我們愛老師 老師愛我們

New abbreviations:

[dC] do Carmo: Riemannian Geometry

[GHL] Gallot, Hulin, Lafontaine: Riemannian Geometry

(一)

The only thing I did last week is trying Schauder interior estimates. I decided long before the boundary estimate would not be in my syllabus. I could never do it right. (I hope I don't have to treat manifolds with boundary in my life.) The version I want is GT Theorem 6.2, not just Corollary 6.3. But I relax the assumption so that the proof can be carried out myself. First, I let $|a^{ij}|_{0, \alpha}, |b^{i}|_{0, \alpha}, |c|_{0, \alpha} \leq \Lambda$, not the weighted norms of the coefficients. Second, I allow the constant $C$ depend on diam($\Omega$). The proof is almost the same. I still need the scaling-invariant interior estimate of Poisson equation, Theorem 4.8 and the interpolation, Lemma 6.32. I feel that these two results are indispensable. If you didn't get this accurate version, the proof should contain gaps. For example, the exposition in Jost's book. (to be continued.)

I was used to think only ellipticity gives us a priori estimates. That's why I was surprised seeing Chen-Yun use the Schauder estimates and Holder estimates for parabolic equation last week. Although we could only get $C^{1, \frac{\alpha}{2}}$-estimates in $t$-variable(by the twice-space-derivative-equal-one-time-derivative principle), but in the equation we only have the information of $\frac{\partial u}{\partial t}.$ Does some miracle happen in parabolic equation? Unfortunately, Liebmann looks unreadable.

(二)

Calculate the curvature of $\mathds{C}P^{n},$ viewed as $SU(n+1)/U(n)$ or $S^{2n+1}/S^{1}.$ This should be the most intimidating example for me. It's frustrating while everybody is studying Weil Conjecture but I'm still learning to calculate curvatures, although it's not easy at all. We could make the problem more difficult. Can you show the curvature tensor of $\mathds{C}P^{n}$ is parallel(an exercise in P)?

I also tried exercise 2.90 in GHL about geodesics of Heisenberg group. As before, I finally give up and turn to the answer at the end of the book. The ODE here is not that hard to solve, but I can't think of using spherical coordinate to represent the initial conditions.

(三)

For some reason, I deviate from the orals and study elliptic functions in Stein and Shakarchi's book. Both the topic and their presentation are beautiful. I learned this stuff before but again feel 嘆為觀止 after every ingenious trick of Weierstrass. I would put him as my second favorite Mathematician temporarily. The first place is still Riemann.

The best thing is

Every elliptic function can be written as the rational function of $\wp$ and $\wp'$

I also love the construction of Weierstrass $\wp$ function so much.