2. Short-time existence

At this point, we briefly justify the short-time existence of unique smooth solutions to Equation (1.1), using results on nonlinear parabolic PDEs from [Reference Gerhardt7]. We begin by writing ${{{\mathcal {M}}}}^{n}$ locally in coordinate patches as a graph over $\Omega \subset {{{\mathbb {R}}}}^{n}$. More precisely, suppose that locally ${{{\mathcal {M}}}}^{n} = \text {graph}(u_0)$, $u_0 : \Omega \longrightarrow {{{\mathbb {R}}}}$, and $F$ is given by $F(p,\,t) = (p,\,u(p,\,t))$. In this setting (1.1) reduces to the initial value problem

(2.1)\begin{equation} \begin{cases} {{{\partial}}}_t u ={-}\sqrt{1 + |Du|^{2}} \, f\left({-}D\cdot\displaystyle\frac{Du}{\sqrt{1+ |Du|^{2}}}\right) \\ u(p,0) = u_0(p) , \end{cases} \end{equation}

where $D$ and $D\cdot$ are the usual gradient and divergence on ${{{\mathbb {R}}}}^{n}$ and $u_0$ is smooth. This comes from the expressions given in Appendix A of [Reference Ecker6] for the mean curvature of a graph $u$,

\[ H ={-}D\cdot\frac{Du}{\sqrt{1+ |Du|^{2}}} ={-}\frac{1}{\sqrt{1 + |Du|^{2}}}\left( \Delta u - \frac{Du\cdot (D^{2} u) Du}{1+ |Du|^{2}} \right) , \]

where $D^{2}u$ is the Hessian matrix of $u$, and the outer (upward pointing) normal

\[ {{{\vec{n}}}} = \frac{({-}Du , 1)}{\sqrt{1 + |Du|^{2}}} . \]

Moreover, the metric and inverse metric on the graph $u$ are given by

\[ g_{ij} = \delta_{ij} + {{{\partial}}}_i u {{{\partial}}}_j u , \qquad g^{ij} = \delta^{ij} - \frac{{{{\partial}}}_i u {{{\partial}}}_j u}{1 + |Du|^{2}} , \]

so we can rewrite (2.1) as

(2.2)\begin{equation} {{{\partial}}}_t u ={-}\sqrt{1 + |Du|^{2}} \ f\left( \displaystyle\frac{-g^{ij}D^{2}_{ij}u}{\sqrt{1 + |Du|^{2}}}\right) . \end{equation}

In particular, note that

\[ \frac{d}{d(g^{ij}D^{2}_{ij}u)} \sqrt{1 + |Du|^{2}} \, f\left(\frac{-g^{ij}D^{2}_{ij}u}{\sqrt{1 + |Du|^{2}}} \right) ={-}f'\left(\frac{-g^{ij}D^{2}_{ij}u}{\sqrt{1 + |Du|^{2}}} \right) < 0 \]

by virtue of (1.2b). We can, therefore, invoke theorems 2.5.7 and 2.5.9 from [Reference Gerhardt7] which guarantee the existence of a unique smooth solution to (2.1) on a short-time interval. This can then be extended in the usual way via a compactness argument to a full solution of (1.1).

3. Evolution equations for geometric quantities

The following proposition is well known for flows of the type (1.1). See for example Appendix A of [Reference Ecker6] or [Reference Zhu14] for outlines of the proofs in the mean curvature flow case.

Proposition 3.1 For surfaces evolving under (1.1) we have the following evolution equations for geometric quantities:

(3.1a)\begin{align} \displaystyle\frac{{{{\partial}}}}{{{{\partial}}}t} g_{ij} & ={-}2f(H)h_{ij} , \end{align}

(3.1b)\begin{align} \displaystyle\frac{{{{\partial}}}}{{{{\partial}}}t} g^{ij} & = 2f(H)g^{ik}g^{jl}h_{kl} , \end{align}

(3.1c)\begin{align} \frac{{{{\partial}}}}{{{{\partial}}}t} h_{ij} & = f'(H)\Delta_{{{{\mathcal{M}}}}_t}h_{ij} + f''(H)\nabla_i H \nabla_j H \nonumber\\ & \quad- (f(H) + Hf'(H))h_{ik}g^{kl}h_{lj} + f'(H)|A|^{2}h_{ij} , \end{align}

(3.1d)\begin{align} \frac{{{{\partial}}}}{{{{\partial}}}t} H & = f'(H)\Delta_{{{{\mathcal{M}}}}_t} H + f''(H) |\nabla H|^{2} + f(H)|A|^{2} , \end{align}

(3.1e)\begin{align} & \frac{{{{\partial}}}}{{{{\partial}}}t} |A|^{2} = f'(H)[ \Delta_{{{{\mathcal{M}}}}_t}|A|^{2} - 2|\nabla A|^{2} + 2|A|^{4} ]\nonumber\\ & \quad + 2f''(H)g^{ij}g^{kl}h_{ik}(\nabla_j H)(\nabla_l H) + 2\text{tr}(A^{3}) [ f(H) - Hf'(H) ], \end{align}

(3.1f)\begin{align} \frac{{{{\partial}}}}{{{{\partial}}}t} \sqrt{g} & ={-}\sqrt{g}Hf(H) . \end{align}

By applying Hamilton's Maximum Principle for tensors (Theorem 4.1 in [Reference Huisken8]) to the evolution equation for $h_{ij}$ (3.1c), and the usual parabolic maximum principle to the evolution equation for $H$ (3.1d), we find that both convexity and strict mean-convexity are preserved by the flow:

Proof. The result $H(F(\cdot,\,t)) \geq \delta$ for all $t$ follows from applying the parabolic maximum principle to (3.1d) in Proposition 3.1. To prove that convexity is preserved, we use Equation (3.1c). In order to use Proposition 4.1 of [Reference Huisken8], set

\begin{align*} M_{ij} & = h_{ij} , \quad u^{k} = 0 \ \text{ and } \\ N_{ij} & = f''(H)\nabla_i H \nabla_j H - \big(f(H) + Hf'(H)\big)h_{ik}g^{kl}h_{lj} + f'(H)|A|^{2}h_{ij} . \end{align*}

This $N_{ij}$ satisfies the null eigenvector condition since if $h_{ij}X^{j} = 0$ then

\[ N_{ij}X^{i}X^{j} = f''(H)\nabla_i H \nabla_j HX^{i}X^{j} = f''(H)(X^{i}\nabla_i H)^{2} \geq 0 . \]

The result follows from this.

