Theorem $2.10$ - A course in minimal surfaces by Colding and Minicozzi

Question

This theorem is on page $78$.

I would like to know how the inequalities and the equality highlighted in the fina of the proof are obtained and how the hypothesis of $k$ in the statement of the theorem was used.

Thanks in advance!

Answer 1

I think the first inequality in the red box should be equality because it is just the decomposition of ring into finite smaller rings. The second inequality is because for $x\in B_{e^{l+1}r_0}\setminus B_{e^lr_0}$ $$1/r^2\leq 1/(e^lr_0)^2=e^{-2l}r_0^{-2}$$ Thus we can estimate the integral in a smaller ring like $$\int_{B_{e^{l+1}r_0}\setminus B_{e^lr_0}}\leq e^{-2l}r_0^{-2}\text{Volume}(B_{e^{l+1}r_0}\setminus B_{e^lr_0})\leq e^{-2l}r_0^{-2}\text{Volume}(B_{e^{l+1}r_0})\leq \frac{4\pi}{3} e^{2}$$ I used the monotonicity formula above: $\text{Area}(B_s)\leq \frac{4\pi}{3}s^2$).