Hello. I'm reading the paper "Proof of the positive mass theorem II" by Schoen and Yau. I have some questions.

Question 1

In page 236, they used the Simons' identity $$ \begin{aligned} \Delta h_{ij} = & \bar{D}_{i} \bar{D}_{j} H+ D_{k} R_{4ijk} + D_{j} R_{4kik} - |A|^2 h_{ij} + H h_{im} h_{mj} \\ & -2 h_{mk} R_{mikj} - h_{im} R_{mkkj} - H R_{4ij4} - R_{4k4k} h_{ij} + R_{mkik} h_{mj} \end{aligned} $$ and the falloff conditions (1.1) (that is, $|g_{ij}-\delta_{ij}|+r|\partial g_{ij}|+r|\partial g_{ij}|+r^2|\partial\partial g_{ij}|\leq c$) to obtain the following inequality (2.13) (see P 237): $$ \frac{1}{2} \Delta |A|^2 \geq |\bar {D} h|^2 - |A|^4 - |H||A|^3 + h_{ij} \bar {D}_{i} \bar {D}_{j} H - c_{2} (|A|^2 + 2) $$

Note that the derivatives of the Riemannian tensors appear in the Simons' identity, while the falloff conditions do not say anything on the third derivatives of the metric. My first question is how do they control the terms $D_{i} R_{4ijk} + D_{j} R_{4kik}$ ?

Question 2

They used the standard continuity method to solve the auxiliary Jang's equation $H(f)-sP(f)=tf$ in Lemma 3. In this proof, they constructed the operator from $B^{2, \beta}\times R$ to $B^{0, \beta}\times R$ defined by $T(f,s) = (H(f)-tf-sP(f),s)$. Here, $f$ belongs to $B^{2, \beta}$ or $B^{0, \beta}$ if the following norms are finite respectively. $$||f||_{2,\beta}= \sup_{N} ( \tau ^{\beta}(x)|f(x)|+\tau ^{1+\beta}(x)|Df(x)|+\tau ^{2+\beta}(x)|DDf(x)|+\tau ^{2+2\beta}(x)||DDf(x)||_{\beta,x});$$ $$||f||_{0,\beta}= \sup_{N} ( \tau ^{2+\beta}(x)|f(x)|+\tau ^{2+2\beta}(x)||f(x)||_{\beta,x})$$ These weighted Holder norms are convenient to write down the Schauder estimates. If $T$ is well defined, it implies $f \in B^{0, \beta}$ and hence $ B^{2, \beta} \subseteq B^{0, \beta}$. But this is impossible. What's wrong?

Question 3

In the proof of proposition 4, I can not understand why local parametric estimates immediately implies that a sequence of graphs converges to a properly embedded limiting submanifold?


  • $\begingroup$ To get your latex to work, I added backticks ` ` around your latex, see the little box on the right of the page titled "How to write math". $\endgroup$ – j.c. Oct 28 '11 at 16:46
  • 4
    $\begingroup$ Maybe you should post 3 separate question? And state all the equations and inequalities explicitly, instead of citing page and equation numbers from the paper? $\endgroup$ – Deane Yang Oct 28 '11 at 17:16
  • $\begingroup$ Totally agree with @Deane, though I do think the question is good in principle. $\endgroup$ – Igor Rivin Oct 28 '11 at 18:41
  • $\begingroup$ I think I might be able to answer question 1, but I don't really want to download the paper and look everything up. $\endgroup$ – Deane Yang Oct 28 '11 at 21:22
  • $\begingroup$ Also, please define your notation. For example, what is $B^{2,\beta}$? $\endgroup$ – Deane Yang Oct 29 '11 at 2:23

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