The importance of the above result is twofold. Firstly, it implies $H_{\min }(t)$ is increasing and the uniform parabolicity of (1.1) is not lost. Secondly, by writing $|A|^{2}$ and $H^{2}$ in terms of the principal curvatures and expanding the square, we see that $|A|^{2} \leq H^{2}$ for any convex surface. The preservation of convexity means that $|A|^{2} \leq H^{2}$ on $F(\cdot,\,t)$ for all $t$, which will be used later.

The evolution equation for $H$ also leads to a finite upper bound on the maximal time of existence.

Proposition 3.3 For surfaces evolving according to Equation (1.1) , we have

(3.2)\begin{equation} H_{\min}(t) \geq \left(\displaystyle\frac{1}{H_{\min}^{2}(0)} - \frac{2t}{n}\right)^{{-}1/2} . \end{equation}

This in turn gives an upper bound on the maximal time of existence:

\[ T \leq \frac{n}{2H_{\min}^{2}(0)} < \infty . \]

Proof. Since $\ln \hat {H} \geq 1$ and $|A|^{2} \geq \frac {1}{n}H^{2}$, we have $f(H)|A|^{2} \geq \frac {1}{n}H^{3}$. We can, therefore, compare (3.1d) in Proposition 3.1 with the ODE

\[ \frac{d}{dt}\psi = \frac{1}{n}\psi^{3} , \quad \psi(0) = H_{\min}(0) . \]

This has solution

\[ \psi(t) = \left(\frac{1}{H_{\min}^{2}(0)} - \frac{2t}{n} \right)^{{-}1/2} , \]

and so the lower bound on $H_{\min }$ (3.2) follows from the comparison principle. The upper bound on the maximal existence time $T$ follows directly from (3.2).

In exactly the same manner as [Reference Schulze12], we obtain the result that $\max _{{{{\mathcal {M}}}}_t}|A|^{2}$ and $\max _{{{{\mathcal {M}}}}_t}H^{2}$ must both tend to infinity as the maximal time of existence $T$ is approached. If this were not so, we obtain bounds on the derivatives of $|A|^{2}$ and $H$, giving a smooth limiting surface, contradicting the maximality of $T$.

4. Pinching with Gauss curvature

Our aim during this section is to show that even though $H$ blows up near a singularity of the flow, the ratios between the principal curvatures approach 1. Following the method of Schulze in [Reference Schulze13], we first derive an evolution equation for the quantity

\[ \gamma := \frac{K}{H^{n}} . \]

By the arithmetic-geometric mean inequality, we have

\[ 0 \leq \gamma \leq \frac{1}{n^{n}} , \]

with equality on the right-hand side if and only if all the principal curvatures are equal.

Throughout this section, if $\zeta$ and $\xi$ are scalars we use the notation $\left \langle {\nabla \zeta } , \, {\nabla \xi } \right \rangle := g^{ab}\nabla _a\zeta \nabla _b\xi$, and $|\nabla \xi |^{2} = \left \langle {\nabla \xi } , \, {\nabla \xi } \right \rangle$. The first step is to find the evolution equation for $K$:

Lemma 4.1 Let $K = \det [g^{ik}h_{kj}]$ be the Gauss curvature of the strictly convex surface ${{{\mathcal {M}}}}_t$. Then we have:

(4.1a)\begin{align} {{{\partial}}}_t K & = K [ H(f-Hf') + f'b^{jk}\Delta_{{{{\mathcal{M}}}}_t} h_{jk} + f''b^{jk}\nabla_j H \nabla_k H + n|A|^{2}f' ] , \end{align}

(4.1b)\begin{align} \nabla_a K & = Kb^{jk}\nabla_a h_{jk} , \end{align}

(4.1c)\begin{align} \Delta_{{{{\mathcal{M}}}}_t} K & = Kb^{jk}\Delta_{{{{\mathcal{M}}}}_t}h_{jk} + \displaystyle\frac{n-1}{nK}|\nabla K|^{2} - \frac{K}{H^{2}}Y^{2} + \frac{H^{2n}}{nK}\left| \nabla\gamma \right|^{2} , \end{align}

(4.1d)\begin{align} {{{\partial}}}_t K & = f'\Delta_{{{{\mathcal{M}}}}_t} K - f'\frac{n-1}{n}\frac{|\nabla K|^{2}}{K} - f'\frac{H^{2n}}{nK}\left| \nabla\gamma \right|^{2} + f'\frac{K}{H^{2}}Y^{2} \nonumber\\ & \quad+ f''Kb^{jk}\nabla_jH\nabla_kH + HK(f-Hf') + f'nK|A|^{2} , \end{align}

where $b^{i}_j$ denotes the components of the inverse of the Weingarten map so that $h^{i}_kb^{k}_j = \delta ^{i}_j$ and $Y^{2} = g^{ij}b^{lp}b^{mq}(H\nabla _i h_{lm} - h_{lm}\nabla _i H)(H\nabla _j h_{pq} - h_{pq}\nabla _j H)$.

