Abstract
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in \({{\mathbb {R}}}^3\).
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Acknowledgements
H. Z. Li would like to thank Professors T. H. Colding, W. P. Minicozzi II and X. Zhou for insightful discussions. Part of this work was done while he was visiting MIT and he wishes to thank MIT for their generous hospitality. B. Wang would like to thank Professors T. Ilmanen, L. Wang and O. Hershkovits for helpful discussions. Both authors are grateful to the anonymous referees for many useful suggestions to improve the exposition of this paper.
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Haozhao Li: Supported by NSFC Grant No. 11671370 and the Fundamental Research Funds for the Central Universities. Bing Wang: Supported by NSF Grant DMS1510401.
Appendices
Appendix A The parabolic Harnack inequality
In this “Appendix”, we include the parabolic Harnack inequality from KrylovSafonov [31, 32]. First, we introduce some notations. Let \(x=(x^1, x^2, \ldots , x^n)\in {{\mathbb {R}}}^n\). Denote
Consider the parabolic operator
where the coefficients are measurable and satisfy the conditions
Here \(b(x, t)=(b^1(x, t), \ldots , b^n(x, t))\). Then we have
Theorem A.1
(Theorem 1.1 of [31, 32]) Suppose the operator L in (A.1) satisfies the conditions (A.2)–(A.4). Let \(\theta >1, R\le 2, u\in W^{1, 2}_{n+1}(Q(\theta , R)), u\ge 0\) in \(Q(\theta , R)\), and \(Lu=0\) on \(Q(\theta , R)\). Then there exists a constant C, depending only on \(\theta , \mu \) and n, such that
Moreover, when \(\frac{1}{\theta 1}\) and \(\frac{1}{\mu }\) vary within finite bounds, C also varies within finite bounds.
Note that in our case the Eq. (4.48) doesn’t satisfy the assumption that \(c(x, t)\ge 0\) in (A.4). Therefore, we cannot use Theorem A.1 directly. However, the Harnack inequality still works when c(x, t) is a constant. Namely, we have
Theorem A.2
Let \(\theta >1, R\le 2\). Suppose that \(u(x, t)\in W^{1, 2}_{n+1}(Q(\theta , R))\) is a nonnegative solution to the equation
where c is a constant and the coefficients satisfy (A.2)–(A.3). Then there exists a constant C, depending only on \(\theta , \mu , c\) and n, such that
Proof
Since u(x, t) is a solution of (A.6) and c is a constant, the function \(v(x, t)=e^{ct}u\) satisfies
Applying Theorem A.1 to the Eq. (A.8), we have
where C depends only on \(\theta , \mu \) and n. Thus, for any \(x\in B_{\frac{R}{2}}(0)\) we have
where \(C'\) depends only on \(\theta , \mu , c\) and n. Here we used \(R\le 2\) by the assumption. The theorem is proved. \(\square \)
We generalize Theorem A.2 to a general bounded domain in \({{\mathbb {R}}}^n\).
Theorem A.3
Let \(\Omega \) be a bounded domain in \({{\mathbb {R}}}^n\). Suppose that \(u(x, t)\in W^{1, 2}_{n+1}(\Omega \times (0, T))\) is a nonnegative solution to the equation
where c is a constant and the coefficients satisfy (A.2)–(A.3) for a constant \(\mu >0\). For any s, t satisfying \(0<s<t<T\) and any \(x, y\in \Omega \) with the following properties

(1).
x and y can be connected by a line segment \(\gamma \) with the length \(xy\le l;\)

(2).
Each point in \(\gamma \) has a positive distance at least \(\delta >0\) from the boundary of \(\Omega ;\)

(3)
s and t satisfy \(T_1\le ts\le T_2\) for some \(T_1, T_2>0;\)
we have
where C depends only on \(c, n, \mu , \min \{s, \delta ^2\}, l, T_1\) and \(T_2\).
Proof
Let \(\gamma \) be the line segment with the property (1) and (2) connecting x and y. We set
for any \(0\le i\le N\). Here we choose N to be the smallest integer satisfying
We define
We can check that \(R\le \frac{\delta }{2}\). For any \(s, t\in (0, T)\), we choose \(\{t_i\}_{i=0}^N\) such that \(t_0=s, t_N=t\) and
for all integers \(1\le i\le N\). Note that (A.11)–(A.13) imply that for any \(0\le i\le N1\),
and
Therefore, for any \(0\le i\le N1\) we have \((t_{i+1}\theta R^2, t_{i+1})\subset (0, T) \) and \(p_{i+1}\in B_{\frac{R}{2}}(p_i)\). Applying Theorem A.2 on \(B_R(p_i)\times (t_{i+1}\theta R^2, t_{i+1})\subset \Omega \times (0, T)\), we have
where C depends only on \(c, n, \mu \) and \(\frac{1}{\theta 1}=\frac{R^2N}{ts}. \) Here we used the fact that \(t_i=(t_{i+1}\theta R^2)+R^2\). Therefore,
where the constant \(C'\) in (A.15) depends only on \(c, n, \mu , \min \{s, \delta ^2\}, l, T_1\) and \(T_2\). The theorem is proved. \(\square \)
A direct corollary of Theorem A.3 is the following result.
Theorem A.4
Let \(\Omega \) be a bounded domain in \({{\mathbb {R}}}^n\). Suppose that \(u(x, t)\in W^{1, 2}_{n+1}(\Omega \times (0, T))\) is a nonnegative solution to the equation
where c is a constant and the coefficients satisfy (A.2)–(A.3) for a constant \(\mu >0\), and \(\Omega ', \Omega ''\) are subdomains in \(\Omega \) satisfying the following properties:

