Skip to main content
Log in

The extension problem of the mean curvature flow (I)

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in \({{\mathbb {R}}}^3\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Bamler, R.H., Zhang, Q.S.: Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature. Adv. Math. 319, 396–450 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brendle, S., Huisken, G.: Mean curvature flow with surgery of mean convex surfaces in \({\mathbb{R}}^3\). Invent. Math. 203(2), 615–654 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, B.L., Yin, L.: Uniqueness and pseudolocality theorems of the mean curvature flow. Commun. Anal. Geom. 15(3), 435–490 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cooper, A.: A characterization of the singular time of the mean curvature flow. Proc. Am. Math. Soc. 139(8), 2933–2942 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Choi, H., Schoen, R.: The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature. Invent. Math. 81(3), 387–394 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, X.X., Sun, S., Wang, B.: Kähler Ricci flow, Kähler Einstein metric, and \(K\)-stability. arXiv:1508.04397. To appear in Geometry and Topology

  7. Chen, X.X., Wang, B.: Space of Ricci flows I. Commun. Pure Appl. Math. 65(10), 1399–1457 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, X.X., Wang, B.: Space of Ricci flow II. arXiv:1405.6797

  9. Chen, X.X., Wang, B.: Space of Ricci flow II–Part A: moduli of singular Calabi-Yau spaces. Forum Math. Sigma. 5, e32 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, X.X., Wang, B.: Space of Ricci flow II—Part B: weak compactness of the flows. To appear in Journal of Differential Geometry

  11. Chen, X.X., Wang, B.: Remarks of weak compactness along Kähler Ricci flow. arXiv:1605.01374. To appear in Proceedings of the seventh International Congress of Chinese Mathematicians

  12. Chen, X.X., Wang, B.: On the conditions to extend Ricci flow(III). Int. Math. Res. Not. IMRN 10, 2349–2367 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Colding, T.H., Minicozzi II, W.P.: A course in minimal surfaces. Graduate Studies in Mathematics, 121. American Mathematical Society, Providence, RI (2011). xii+313 pp

  14. Colding, T.H., Minicozzi II, W.P.: Embedded minimal surfaces without area bounds in 3-manifolds. Geometry and topology: Aarhus (1998), 107–120, Contemporary Mathematics, 258, American Mathematical Society, Providence, RI, 2000

  15. Colding, T.H., Minicozzi II, W.P.: Smooth compactness of self-shrinkers. Comment. Math. Helv. 87(2), 463–475 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Colding, T.H., Minicozzi II, W.P.: Generic mean curvature flow I: generic singularities. Ann. of Math. 175(2), 755–833 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Colding, T.H., Minicozzi II, W.P.: The singular set of mean curvature flow with generic singularities. Invent. Math. 204(2), 443–471 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ecker, K.: Regularity Theory for Mean Curvature Flow. Progress in Nonlinear Differential Equations and Their Applications, vol. 57. Birkhäuser, Boston (2004)

    Google Scholar 

  19. Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. of Math. 130(3), 453–471 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ecker, K., Huisken, G.: Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105(3), 547–569 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gulliver, R., Lawson, B.: The structure of stable minimal hypersurfaces near a singularity. Proc. Symp. Pure Math. 44, 213–237 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  23. Haslhofer, R., Kleiner, B.: Mean curvature flow of mean convex hypersurfaces. Commun. Pure Appl. Math. 70(3), 511–546 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Haslhofer, R., Kleiner, B.: Mean curvature flow with surgery. Duke Math. J. 166(9), 1591–1626 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237–266 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  26. Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31(1), 285–299 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  27. Huisken, G., Sinestrari, C.: Mean curvature flow singularities for mean convex surfaces. Calc. Var. Partial Differ. Equ. 8(1), 1–14 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  28. Huisken, G., Sinestrari, C.: Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183(1), 45–70 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Huisken, G., Sinestrari, C.: Mean curvature flow with surgeries of two-convex hypersurfaces. Invent. Math. 175(1), 137–221 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ilmanen, T.: Singularities of mean curvature flow of surfaces, preprint. http://www.math.ethz.ch/~/papers/pub.html (1995)

  31. Krylov, N., Safonov, M.: A certain property of solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR Ser. Mat. 44(1), 161–175 (1980)

    MathSciNet  Google Scholar 

  32. Krylov, N., Safonov, M.: A certain property of solutions of parabolic equations with measurable coefficients. Math. USSR-Izv. 16(1), 151–164 (1981)