The previous lemma agrees with Lemma 2.1 in [Reference Schulze13].

Proof. Given that $K = \text {det}[h^{i}_j] = \det [g^{ik}h_{kj}]$, we make use of the following identity: If $A(t)$ is a positive definite matrix then

\[ {{{\partial}}}_t\det[A(t)] = \det[A(t)]\text{tr}[A^{{-}1}(t){{{\partial}}}_tA(t)] . \]

The first two equations follow from this and substituting in the evolution equation for $h_{ij}$ (3.1c).

For the third, taking the covariant derivative of (4.1b) gives

\begin{align*} \Delta_{{{{\mathcal{M}}}}_t}K& = g^{ac}\nabla_a \nabla_c K \\ & = g^{ac}(\nabla_aK)b^{jk}\nabla_ch_{jk} + Kg^{ac}(\nabla_ab^{jk})(\nabla_ch_{jk}) + Kb^{jk}\Delta_{{{{\mathcal{M}}}}_t}h_{jk} . \end{align*}

Using $b^{jk}\nabla _ch_{jk} = \frac {1}{K}\nabla _c K$ and the identity

\[ Kg^{ac}(\nabla_ab^{jk})(\nabla_ch_{jk}) ={-}\frac{1}{nK}|\nabla K|^{2} - \frac{K}{H^{2}}Y^{2} + \frac{H^{2n}}{nK}|\nabla\gamma|^{2} , \]

and substituting $Kb^{jk}\Delta _{{{{\mathcal {M}}}}_t}h_{jk}$ from (4.1c) into (4.1a) , we obtain (4.1d).

Lemma 4.2 We have the following equations for $\gamma$:

(4.2a)\begin{align} \nabla_a\gamma & = \displaystyle\frac{\nabla_a K}{H^{n}} - \frac{nK\nabla_a H}{H^{n+1}} , \end{align}

(4.2b)\begin{align} \Delta_{{{{\mathcal{M}}}}_t}\gamma & = \displaystyle\frac{\Delta_{{{{\mathcal{M}}}}_t}K}{H^{n}} - \frac{2n}{H^{n+1}}\left\langle {\nabla K} , \, {\nabla H} \right\rangle - \frac{n\gamma}{H}\Delta_{{{{\mathcal{M}}}}_t}H + \frac{n(n+1)\gamma}{H^{2}}|\nabla H|^{2} , \end{align}

(4.2c)\begin{align} {{{\partial}}}_t\gamma & = f'\Delta_{{{{\mathcal{M}}}}_t}\gamma + f'\frac{n+1}{nH^{n}}\left\langle {\nabla\gamma} , \, {\nabla H^{n}} \right\rangle - f'\frac{n-1}{nK}\left\langle {\nabla\gamma} , \, {\nabla K} \right\rangle\nonumber\\ & \quad- f'\frac{H^{n}}{nK}|\nabla\gamma|^{2} + \frac{Hf' - f}{H}(n|A|^{2} - H^{2})\gamma + f'\frac{Y^{2}}{H^{2}}\gamma \nonumber\\ & \quad+ f''[ ( b^{ij} - \frac{n}{H}g^{ij} )\nabla_i H\nabla_j H ]\gamma . \end{align}

Proof. The first two equations follow by directly differentiating $\gamma$. Using (4.2a), we derive the identities

(4.3a)\begin{align} \displaystyle\frac{nK}{H^{n+1}}|\nabla H|^{2} {} & = \frac{\left\langle {\nabla K} , \, {\nabla H} \right\rangle}{H^{n}} - \frac{\left\langle {\nabla \gamma} , \, {\nabla H^{n}} \right\rangle}{nH^{n-1}} \end{align}

(4.3b)\begin{align} \left\langle {\nabla K} , \, {\nabla H} \right\rangle {} & = \displaystyle\frac{H^{n+1}}{nK}\left(\frac{|\nabla K|^{2}}{H^{n}} - \left\langle {\nabla\gamma} , \, {\nabla K} \right\rangle \right) . \end{align}

Applying these and $\Delta _{{{{\mathcal {M}}}}_t}H^{n} = nH^{n-1}\Delta _{{{{\mathcal {M}}}}_t}H + n(n-1)H^{n-2}|\nabla H|^{2}$ to (4.2b) gives

\begin{align*} \Delta_{{{{\mathcal{M}}}}_t}\gamma & = \frac{\Delta_{{{{\mathcal{M}}}}_t} K}{H^{n}} - \frac{\gamma\Delta_{{{{\mathcal{M}}}}_t} H^{n}}{H^{n}} - \frac{n+1}{nH^{n}} \left\langle {\nabla\gamma} , \, {\nabla H^{n}} \right\rangle +\frac{n-1}{nK}\left\langle {\nabla\gamma} , \, {\nabla K} \right\rangle \\ & \quad+ \frac{n(n-1)\gamma}{H^{2}}|\nabla H|^{2} - \frac{n-1}{nK}\frac{|\nabla K|^{2}}{H^{n}} . \end{align*}

We also have

\[ {{{\partial}}}_t H^{n} = f'\Delta_{{{{\mathcal{M}}}}_t}H^{n} + nH^{n-1}|A|^{2}f + nH^{n-2}\left( Hf'' - (n-1)f' \right)|\nabla H|^{2} . \]

Now (4.2c) follows by substituting the above and (4.1d) into

\[ {{{\partial}}}_t\gamma = \frac{{{{\partial}}}_t K}{H^{n}} - \frac{K}{H^{2n}}{{{\partial}}}_t H^{n}. \]

Proposition 4.3 There exists a constant $0 < C(n,\,\alpha ) < 1/n^{n}$ such that if the initial hypersurface ${{{\mathcal {M}}}}_0$ is pinched in the sense that

(4.4)\begin{equation} \displaystyle\frac{K(p)}{H(p)^{n}} \geq C(n,\alpha) \quad \forall p \in {{{\mathcal{M}}}}_0 , \end{equation}

then the pinching (4.4) is preserved by the flow.