(1).
\(\Omega '\subset \Omega ''\subset \Omega \), and \(\Omega ''\) has a positive distance \(\delta >0\) from the boundary of \(\,\Omega \);

(2).
\(\Omega '\) can be covered by k balls with radius r, and all balls are contained in \(\Omega ''\).
Then for any s, t satisfying \(0<s<t<T\) and any \(x, y\in \Omega '\), we have
where C depends only on \(c, n, \mu , \min \{s, \delta ^2\}, ts, r\) and k.
Proof
By the assumption, we can find finite many points \({{\mathcal {A}}}=\{q_1, q_2, \ldots , q_k\}\) such that
For any \(x, y\in \Omega '\), there exists two points in \({{\mathcal {A}}}\), which we denote by \(q_1\) and \(q_2\), such that \(x\in B_{r}(q_1)\) and \(y\in B_{r}(q_2)\). Then x and y can be connected by a polygonal chain \(\gamma \), which consists of two line segments \(\overline{xq_1}, \overline{yq_2}\) and a polygonal chain with vertices in \({{\mathcal {A}}}\) connecting \(q_1\) and \(q_2\). Clearly, the number of the vertices of \(\gamma \) is bounded by \(k+2\) and the total length of \(\gamma \) is bounded by \((k+2) r\). Moreover, by the assumption we have \(\gamma \subset \Omega ''\) and each point in \(\gamma \) has a positive distance at least \(\delta >0\) from the boundary of \(\Omega \).
Assume that the polygonal chain \(\gamma \) has consecutive vertices \(\{p_0, p_1, \ldots , p_N\}\) with \(p_0=y, p_N=x\) and \(1\le N\le k+2\). We apply Theorem A.3 for each line segment \(\overline{p_ip_{i+1}}\) and the interval \([t_i, t_{i+1}]\), where \(\{t_i\}\) is chosen as in (A.13). Note that
Thus, for any \(0\le i\le N1\) we have
where C depends only on \(c, n, \mu , \min \{s, \delta ^2\}, r, k\) and \(ts\), and (A.19) implies (A.17). This finishes the proof of Theorem A.4. \(\square \)
Appendix B The interior estimates of parabolic equations
In this “Appendix”, we present the interior estimates of parabolic equations from G. Lieberman’s book [37] for the reader’s convenience.
Let \(X=(x, t)\) be a point in \({{\mathbb {R}}}^{n+1}\) and \(x=(x_1, x_2, \ldots , x_n)\in {{\mathbb {R}}}^n\). The norms on \({{\mathbb {R}}}^n\) and \({{\mathbb {R}}}^{n+1}\) are given by
Let \(\Omega \) be a domain in \({{\mathbb {R}}}^{n+1}\). Let \(d(X, Y)=\min \{d(X), d(Y)\}\) where \(d(X)=\mathrm {dist}(X, {{\mathcal {P}}}\Omega \cap \{t<t_0\})\). Here \({{\mathcal {P}}}\Omega \) denotes the parabolic boundary of \(\Omega \). We define
If \(b\ge 0\), we define
If \(a=k+\alpha >0\) and \(a+b\ge 0\), where k is a nonnegative integer and \(\alpha \in (0, 1]\), we define
We also define \(f_a^*=f_a^{(0)}\), and define the spaces
With these notations, we have the following result.
Theorem B.1
(Theorem 4.9 of [37]) Let \(\Omega \) be a bounded domain in \({{\mathbb {R}}}^{n+1}\), and let \(a^{ij}\in H_{\alpha }^{(0)}\) and \(b^i\in H_{\alpha }^{(1)}\) satisfy
for some constants \(A, B, \lambda \) and \(\Lambda \). Let \(c\in H_{\alpha }^{(2)}\) satisfy
for some constant \(c_1\) and let \(f\in H_{\alpha }^{(2)}\). If \(u\in H^*_{2+\alpha }\) is a solution of
in \(\Omega \), then there is a constant C determined only by \(A, B, c_1, n, \lambda \) and \(\Lambda \) such that
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Li, H., Wang, B. The extension problem of the mean curvature flow (I). Invent. math. 218, 721–777 (2019). https://doi.org/10.1007/s00222019008932
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DOI: https://doi.org/10.1007/s00222019008932