## Uniqueness of self-similar shrinkers with asymptotically conical ends

HTML articles powered by AMS MathViewer

- by Lu Wang;
- J. Amer. Math. Soc.
**27**(2014), 613-638 - DOI: https://doi.org/10.1090/S0894-0347-2014-00792-X
- Published electronically: March 19, 2014
- PDF | Request permission

## Abstract:

Let $C\subset \mathbb {R}^{n+1}$ be a regular cone with vertex at the origin. In this paper, we show the uniqueness for smooth properly embedded self-shrinking ends in $\mathbb {R}^{n+1}$ that are asymptotic to $C$. As an application, we prove that not every regular cone with vertex at the origin has a smooth complete properly embedded self-shrinker asymptotic to it.## References

- U. Abresch and J. Langer,
*The normalized curve shortening flow and homothetic solutions*, J. Differential Geom.**23**(1986), no. 2, 175–196. MR**845704** - Sigurd B. Angenent,
*Shrinking doughnuts*, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989) Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992, pp. 21–38. MR**1167827** - Xu Cheng and Detang Zhou,
*Volume estimate about shrinkers*, Proc. Amer. Math. Soc.**141**(2013), no. 2, 687–696. MR**2996973**, DOI 10.1090/S0002-9939-2012-11922-7 - David L. Chopp,
*Computation of self-similar solutions for mean curvature flow*, Experiment. Math.**3**(1994), no. 1, 1–15. MR**1302814** - Tobias H. Colding and William P. Minicozzi II,
*Generic mean curvature flow I: generic singularities*, Ann. of Math. (2)**175**(2012), no. 2, 755–833. MR**2993752**, DOI 10.4007/annals.2012.175.2.7 - Tobias H. Colding and William P. Minicozzi II,
*Smooth compactness of self-shrinkers*, Comment. Math. Helv.**87**(2012), no. 2, 463–475. MR**2914856**, DOI 10.4171/CMH/260 - Tobias H. Colding and William P. Minicozzi II,
*Minimal surfaces*, Courant Lecture Notes in Mathematics, vol. 4, New York University, Courant Institute of Mathematical Sciences, New York, 1999. MR**1683966** - Celso J. Costa,
*Example of a complete minimal immersion in $\textbf {R}^3$ of genus one and three embedded ends*, Bol. Soc. Brasil. Mat.**15**(1984), no. 1-2, 47–54. MR**794728**, DOI 10.1007/BF02584707 - Q. Ding and Y.L. Xin,
*Volume growth, eigenvalue and compactness for self-shrinkers*, Asian J. Math.**17**(2013), no. 3, 443–456. - Klaus Ecker,
*Regularity theory for mean curvature flow*, Progress in Nonlinear Differential Equations and their Applications, vol. 57, Birkhäuser Boston, Inc., Boston, MA, 2004. MR**2024995**, DOI 10.1007/978-0-8176-8210-1 - Klaus Ecker and Gerhard Huisken,
*Mean curvature evolution of entire graphs*, Ann. of Math. (2)**130**(1989), no. 3, 453–471. MR**1025164**, DOI 10.2307/1971452 - Klaus Ecker and Gerhard Huisken,
*Interior estimates for hypersurfaces moving by mean curvature*, Invent. Math.**105**(1991), no. 3, 547–569. MR**1117150**, DOI 10.1007/BF01232278 - Luis Escauriaza and Francisco Javier Fernández,
*Unique continuation for parabolic operators*, Ark. Mat.**41**(2003), no. 1, 35–60. MR**1971939**, DOI 10.1007/BF02384566 - L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega,
*Decay at infinity of caloric functions within characteristic hyperplanes*, Math. Res. Lett.**13**(2006), no. 2-3, 441–453. MR**2231129**, DOI 10.4310/MRL.2006.v13.n3.a8 - L. Escauriaza, G. Seregin, and V. Šverák,
*Backward uniqueness for parabolic equations*, Arch. Ration. Mech. Anal.**169**(2003), no. 2, 147–157. MR**2005639**, DOI 10.1007/s00205-003-0263-8 - L. Escauriaza, G. Seregin, and V. Šverák,
*On backward uniqueness for parabolic equations*, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)**288**(2002), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 32, 100–103, 272 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.)**123**(2004), no. 6, 4577–4579. MR**1923546**, DOI 10.1023/B:JOTH.0000041475.11233.d8 - Nicola Garofalo and Fang-Hua Lin,
*Unique continuation for elliptic operators: a geometric-variational approach*, Comm. Pure Appl. Math.**40**(1987), no. 3, 347–366. MR**882069**, DOI 10.1002/cpa.3160400305 - Nicola Garofalo and Fang-Hua Lin,
*Monotonicity properties of variational integrals, $A_p$ weights and unique continuation*, Indiana Univ. Math. J.**35**(1986), no. 2, 245–268. MR**833393**, DOI 10.1512/iumj.1986.35.35015 - David Hoffman and William H. Meeks III,
*Embedded minimal surfaces of finite topology*, Ann. of Math. (2)**131**(1990), no. 1, 1–34. MR**1038356**, DOI 10.2307/1971506 - Gerhard Huisken,
*Local and global behaviour of hypersurfaces moving by mean curvature*, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 175–191. MR**1216584**, DOI 10.1090/pspum/054.1/1216584 - Gerhard Huisken,
*Asymptotic behavior for singularities of the mean curvature flow*, J. Differential Geom.**31**(1990), no. 1, 285–299. MR**1030675** - T. Ilmanen, Personal communication.