Before proving Proposition 4.3, we need the following lemma:

Lemma 4.4 There exists a constant $0 < C < 1/n^{n}$ such that if a convex hypersurface $\mathcal {N}$ satisfies (4.4) with this constant $C,$ then there exists $0 < \varepsilon = \varepsilon (C) \leq 1/n$ such that

(4.5)\begin{equation} h_{ij} \geq \varepsilon Hg_{ij} \end{equation}

and

(4.6)\begin{equation} \left| b^{ij} - \displaystyle\frac{n}{H}g^{ij} \right| \leq \frac{\varepsilon^{2}}{8\alpha H}\end{equation}

on $\mathcal {N}$. Moreover, $\varepsilon \longrightarrow 1/n$ as $C \longrightarrow 1/n^{n}$.

Proof. Suppose $\mathcal {N}$ satisfies (4.4), and let $\lambda _1,\,\ldots,\,\lambda _n$ be the principal curvatures of $\mathcal {N}$ with the ordering $0 < \lambda _1 \leq \ldots \leq \lambda _n$. Define the quantities

\[ x_k = \frac{\lambda_k}{\sum_{l=1}^{n} \lambda_l} . \]

Note that we also have $0 < x_1 \leq \ldots \leq x_n \leq 1$. To prove (4.5), we must first show that $x_1$ is bounded below. This is the case, since $\sum \nolimits \lambda _l \geq \lambda _k$ for all $k$, and so

\[ 0 < C \leq \frac{K}{H^{n}} = \prod_{l=1}^{n} x_l = \frac{\lambda_1\ldots\lambda_n}{\left(\sum_{l=1}^{n} \lambda_l\right)^{n}} \leq \frac{\lambda_1\ldots\lambda_n}{\left(\sum_{l=1}^{n} \lambda_l\right)\lambda_2\ldots\lambda_n} = x_1 . \]

Now take $\varepsilon$ to be the maximal constant such that $x_1 \geq \varepsilon$, which is equivalent to (4.5). The fact $\varepsilon \longrightarrow 1/n$ as $C \longrightarrow 1/n^{n}$ comes from the arithmetic-geometric mean inequality: as $C \longrightarrow 1/n^{n}$, all the principal curvatures tend to being equal, so $x_1 \longrightarrow 1/n$, and therefore, the maximal $\varepsilon$ must itself tend to $1/n$.

Now consider (4.6). By writing the inequality in terms of the principal curvatures, we find that it is equivalent to show

\[ \left| \frac{1}{n} - x_k \right| \leq \frac{\varepsilon^{2}}{8\alpha n}x_k \quad \forall k . \]

To prove this, set $\varepsilon = 1/n - \delta$. We deal with the cases $x_k < 1/n$ and $x_k > 1/n$ separately (if $x_k = 1/n$ then the statement holds immediately). First, assume that $x_k < 1/n$. By choosing $C$ close enough to $1/n^{n}$, we can make $\delta$ small enough so that $\delta + \sqrt [3]{8\alpha n\delta } \leq 1/n$. We therefore have

\[ \frac{1}{n} - \delta \leq x_k \implies \left| \frac{1}{n} - x_k \right| \leq \delta \leq \frac{1}{8\alpha n}\left(\frac{1}{n} - \delta \right)^{3} = \frac{\varepsilon^{2}}{8\alpha n}\left(\frac{1}{n} - \delta \right) \leq \frac{\varepsilon^{2}}{8\alpha n}x_k , \]

as required. Secondly, assume that $x_k > 1/n$. We first need to bound $x_k$ above by something depending on $\delta$. Since $\varepsilon$ is a lower bound for all the $x_l$,

\[ \varepsilon^{n-1} x_k \leq \prod_{l=1}^{n} x_l = \frac{K}{H^{n}} \leq \frac{1}{n^{n}} \implies x_k \leq \frac{1}{n^{n}\varepsilon^{n-1}} = \frac{1}{n(1-n\delta)^{n-1}} . \]

By the binomial theorem, it is possible to choose $C$ large enough (or equivalently, $\delta$ small enough) that

\[ x_k \leq \frac{1}{n(1-n\delta)^{n-1}} \leq \frac{1}{n} + n\delta . \]

Thus, we have

\[ \left| \frac{1}{n} - x_k \right| \leq n\delta \leq \frac{1}{8\alpha n} \left(\frac{1}{n} - \delta \right)^{3} \leq \frac{\varepsilon^{2}}{8\alpha n} x_k , \]

provided $\delta + \sqrt [3]{8\alpha n^{2}\delta } \leq 1/n$. Thus, we have shown that for $C$ close enough to $1/n^{n}$, (4.6) holds.

We are now ready to prove Proposition 4.3.