    Article  MATH  Google Scholar 

  33. Le, N.Q., Sesum, N.: The mean curvature at the first singular time of the mean curvature flow. Ann. Inst. H. Poincare Anal. Non Lineaire 27(6), 1441–1459 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Le, N.Q., Sesum, N.: On the extension of the mean curvature flow. Math. Z. 267(3–4), 583–604 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Le, N.Q., Sesum, N.: Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers. Commun. Anal. Geom. 19(4), 633–659 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lin, L.Z., Sesum, N.: Blow-up of the mean curvature at the first singular time of the mean curvature flow. Calc. Var. Partial Differ. Equ. 55(3), 16 (2016). Art. 65

    Article  MathSciNet  MATH  Google Scholar 

  37. Lieberman, G.M.: Second Order Parabolic Differential Equations. World Scientific Publishing Co., Inc., River Edge (1996). xii+439 pp

    Book  MATH  Google Scholar 

  38. Mantegazza, C.: Lecture Notes on Mean Curvature Flow, Progress in Mathematics, 290. Birkhäuser/Springer Basel AG, Basel (2011). xii+166 pp

  39. Sesum, N.: Curvature tensor under the Ricci flow. Am. J. Math. 127(6), 1315–1324 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Simon, M.: Extending four dimensional Ricci flows with bounded scalar curvature. arXiv:1504.02910

  41. Wang, B.: On the conditions to extend Ricci flow. Int. Math. Res. Not. IMRN 2008, no. 8, Art. ID rnn012, p. 30

  42. Wang, B.: On the conditions to extend Ricci flow(II). Int. Math. Res. Not. IMRN 14, 3192–3223 (2012)

    MathSciNet  MATH  Google Scholar 

  43. Wang, L.: Geometry of two-dimensional self-shrinkers. arXiv:1505.00133

  44. Wang, M.T.: The mean curvature flow smoothes Lipschitz submanifolds. Commun. Anal. Geom. 12(3), 581–599 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  45. White, B.: The size of the singular set in mean curvature flow of mean-convex sets. J. Am. Math. Soc. 13(3), 665–695 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  46. White, B.: The nature of singularities in mean curvature flow of mean-convex sets. J. Am. Math. Soc. 16(1), 123–138 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  47. White, B.: A local regularity theorem for mean curvature flow. Ann. of Math. 161(3), 1487–1519 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  48. White, B.: Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals. Invent. Math. 88(2), 243–256 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  49. White, B.: Lectures on Minimal Surface Theory. arXiv:1308.3325

  50. Xu, H.W., Ye, F., Zhao, E.T.: Extend mean curvature flow with finite integral curvature. Asian J. Math. 15(4), 549–556 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  51. Zhang, Z.: Scalar curvature behavior for finite-time singularity of Kähler–Ricci flow. Mich. Math. J. 59(2), 419–433 (2010)

    Article  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Haozhao Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Haozhao Li: Supported by NSFC Grant No. 11671370 and the Fundamental Research Funds for the Central Universities. Bing Wang: Supported by NSF Grant DMS-1510401.

Appendices

Appendix A The parabolic Harnack inequality

In this “Appendix”, we include the parabolic Harnack inequality from Krylov-Safonov [31, 32]. First, we introduce some notations. Let \(x=(x^1, x^2, \ldots , x^n)\in {{\mathbb {R}}}^n\). Denote

$$\begin{aligned} |x|= & {} \Big (\sum _{i=1}^n(x^i)^2\Big )^{\frac{1}{2}},\quad B_R(x)=\{y\in {{\mathbb {R}}}^n\;|\; |x-y|<R\},\\ Q(\theta , R)= & {} B_R(0)\times (0, \theta R^2). \end{aligned}$$

Consider the parabolic operator

$$\begin{aligned} L u=-\frac{\partial u}{\partial t}+a^{ij}(x, t)u_{ij}+b^i(x, t)u_i-c(x, t)u, \end{aligned}$$
(A.1)

where the coefficients are measurable and satisfy the conditions

$$\begin{aligned} \mu |\xi |^2\le & {} a^{ij}(x, t)\xi _i\xi _j\le \frac{1}{\mu }|\xi |^2, \end{aligned}$$
(A.2)
$$\begin{aligned} |b(x, t)|\le & {} \frac{1}{\mu },\end{aligned}$$
(A.3)
$$\begin{aligned} 0\le & {} c(x, t)\le \frac{1}{\mu }. \end{aligned}$$
(A.4)

