I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken. I'm stuck in the theorem $2.3$.

Some definitions that are importants for the theorem are the height of $M_t$ with respect to the hyperplane orthogonal to $\omega$ by $u = \langle x, \omega \rangle$ and the height function $w(y, t)$ which is defined to be the height at time $t$ over a fixed point $y$ in this hyperplane.

$\textbf{Theorem 2.3}$ The gradient of the height function $w$ satisfies the estimate

$$\sqrt{1 + |Dw(y_0,t)|^2} \leq C_1(n) \sup_{B_R(y_0)} \sqrt{1 + |Dw_0|^2} \cdot \exp \left[ C_2(n) \ R^{-2} \ \sup_{[0,T]} \left( \sup_{B_R(y_0) \times [0,T]} w - w(y_0,t) \right)^2 \right],$$

where $0 \leq t \leq T$, $B_R(y_0)$ is a ball in the hyperplane orthogonal to $\omega$ and $w_0$ denotes the initial height function over this hyperplane.

I'm stuck on the final of the proof in two points:

$\textbf{Point 1.}$ The authors found

$$(15) \ v \eta \leq \sup_{\overline{M}_0} v \eta + (4 + 16 \beta) e^{64n\beta^2}.$$

The definitions and conditions necessary to understand the inequality above are $v := \left( \langle \nu, \omega \rangle \right)^{-1}$, $\nu$ is the unit normal vector of the hypersurface, $\omega$ is some fixed unit vector such that $\langle \nu, \omega \rangle > 0$, $\eta:= -1 + \exp (\lambda \varphi)$ (it's assumed that $\eta \geq 0$ and $\eta(x,t)$ is a smooth map which vanishes outside some compact subset of $\mathbb{R}^{n+1}$ in the proof), $\lambda := 64n\beta^2$,

$\varphi := \left( \frac{1}{2 \beta} u + 1 - (|x|^2 - u^2) \right)_+$, $x \in \overline{M}_t, \beta > 0$ to be chosen and $\varphi := 0$ otherwise

and $\overline{M}_t := \{ x \in M_t \ ; \ |x|^2 - u^2 \leq 1 \}$ for $t \in [0,T]$ and it's assumed $u < 0$ in this set, where $u := \langle x,\omega \rangle$.

Keeping these definitions and considerations in mind, the authors stated

As described in Sect. $1$ this estimate implies an equivalent bound for the height function $w$. At the point $y=0$ in the hyperplane orthogonal to $\omega$ we obtain in equivalence to $(15)$ for $t \in [0,T]$ and arbitrary $\beta > 0$

$$(16) \ [e^{64n\beta^2((2 \beta)^{-1}w(0,t) + 1)_+} - 1] \sqrt{1 + |Dw(0,t)|^2} \leq e^{64n\beta^2} \left( \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} + 4 + 16 \beta \right).$$

I didn't understand how the inequality $(15)$ imply the inequality $(16)$ since the section $1$ states that $\left( \frac{d}{dt} - \triangle \right) |x|^2 = -2n$ and $\left( \frac{d}{dt} - \triangle \right) u = 0.$

Furthermore, it is proved in this section that, when $M_t = \text{graph} \ w_t$, the evolution equation of the MCF is equivalent to the following evolution equation:

$$\frac{d}{dt} w(y,t) = \sqrt{1 + |Dw(y,t)|^2} \text{div}_y \left( \frac{Dw}{\sqrt{1 + |Dw|^2}} \right) (y,t)$$

$\textbf{Point 2.}$ The authors stated

Now choose $\beta = \sup_{t \in [0,T]} -w(0,t)$. We then infer from $(16)$

$$\sqrt{1 + |Dw(0,t)|^2} \leq C_1(n) \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} \exp [C_2(n) \sup_{[0,T]} (-w(0,t))^2].$$

I didn't understand how this inequality was obtained from the choose of $\beta$.

I will be grateful if someone can explain me how obtain these two inequalities.

$\textbf{EDIT:}$

After do some research on the internet, I found some clue about my doubt in point $1$ in this paper by Colding and Minicozzi where explain that the theorem $2.3$ is an application of Koorevar's argument to mean curvature flow. Keeping the work of Koorevar in mind, I wouldn't be able to find the work of Koorevar, but I find this other paper which states that the porpuse is generalize Koorevar's argument for Killing graphs and I figure out that the Koorevar's argument is known as $\textbf{interior gradient estimate for mean curvature equation}$ on the parabolic PDE's literature by this last paper too. Knowing this, I found this presentation which tells about the history of the study of the interior gradient estimate for mean curvature equation.

I will read the proof given on the last paper that I cited and I will try do the argument for the $\mathbb{R}^{n+1}$, where the argument is applied on Ecker and Huisken's paper, but I don't know if I'm able to understand the argument since I don't have familiarity with Killing graphs, so I will appreciate if anyone who knows about the argument of the interior gradient estimate for mean curvature equation can give me any hints.

I managed to develop something for the point $2$, but it is not the estimate pointed out on paper yet.

My attempt:

Firstly, I did some observations:

$\textbf{(a)}$ $1 \leq \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2}$,

$\textbf{(b)}$ $e^x \geq x + 1$ for every $x \geq 0$,

$\textbf{(c)}$ $x < x^2 + 1 \Longrightarrow e^x < e^{x^2 + 1}$ for every $x \geq 0$,

By $b$ and $c$, follows that

$\textbf{(d)}$ $x + 1 < e^{x^2 + 1}$ for every $x \geq 0$,

Using $d$, $(16)$ and $w(0,t) \geq -\beta$ by definition of $\beta$, we observe that

$\begin{align} [e^{32n\beta^2} - 1] \sqrt{1 + |Dw(0,t)|^2} & = [e^{64n\beta^2((2 \beta)^{-1}(-\beta) + 1)_+} - 1] \sqrt{1 + |Dw(0,t)|^2} \\ & = [e^{64n\beta^2((2 \beta)^{-1}(-\beta) + 1)_+} - 1] \sqrt{1 + |Dw(0,t)|^2} \\ & \leq [e^{64n\beta^2((2 \beta)^{-1}w(0,t) + 1)_+} - 1] \sqrt{1 + |Dw(0,t)|^2} \\ & \leq e^{64n\beta^2} \left( \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} + 4 + 16 \beta \right) \\ & < e^{64n\beta^2} \left( \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} + 15 + 16 \beta \right) \\ & \overset{\textbf{(a)}}{<} e^{64n\beta^2} \left( \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} + (15 + 16 \beta) \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} \right) \\ & = e^{64n\beta^2} 16 \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} \left( 1 + \beta \right) \\ & \overset{\textbf{(d)}}{<} e^{64n\beta^2} 16 \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} \exp(\beta^2 + 1) \\ & = e^{64n\beta^2 + 1} 16 \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} \exp(\beta^2) \\ & = e^{64n\beta^2 + 1} 16 \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} \exp \left( \left( \sup_{t \in [0,T]} -w(0,t) \right)^2 \right), \end{align}$

which imply

$$\sqrt{1 + |Dw(0,t)|^2} < \left( 16 \frac{e^{64n\beta^2 + 1}}{[e^{32n\beta^2} - 1]} \right) \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} \exp \left( \left( \sup_{t \in [0,T]} -w(0,t) \right)^2 \right)$$