Proof of Proposition 4.3. Proof of Proposition 4.3

We use the parabolic maximum principle on the evolution Equation (4.2c) in Lemma 4.2. Since $Hf' - f \geq 0$ by (1.3d), and the identity $H^{2} \leq n|A|^{2}$ holds for any hypersurface by the Cauchy–Schwartz inequality, we automatically have

\[ \frac{Hf' - f}{H}(n|A|^{2} - H^{2}) \geq 0 . \]

We now show that for the right choice of initial surface,

\[ \frac{Y^{2}}{H^{2}} + \frac{f''}{f'}\left( b^{ij} - \frac{n}{H}g^{ij} \right)\nabla_i H\nabla_j H \geq 0 . \]

As in Corollary 4.3 of [Reference Chow5] and Lemma 2.2 of [Reference Schulze13], we take the number $C(n,\,\alpha )$ to be the minimal constant such that $0 \leq C(n,\,\alpha ) < 1/n^{n}$ and such that there exists $\varepsilon >0$ satisfying

\[ h_{ij} \geq \varepsilon Hg_{ij} \text{ and } \left| b^{ij} - \frac{n}{H}g^{ij} \right| \leq \frac{\varepsilon^{2}}{8\alpha H} . \]

We can do this by Lemma 4.4. Since on convex surfaces $b^{ij} \geq g^{ij}/H$ always holds, due to $|A|^{2} \leq H^{2}$ and the fact that $h_{ij}$ is positive definite, we can also estimate

\[ Y^{2} \geq \frac{|H\nabla_i h_{jl} - h_{jl}\nabla_i H|^{2}}{H^{2}} \geq \frac{1}{2}\varepsilon^{2}|\nabla H|^{2} \]

by Lemma 2.3(ii) in [Reference Huisken8]. Hence we have

\[ \frac{Y^{2}}{H^{2}} + \frac{f''}{f'}\left[ \left( b^{ij} - \frac{n}{H}g^{ij} \right)\nabla_i H\nabla_j H \right] \geq \left[ \frac{\varepsilon^{2}}{2} - 2\alpha\frac{\varepsilon^{2}}{8\alpha} \right]\frac{|\nabla H|^{2}}{H^{2}} \geq \frac{\varepsilon^{2}}{4}\frac{|\nabla H|^{2}}{H^{2}} > 0 . \]

The desired result therefore follows from the maximum principle because the coefficient of $\gamma$ in (4.2c) is positive.

Following [Reference Schulze13], we now want to show that at points where the mean curvature is large, the ratios between the principal curvatures approach 1. To do this, define the auxiliary functions

(4.7)\begin{equation} g := \displaystyle\frac{1}{n^{n}} - \frac{K}{H^{n}} \text{ and } g_\sigma := g\phi(H), \end{equation}

where $\phi (H)$ is a function to be chosen later with dependence on a small parameter $\sigma > 0$. Note that $0 \leq g \leq 1/n^{n}$ and $g(p,\,t) = 0$ if and only if the principal curvatures at $(p,\,t)$ are all equal.

Lemma 4.5 The function $g_\sigma$ satisfies the evolution equation

\begin{align*} \displaystyle\frac{{{{\partial}}}g_\sigma}{{{{\partial}}}t} & = f'\Delta g_\sigma + \frac{f'}{\phi}\frac{H^{n}}{K}|\nabla g_\sigma|^{2} + 2f' \left[1 - \frac{H\phi'}{\phi}\left(1 + \frac{H^{n}}{K}g\right)\right]\left\langle {\nabla g_\sigma} , \, {\frac{\nabla H}{H}} \right\rangle \\ & \quad+ f'\displaystyle\frac{H\phi'}{\phi}\left[\frac{Hf''}{f'} - \frac{H\phi''}{\phi'} - 2\left( 1 - \frac{H\phi'}{\phi}\right) + \frac{H^{n}}{K}\frac{H\phi'}{\phi}\right]\frac{|\nabla H|^{2}}{H^{2}}g_\sigma \\ & \quad- \phi\gamma \left[f'\frac{Y^{2}}{H^{2}} + f''\left( b^{ij} - \frac{n}{H}g^{ij}\right)\nabla_i H \nabla_j H\right] \\ & \quad- \phi\gamma \frac{Hf' - f}{H}(n|A|^{2} - H^{2}) + fg\phi'|A|^{2} . \end{align*}

Proof. We compute

(4.8a)\begin{align} \nabla g & ={-}\nabla\gamma , \end{align}

(4.8b)\begin{align} \nabla g_\sigma & = \phi\nabla g + g\phi'\nabla H , \end{align}

(4.8c)\begin{align} \Delta g_\sigma & ={-}\phi\Delta\gamma + g(\phi''|\nabla H|^{2} + \phi'\Delta H) + 2\phi'\left\langle {\nabla g} , \, {\nabla H} \right\rangle , \end{align}

(4.8d)\begin{align} \left\langle {\nabla\gamma} , \, {\nabla K} \right\rangle & = H^{n}|\nabla\gamma|^{2} + \displaystyle\frac{nK\left\langle {\nabla\gamma} , \, {\nabla H} \right\rangle}{H} = H^{n}|\nabla g|^{2} - \frac{nK\left\langle {\nabla g} , \, {\nabla H} \right\rangle}{H} , \end{align}

(4.8e)\begin{align} |\phi\nabla g|^{2} & = |\nabla g_\sigma|^{2} - 2g\phi'\left\langle {\nabla g_\sigma} , \, {\nabla H} \right\rangle + g^{2} (\phi')^{2}|\nabla H|^{2} . \end{align}

