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?

Thanks!