- T. Ilmanen,
*Lectures on mean curvature flow and related equations*(1995). unpublished notes. - T. Ilmanen,
*Singularities of mean curvature flow of surfaces*(1995). preprint. - David Jerison,
*Carleman inequalities for the Dirac and Laplace operators and unique continuation*, Adv. in Math.**62**(1986), no. 2, 118–134. MR**865834**, DOI 10.1016/0001-8708(86)90096-4 - David Jerison and Carlos E. Kenig,
*Unique continuation and absence of positive eigenvalues for Schrödinger operators*, Ann. of Math. (2)**121**(1985), no. 3, 463–494. With an appendix by E. M. Stein. MR**794370**, DOI 10.2307/1971205 - Nikolaos Kapouleas,
*Complete embedded minimal surfaces of finite total curvature*, J. Differential Geom.**47**(1997), no. 1, 95–169. MR**1601434** - Nikolaos Kapouleas, S.J. Kleene, and N.M. Møller,
*Mean curvature self-shrinkers of high genus: non-compact examples*, J. Reine Angew. Math.,. to appear, arXiv: 1106.5454. - S.J. Kleene and N.M. Møller,
*Self-shrinkers with a rotational symmetry*, Trans. Amer. Math. Soc.,. to appear, arXiv: 1008.1609. - B. Kotschwar, Personal communication.
- B. Kotschwar,
*Ricci flow and the holonomy group*. J. Reine Angew. Math., to appear, arXiv: 1105.3722. - Brett L. Kotschwar,
*Backwards uniqueness for the Ricci flow*, Int. Math. Res. Not. IMRN**21**(2010), 4064–4097. MR**2738351**, DOI 10.1093/imrn/rnq022 - Lu Li and Vladimír Šverák,
*Backward uniqueness for the heat equation in cones*, Comm. Partial Differential Equations**37**(2012), no. 8, 1414–1429. MR**2957545**, DOI 10.1080/03605302.2011.635323 - Fang-Hua Lin,
*A uniqueness theorem for parabolic equations*, Comm. Pure Appl. Math.**43**(1990), no. 1, 127–136. MR**1024191**, DOI 10.1002/cpa.3160430105 - S. Micu and E. Zuazua,
*On the lack of null-controllability of the heat equation on the half space*, Port. Math. (N.S.)**58**(2001), no. 1, 1–24. MR**1820835** - N.M. Møller,
*Closed self-shrinking surfaces in $\mathbb {R}^3$ via the torus*(2001). preprint. - Tu A. Nguyen,
*On a question of Landis and Oleinik*, Trans. Amer. Math. Soc.**362**(2010), no. 6, 2875–2899. MR**2592940**, DOI 10.1090/S0002-9947-10-04733-1 - Xuan Hien Nguyen,
*Construction of complete embedded self-similar surfaces under mean curvature flow. III*, Duke Math. J.,. to appear, arXiv: 1106.5272. - Xuan Hien Nguyen,
*Construction of complete embedded self-similar surfaces under mean curvature flow. II*, Adv. Differential Equations**15**(2010), no. 5-6, 503–530. MR**2643233** - Xuan Hien Nguyen,
*Construction of complete embedded self-similar surfaces under mean curvature flow. I*, Trans. Amer. Math. Soc.**361**(2009), no. 4, 1683–1701. MR**2465812**, DOI 10.1090/S0002-9947-08-04748-X - Yifei Pan and Thomas Wolff,
*A remark on unique continuation*, J. Geom. Anal.**8**(1998), no. 4, 599–604. MR**1724207**, DOI 10.1007/BF02921714 - Chi-Cheung Poon,
*Unique continuation for parabolic equations*, Comm. Partial Differential Equations**21**(1996), no. 3-4, 521–539. MR**1387458**, DOI 10.1080/03605309608821195 - David L. Russell,
*A unified boundary controllability theory for hyperbolic and parabolic partial differential equations*, Studies in Appl. Math.**52**(1973), 189–211. MR**341256**, DOI 10.1002/sapm1973523189 - Gregory Seregin and Vladimir Šverák,
*The Navier-Stokes equations and backward uniqueness*, Nonlinear problems in mathematical physics and related topics, II, Int. Math. Ser. (N. Y.), vol. 2, Kluwer/Plenum, New York, 2002, pp. 353–366. MR**1972005**, DOI 10.1007/978-1-4615-0701-7_{1}9 - C. D. Sogge,
*A unique continuation theorem for second order parabolic differential operators*, Ark. Mat.**28**(1990), no. 1, 159–182. MR**1049649**, DOI 10.1007/BF02387373 - L. Wang,
*Uniqueness of self-similar shrinkers with asymptotically cylindrical ends*, J. Reine Angew. Math. (2013). in press. - Lu Wang,
*A Bernstein type theorem for self-similar shrinkers*, Geom. Dedicata**151**(2011), 297–303. MR**2780753**, DOI 10.1007/s10711-010-9535-2 - Brian White,
*Partial regularity of mean-convex hypersurfaces flowing by mean curvature*, Internat. Math. Res. Notices**4**(1994), 186 ff., approx. 8 pp.}, issn=1073-7928, review= MR**1266114**, doi=10.1155/S1073792894000206, DOI 10.1155/

## Bibliographic Information

**Lu Wang**- Affiliation:
Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore

, Maryland 21218

- Email: coral0426@gmail.com
- Received by editor(s): October 3, 2011
- Received by editor(s) in revised form: June 27, 2013, and October 19, 2013
- Published electronically: March 19, 2014
- © Copyright 2014 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**27**(2014), 613-638 - MSC (2010): Primary 53C44, 53C24, 35J15; Secondary 35B60
- DOI: https://doi.org/10.1090/S0894-0347-2014-00792-X
- MathSciNet review: 3194490