Using these,

\begin{align*} \frac{{{{\partial}}}g_\sigma}{{{{\partial}}}t} & = \phi\frac{{{{\partial}}}g}{{{{\partial}}}t} + g\phi'\frac{{{{\partial}}}H}{{{{\partial}}}t} ={-}\phi\frac{{{{\partial}}}\gamma}{{{{\partial}}}t} + g\phi'\frac{{{{\partial}}}H}{{{{\partial}}}t} \\ & \stackrel{(4.2c)}{=} - f'\phi\Delta\gamma - f'\phi(n+1) \left\langle {\nabla\gamma} , \, {\frac{\nabla H}{H}} \right\rangle + f'\phi\frac{n-1}{nK}\left\langle {\nabla\gamma} , \, {\nabla K} \right\rangle + f'\phi\frac{H^{n}}{nK}|\nabla\gamma|^{2} \\ & \quad- \phi\gamma\frac{Hf' - f}{H}(n|A|^{2} - H^{2}) - f'\phi\gamma\frac{Y^{2}}{H^{2}} - f''\phi\gamma\left[ \left( b^{ij} - \frac{n}{H}g^{ij} \right)\nabla_i H\nabla_j H \right] \\ & \quad+ g\phi' \left[ f'\Delta H + f'' |\nabla H|^{2} + f|A|^{2} \right] \\ & \mathop{=}\limits_{(4.8c)}^{(4.8d)} f' \Delta g_\sigma - f'g\phi''|\nabla H|^{2} - 2f'\phi' \left\langle {\nabla g} , \, {\nabla H} \right\rangle + 2f'\left\langle {\phi\nabla g} , \, {\frac{\nabla H}{H}} \right\rangle \\ & \quad+ \frac{f'}{\phi}\frac{H^{n}}{K}|\phi\nabla g|^{2} - \phi\gamma\frac{Hf' - f}{H}(n|A|^{2} - H^{2}) - f'\phi\gamma\frac{Y^{2}}{H^{2}} \\ & \quad- f''\phi\gamma\left[ \left( b^{ij} - \frac{n}{H}g^{ij} \right)\nabla_i H\nabla_j H \right] + g\phi'\left[ f'' |\nabla H|^{2} + f|A|^{2} \right] \\ & \stackrel{(4.8e)}{=} f' \Delta g_\sigma + \frac{f'}{\phi}\frac{H^{n}}{K}|\nabla g_\sigma|^{2} + 2f'\left( 1 - \frac{H\phi'}{\phi} \right)\left\langle {\phi\nabla g} , \, {\frac{\nabla H}{H}} \right\rangle \\ & \quad- 2f'g\frac{H^{n}}{K}\frac{H\phi'}{\phi}\left\langle {\nabla g_\sigma} , \, {\frac{\nabla H}{H}} \right\rangle - f'g\phi''|\nabla H|^{2} + f''g\phi'|\nabla H|^{2} \\ & \quad+ \frac{f'g^{2}(\phi')^{2}}{\phi}\frac{H^{n}}{K}|\nabla H|^{2} - \phi\gamma\frac{Hf' - f}{H}(n|A|^{2} - H^{2}) + fg\phi'|A|^{2} \\ & \quad- f'\phi\gamma\frac{Y^{2}}{H^{2}} - f''\phi\gamma\left[ \left( b^{ij} - \frac{n}{H}g^{ij} \right)\nabla_i H\nabla_j H \right]. \end{align*}

The evolution equation now follows by applying (4.8b) to the $\left \langle {\phi \nabla g} , \, {\nabla H/H} \right \rangle$ term.

In order to prove the main theorem about $g_\sigma$, we also need the following result, proved as Lemma 2.5 in [Reference Schulze13], which holds for any convex surface with $h_{ij} \geq \varepsilon Hg_{ij}$:

Lemma 4.6 If a convex surface is such that $\lambda _i \geq \varepsilon H > 0$ for some $\varepsilon > 0$ and all $i = 1,\,\ldots,\,n$, then there exists $\delta > 0$ such that

\[ \frac{n|A|^{2} - H^{2}}{H^{2}} \geq \delta \left(\frac{1}{n^{n}} - \frac{K}{H^{n}} \right) . \]

We now make the choice

\[ \phi(H) = (\ln\hat{H})^{\sigma} . \]

From this, and by applying the maximum principle, we can show the following theorem.

Theorem 4.7 Let the initial hypersurface ${{{\mathcal {M}}}}_0$ satisfy the pinching condition (4.4), and $g_\sigma$ be as in (4.7). Then there exists $\sigma > 0$ such that

(4.9)\begin{equation} g_\sigma(p,t) \leq \max_{p\in {{{\mathcal{M}}}}_0}g_\sigma(p,0) \quad \forall(p,t) \in {{{\mathcal{M}}}}\times[0,T) . \end{equation}

Proof. We deal with each term in the evolution equation for $g_\sigma$ which was computed in 4.5 separately. Define

\begin{align*} L_1& = \displaystyle\frac{f'}{\phi}\frac{H^{n}}{K}|\nabla g_\sigma|^{2} + 2f' \left[1 - \frac{H\phi'}{\phi}\left( 1 + \frac{H^{n}}{K}g\right)\right]\left\langle {\nabla g_\sigma} , \, {\frac{\nabla H}{H}} \right\rangle \\ L_2 & = f'\displaystyle\frac{H\phi'}{\phi}\left[ \frac{Hf''}{f'} - \frac{H\phi''}{\phi'} - 2\left( 1 - \frac{H\phi'}{\phi}\right) + \frac{H^{n}}{K}\frac{H\phi'}{\phi}\right]\frac{|\nabla H|^{2}}{H^{2}}g_\sigma \\ L_3& ={-} \phi\gamma \left[ f'\frac{Y^{2}}{H^{2}} + f''\left( b^{ij} - \frac{n}{H}g^{ij}\right)\nabla_i H \nabla_j H\right] \\ L_4 = {} & - \phi\gamma \frac{Hf' - f}{H}(n|A|^{2} - H^{2}) + fg\phi'|A|^{2} . \end{align*}

