

This finishes my minor.

P says nothing in the first part.

M: Let's go to PDE.

Now it's M and O who don't ask questions anymore. P takes full charge of my major.

P: Suppose you have a second order elliptic equation with smooth coefficients. Please state the Schauder estimate.

I use the version in GT. So I first defined the weighted norm and the estimate in terms of it. I assumed the domain is $B_1(0).$

P: Please prove it for a compact subset.

To be homest, about one hour before the oral, I realized that I totally forgot how to prove Schauder estimate. As I mentioned in Oral report 1, I adjusted the proof in GT so that it looks much easier. But my proof just disappeared and it hasn't come back after the oral. I started to doubt if I really have a right proof.

But the only thing I could do at the moment is going into the proof and praying P would stop me before he found that I don't have a proof. This is possible. The perturbation seems so natural that it is surprsing how careful we have to be to carry out the detail.

P wanted to make the life easy. He said I only have to prove the estimate for $B_{1/2},$ not the weighted version. But I really have no idea how to get that directly without proving the weighted one.

We spent some time on that. Fortunately, after I said $a^{ij}(x_0) - a^{ij}(x)$ is small when $x - x_0$ is small, P stopped me. I don't know if it's because he didn't want to embarrass me.

Next he asked me to prove the constant coefficient case, $\triangle u = f.$ Another thing I thought they wouldn't ask. I remembered the formula $D_{ij} w = \int_{B_1} D_{ij} \Gamma(x-y)(f(y) - f(x)) - f(x) \int_{\partial B_1} D_i \Gamma(y) \nu_j(y)$ which makes me not so ignorant.

P: Viewing $D_{ij} \Gamma$ as a kernel. It is homogeneous of order -n, smooth except at 0. What's the most important property of it?

The answer is the integral is 0 on any annulus. This is everything I know. I don't remember GT use this property. P is such an authority that everytime he changes the topic is because I said something wrong. Fortunately, he didn't ask me to prove the estimate.

P: Let's turn to $L^p-$ estimate.

This may be the part I performed best in the whole oral, although still not so organized as I expected. After stating the result. The proof is Calderon-Zygmund inequality + perturbation argument.

P: Prove the C-Z inequality for p=2.

I'm a little surprised. Does he forget it's equality for p=2 and integration by parts suffices? Anyway, this part is easy. Of course the main issue is the case 1<p<2. We need Marcinkiewicz interpolation and estimate of distibution function.

P: Since we are running out of time, why don't you quickly go through the proof?

So I proved the inequality $\mu_{D_{ij}w}(t) \leq \frac{C \|f\|_{L^1}}{t}$ and no big trouble happened.

That's all. The exam usually takes 2 hour, but mine only last for 1 hour 50 minutes. Of course staying longer only means I'm more likely to make mistake. In the whole exam, not many problems were completely solved by myself. One tiny regret is they didn't ask me Holder continuity of weak solutions which I spent the most of time to prepare.(Moser iteration, Harnack inequlity, the final argument.) During the whole oral, I pretended to appear nervous. I hope they think the reason I can't answer the questions is I'm too nervous. After all, I wrote the syllabus by myself and it's hard to explain why I forgot preparing certain theorems on it. In this regards, it's not a satisfactory oral. I missed the chance again to change my life.

考完後我問 M 我哪裡需要改進，他說沒什麼問題(屁啦)，藉著口試把基本的東西弄得很熟也不錯。(既然我很熟，那為什麼會卡住？)

但接下來是美好的一刻。他說可以做研究了，做研究跟唸書很不一樣，得非常 flexible，懂得找問題，去 challenge 問題，我們慢慢開始吧。

我不知道以後我的學術生涯會怎樣，很懷疑我是不是做研究的料，但很謝謝他願意講這番話為我開場。