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

$$\begin{aligned} u(0, R^2)\le C\, u(x, \theta R^2),\quad \forall \; x\in B_{\frac{R}{2}}(0). \end{aligned}$$
(A.5)

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(xt) 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

$$\begin{aligned} L u=-\frac{\partial u}{\partial t}+a^{ij}(x, t)u_{ij}+b^i(x, t)u_i+c u=0, \end{aligned}$$
(A.6)

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

$$\begin{aligned} u(0, R^2)\le C\, u(x, \theta R^2),\quad \forall \; |x|<\frac{1}{2} R. \end{aligned}$$
(A.7)

Proof

Since u(xt) is a solution of (A.6) and c is a constant, the function \(v(x, t)=e^{-ct}u\) satisfies

$$\begin{aligned} -\frac{\partial v}{\partial t}+a^{ij}(x, t)v_{ij}+b^i(x, t)v_i=0. \end{aligned}$$
(A.8)

Applying Theorem A.1 to the Eq. (A.8), we have

$$\begin{aligned} v(0, R^2)\le C\, v(x, \theta R^2),\quad \forall \; |x|<\frac{1}{2} R, \end{aligned}$$

where C depends only on \(\theta , \mu \) and n. Thus, for any \(x\in B_{\frac{R}{2}}(0)\) we have

$$\begin{aligned} u(0, R^2)\le Ce^{-c(\theta -1)R^2}u(x, \theta R^2)\le C'u(x, \theta R^2), \end{aligned}$$

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

$$\begin{aligned} L u=-\frac{\partial u}{\partial t}+a^{ij}(x, t)u_{ij}+b^i(x, t)u_i+c u=0, \end{aligned}$$
(A.9)

where c is a constant and the coefficients satisfy (A.2)–(A.3) for a constant \(\mu >0\). For any st satisfying \(0<s<t<T\) and any \(x, y\in \Omega \) with the following properties

  1. (1).

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

  2. (2).

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

  3. (3)

    s and t satisfy \(T_1\le t-s\le T_2\) for some \(T_1, T_2>0;\)

we have

$$\begin{aligned} u(y, s)\le C \,u(x, t) , \end{aligned}$$
(A.10)

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

$$\begin{aligned} p_0=y,\quad p_N=x, \quad p_i=p_0+\frac{x-y}{N}i\in \gamma \end{aligned}$$

for any \(0\le i\le N\). Here we choose N to be the smallest integer satisfying

$$\begin{aligned} N>\max \Big \{\frac{2(t-s)}{s}, \frac{l}{\min \{\frac{\sqrt{s}}{4}, \frac{\delta }{4} \}}\Big \}. \end{aligned}$$
(A.11)

We define

$$\begin{aligned} R=\frac{2l}{N}, \quad \theta =1+\frac{t-s}{R^2N}. \end{aligned}$$
(A.12)

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

$$\begin{aligned} t_{i}-t_{i-1}=\frac{t-s}{N} \end{aligned}$$
(A.13)

for all integers \(1\le i\le N\). Note that (A.11)–(A.13) imply that for any \(0\le i\le N-1\),

$$\begin{aligned} t_{i+1}-\theta R^2\ge s-\theta R^2=s-R^2-\frac{t-s}{N}\ge \frac{s}{4}>0 \end{aligned}$$

and

$$\begin{aligned} |p_{i+1}-p_i|=\frac{|x-y|}{N}\le \frac{l}{N}=\frac{R}{2}. \end{aligned}$$

Therefore, for any \(0\le i\le N-1\) 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

$$\begin{aligned} u(p_i, t_i)\le C\,u(p_{i+1}, t_{i+1}), \end{aligned}$$
(A.14)

where C depends only on \(c, n, \mu \) and \(\frac{1}{\theta -1}=\frac{R^2N}{t-s}. \) Here we used the fact that \(t_i=(t_{i+1}-\theta R^2)+R^2\). Therefore,

$$\begin{aligned} u(y, s)=u(p_0, t_0)\le C^Nu(p_N, t_N)=C' u(x, t) \end{aligned}$$
(A.15)

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

$$\begin{aligned} L u=-\frac{\partial u}{\partial t}+a^{ij}(x, t)u_{ij}+b^i(x, t)u_i+c u=0, \end{aligned}$$
(A.16)

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. (1).

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

  2. (2).

    \(\Omega '\) can be covered by k balls with radius r, and all balls are contained in \(\Omega ''\).