To apply the maximum principle we must have $L_2 + L_3 + L_4 \leq 0$. The term $L_1$ contains only gradient terms and can be ignored since we are interested in the maximum of $g_\sigma$. The fact that $|A|^{2} \leq H^{2}$ and Lemmas 4.4 and 4.6 together imply that

\begin{align*} L_3 & \leq{-} f'\phi\displaystyle\frac{C\varepsilon^{2}}{4}\frac{|\nabla H|^{2}}{H^{2}},\end{align*}

and

\begin{align*} L_4 \leq{-} (C\alpha\delta - \sigma) g\displaystyle\frac{H^{3}}{\hat{H}}(\ln\hat{H})^{\sigma+\alpha-1} . \end{align*}

Moreover, since $H^{n}/K \leq 1/C$ and $g \leq 1/n^{n}$,

\[ L_2 \leq f'\phi\frac{\sigma}{n^{n}}\left( 2\alpha + \sigma\left[ 1 + \frac{1}{Cn^{n}} \right] \right)\frac{|\nabla H|^{2}}{H^{2}} . \]

Therefore, we have

\begin{align*} \displaystyle\frac{{{{\partial}}}g_\sigma}{{{{\partial}}}t} - f'\Delta g_\sigma & \leq L_1 + f'\phi\frac{\sigma}{n^{n}}\left( 2\alpha + \sigma\left[ 1 + \frac{1}{Cn^{n}}\right]\right)\frac{|\nabla H|^{2}}{H^{2}} - f'\phi\frac{C\varepsilon^{2}}{4}\frac{|\nabla H|^{2}}{H^{2}} \\ & \quad - (C\alpha\delta - \sigma) g\displaystyle\frac{H^{3}}{\hat{H}}(\ln\hat{H})^{\sigma+\alpha-1} \\ & = L_1 + \frac{f'\phi}{n^{n}}\left(\sigma\left( 2\alpha + \sigma\left[1 + \frac{1}{Cn^{n}}\right]\right) - \frac{C\varepsilon^{2} n^{n}}{4}\right) \frac{|\nabla H|^{2}}{H^{2}} \\ & \quad - (C\alpha\delta - \sigma) g\frac{H^{3}}{\hat{H}}(\ln\hat{H})^{\sigma+\alpha-1} . \end{align*}

Hence, by choosing

\[ 0 < \sigma \leq \min\left\{ C\alpha\delta , \ \frac{-2\alpha + \sqrt{4\alpha^{2} + C\varepsilon^{2}n^{n} + \varepsilon^{2}}}{2(1 + 1/Cn^{n})} \right\} , \]

we see that the last three terms in the evolution equation for $g_\sigma$ in Lemma 4.5 are negative, which is the correct sign to apply the maximum principle, giving the result.

5. Convergence to a sphere

We now show that under a suitable scaling procedure near a singularity the limiting hypersurface becomes spherical. Assume that a singularity occurs at time $T$. Using the evolution Equation (3.1d) from Proposition 3.1 and $|A|^{2} \leq H^{2}$, we obtain

\[ \frac{{{{\partial}}}}{{{{\partial}}}t}H_{\max} \leq H_{\max}^{3} (\ln\hat{H}_{\max})^{\alpha} . \]

Define the variable $v(t)$ by

\[ v(t) := H_{\max}(t) \quad \text{ so that } \quad \frac{{{{\partial}}}v}{{{{\partial}}}t} \leq v^{3} (\ln\hat{v})^{\alpha} . \]

Let the function $J_\alpha : (0,\,\infty ) \longrightarrow (-\infty,\,0)$ be a solution of the differential equation

\[ J_\alpha'(x) = \frac{1}{x^{3} (\ln\hat{x})^{\alpha}}; \quad J_\alpha(x) \longrightarrow 0 \text{ as } x \longrightarrow \infty. \]

This function $J_\alpha$ has non-zero derivative on $(0,\,\infty )$, so is a bijection, and for large enough $x\in (0,\,\infty )$,

(5.1)\begin{equation} -\displaystyle\frac{1}{2x^{2}} < J_\alpha(x) <{-}\frac{1}{3x^{3}} . \end{equation}

For $0 \leq t < s \leq T$ we, therefore, have

\[ T - t \geq s - t \geq \int_{v(t)}^{v(s)} \frac{1}{v^{3}(\ln\hat{v})^{\alpha}} \equiv J_\alpha(v(s)) - J_\alpha(v(t)) , \]

and since we know that $v(s) = H_{\max }(s) \longrightarrow \infty$ as $s \longrightarrow T$, the defining boundary condition for $J_\alpha$ gives us that $J_\alpha (v(s)) \longrightarrow 0$ as $s \longrightarrow T$. Thus,

\[ \frac{1}{T-t} \leq{-}\frac{1}{J_\alpha(v(t))} . \]

Define the function $G_\alpha : (0,\,\infty ) \longrightarrow (0,\,\infty )$ by

\[ G_\alpha(x) :={-}\frac{1}{J_\alpha(x)} . \]

For large enough $x\in (0,\,\infty )$, inequality (5.1) implies

\[ 2x^{2} < G_\alpha(x) < 3x^{3} . \]

$G_\alpha$ is also a bijection and is, therefore, invertible. If we denote its inverse by $G_\alpha ^{-1}$, then we have $G_\alpha (x) \longrightarrow \infty$ and $G_\alpha ^{-1}(x) \longrightarrow \infty$ as $x \longrightarrow \infty$. Finally, we have $G_\alpha ^{-1}(x) \leq \sqrt {x/2}$.

We refer to a singularity as type-1 if there exists $C_0 > 0$ such that

\[ H_{\max}(t) \leq C_0 G_\alpha^{{-}1}\left(\frac{1}{T-t} \right) \ \text{ for all } \ t \in [0,T) . \]

If no such $C_0$ exists we call it type-2. We consider a different scaling procedure for each type.

Assuming first that we have a type-1 singularity, we can choose a sequence of points and times $(x_k,\,t_k) \in {{{\mathcal {M}}}}^{n} \times [0,\,T-1/k]$ such that

(5.2)\begin{equation} H(x_k,t_k) = \max_{\stackrel{t\leq T-1/k}{x\in{{{\mathcal{M}}}}^{n}}} H(x,t) . \end{equation}

Now we rescale the surfaces by

\[ \tilde{F}_k({\cdot},\tau) = \frac{F\left({\cdot},t_k + \tau(T-t_k-1/k) \right) - F(x_k,t_k)}{\varepsilon_k} \]

where

\[ \tau \in \left[ - \frac{t_k}{T-t_k-1/k} , \ 1 \,\right] \text{ and } \varepsilon_k = \frac{1}{G_\alpha^{{-}1}\left(\frac{1}{T-t_k-1/k} \right)} . \]

With this choice of scaling, we have

\[ \tilde{H}_k({\cdot},\tau) = \frac{H({\cdot},t_k+\tau(T-t_k-1/k))}{G_\alpha^{{-}1}\left(\frac{1}{T-t_k-1/k}\right)} \leq C_0\frac{G_\alpha^{{-}1}\left(\frac{1}{T - t_k -\tau(T-t_k-1/k)} \right)}{G_\alpha^{{-}1}\left(\frac{1}{T - t_k - 1/k} \right)} , \]

which is bounded for large enough $k$ by the continuity of $G_\alpha ^{-1}$.

Now assume we have a type-2 singularity, meaning

\[ \frac{H_{\max}(t)}{G_\alpha^{{-}1}\left(\frac{1}{T-t} \right)} \longrightarrow \infty \ \text{ as } t \longrightarrow T . \]

Choose a sequence of points and times $(x_k,\,t_k) \in {{{\mathcal {M}}}}^{n} \times [0,\,T-1/k]$ such that

(5.3)\begin{equation} \displaystyle\frac{H(x_k,t_k)}{G_\alpha^{{-}1}\left(\frac{1}{T-1/k-t_k}\right)} = \max_{\stackrel{t\leq T-1/k}{x\in{{{\mathcal{M}}}}^{n}}} \frac{H(x,t)}{G_\alpha^{{-}1}\left(\frac{1}{T-1/k-t}\right)} . \end{equation}

Now scale the surfaces according to

\[ \tilde{F}_k({\cdot},\tau) = \frac{F\left({\cdot},t_k + \frac{\tau}{G_\alpha(1/\varepsilon_k)} \right) - F(x_k,t_k)}{\varepsilon_k} \]

where

\[ \tau \in [- G_\alpha(1/\varepsilon_k)t_k , G_\alpha(1/\varepsilon_k)(T-t_k-1/k)] \text{ and }\varepsilon_k = \frac{1}{H(x_k,t_k)} . \]

Since we are considering a type-2 singularity, (5.3) implies that $1/\varepsilon _k \longrightarrow \infty$ as $k \longrightarrow \infty$. We can now consider the mean curvature of the rescaled surfaces, $\tilde {H}_k(\cdot,\,\tau )$. This satisfies

\[ 0 \leq \tilde{H}_k({\cdot},\tau) = \varepsilon_k H({\cdot},t) \leq \frac{G_\alpha^{{-}1}\left(\frac{1}{T - t_k -1/k - \tau/G_\alpha( 1/\varepsilon_k)} \right)}{G_\alpha^{{-}1}\left(\frac{1}{T - t_k - 1/k} \right)} . \]

As before, the right-hand side is bounded for large $k$ by the continuity of $G_\alpha ^{-1}$.

We, therefore, have that in both cases (type-1 and type-2) that $\tilde {H}_k$ is bounded. This implies a bound on $|\tilde {A}_k|^{2}$ for each of the surfaces, since the rescalings are convex, which itself gives bounds on the higher derivatives of curvatures by a standard argument (see [Reference Andrews2, Reference Schulze12]): the local graph representations of the rescaled surfaces satisfy a uniformly parabolic equation (thanks to $f' > 0$), and so by the uniform Hölder estimates of Krylov (Chapter 5 in [Reference Krylov11]) we may conclude that the resulting surface is smooth with a uniform bound on the derivatives of its curvature. Thus, by the Arzela–Ascoli theorem, we can extract a subsequence of $\tilde {F}_k(\cdot,\,t)$ which converges uniformly on compact subsets of ${{{\mathbb {R}}}}^{n+1}\times {{{\mathbb {R}}}}$ to a limit $\tilde {F}_\infty (\cdot,\,\tau )$.

In both cases (type-1 and type-2), the estimate in Theorem 4.7 implies that the scaled limiting surface is spherical. This is because the quantity $K/H^{n}$ is scaling-invariant, and so (4.9) implies

\[ 0 \leq\frac{1}{n^{n}} - \frac{\tilde{K}}{\tilde{H}^{n}} = \frac{1}{n^{n}} - \frac{K}{H^{n}} = \frac{g_\sigma}{(\ln\hat{H})^{\sigma}} \leq \frac{\max_{p\in {{{\mathcal{M}}}}_0}g_\sigma(p,0)}{(\ln\hat{H})^{\sigma}} \longrightarrow 0 \text{ as } t \longrightarrow T , \]

for all $p\in {{{\mathcal {M}}}}$. Therefore, at all points on $\tilde {F}_\infty$, we have

\[ \frac{1}{n^{n}} = \frac{\tilde{K}}{\tilde{H}^{n}} , \]

which implies that $\tilde {F}_\infty$ is spherical.