Then for any st satisfying \(0<s<t<T\) and any \(x, y\in \Omega '\), we have

$$\begin{aligned} u(y, s)\le C \,u(x, t) , \end{aligned}$$
(A.17)

where C depends only on \(c, n, \mu , \min \{s, \delta ^2\}, t-s, r\) and k.

Proof

By the assumption, we can find finite many points \({{\mathcal {A}}}=\{q_1, q_2, \ldots , q_k\}\) such that

$$\begin{aligned} \Omega '\subset \cup _{q\in {{\mathcal {A}}}}B_r(q)\subset \Omega ''. \end{aligned}$$
(A.18)

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

$$\begin{aligned} \frac{t-s}{k+2}\le t_{i+1}-t_i=\frac{t-s}{N}\le t-s. \end{aligned}$$

Thus, for any \(0\le i\le N-1\) we have

$$\begin{aligned} u(p_i, t_i)\le Cu(p_{i+1}, t_{i+1}), \end{aligned}$$
(A.19)

where C depends only on \(c, n, \mu , \min \{s, \delta ^2\}, r, k\) and \(t-s\), 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

$$\begin{aligned} |x|=\Big (\sum _{i=1}^n(x^i)^2\Big )^{\frac{1}{2}},\quad |X|=\max \{|x|, |t|^{\frac{1}{2}}\}. \end{aligned}$$

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

$$\begin{aligned} |f|_0=\sup _{\Omega } |f|. \end{aligned}$$

If \(b\ge 0\), we define

$$\begin{aligned} |f|_0^{(b)}=\sup _{X\in \Omega }d(X)^b|f(X)|. \end{aligned}$$

If \(a=k+\alpha >0\) and \(a+b\ge 0\), where k is a nonnegative integer and \(\alpha \in (0, 1]\), we define

$$\begin{aligned}{}[f]_a^{(b)}= & {} \sup \Big \{\sum _{|\beta |+2j=k}d(X, Y)^{a+b}\frac{|D_x^{\beta }D_t^j f(X)-D_x^{\beta }D_t^j f(Y)|}{|X-Y|^{\alpha }}: \\&\qquad X\ne Y \; \hbox {in}\;\Omega \Big \},\\ \langle f\rangle _a^{(b)}= & {} \sup \Big \{\sum _{|\beta |+2j=k-1}d(X, Y)^{a+b}\frac{|D_x^{\beta }D_t^j f(X)-D_x^{\beta }D_t^j f(Y)|}{|X-Y|^{1+\alpha }}: \\&\qquad X\ne Y \; \hbox {in}\;\Omega ,\;x=y\Big \}, \\ |f|_a^{(b)}= & {} \sum _{|\beta |+2j\le k}|D_x^{\beta }D_t^jf|_0^{(|\beta |+2j+b)}+[f]_a^{(b)}+\langle f\rangle _a^{(b)}. \end{aligned}$$

We also define \(|f|_a^*=|f|_a^{(0)}\), and define the spaces

$$\begin{aligned} H_a^*=\{f:\;|f|_a^*<\infty \},\quad H_a^{(b)}=\{f:\; |f|_a^{(b)}<\infty \}. \end{aligned}$$

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

$$\begin{aligned} \lambda |\xi |^2\le & {} a^{ij}\xi _i\xi _j\le \Lambda |\xi |^2, \quad [a^{ij}]_{\alpha }^{(0)}\le A, \end{aligned}$$
(B.1)
$$\begin{aligned} |b^i|_{\alpha }^{(1)}\le & {} B \end{aligned}$$
(B.2)

for some constants \(A, B, \lambda \) and \(\Lambda \). Let \(c\in H_{\alpha }^{(2)}\) satisfy

$$\begin{aligned} |c|_{\alpha }^{(2)}\le c_1 \end{aligned}$$
(B.3)

for some constant \(c_1\) and let \(f\in H_{\alpha }^{(2)}\). If \(u\in H^*_{2+\alpha }\) is a solution of

$$\begin{aligned} -\frac{\partial u}{\partial t}+a^{ij}u_{ij}+b^i u_i+c u=f \end{aligned}$$
(B.4)

in \(\Omega \), then there is a constant C determined only by \(A, B, c_1, n, \lambda \) and \(\Lambda \) such that

$$\begin{aligned} |u|_{2+\alpha }^*\le C(|u|_0+|f|_{\alpha }^{(2)}). \end{aligned}$$
(B.5)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, H., Wang, B. The extension problem of the mean curvature flow (I). Invent. math. 218, 721–777 (2019). https://doi.org/10.1007/s00222-019-00893-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-019-00893-2

Navigation