Generic mean curvature flow I; generic singularities

Tobias H. Colding; William P. Minicozzi II

doi:https://doi.org/10.4007/annals.2012.175.2.7

Abstract

It has long been conjectured that starting at a generic smooth closed embedded surface in $\mathbf{R}^3$, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere.
The key to showing this conjecture is to show that shrinking spheres, cylinders, and planes are the only stable self-shrinkers under the mean curvature flow. We prove this here in all dimensions. An easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away.

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[AbLa] $Go to document$ U. Abresch and J. Langer, "The normalized curve shortening flow and homothetic solutions," J. Differential Geom., vol. 23, iss. 2, pp. 175-196, 1986.

Show bibtex

@article {AbLa, MRKEY = {0845704}, AUTHOR = {Abresch, U. and Langer, J.}, TITLE = {The normalized curve shortening flow and homothetic solutions}, JOURNAL = {J. Differential Geom.}, FJOURNAL = {Journal of Differential Geometry}, VOLUME = {23}, YEAR = {1986}, NUMBER = {2}, PAGES = {175--196}, ISSN = {0022-040X}, CODEN = {JDGEAS}, MRCLASS = {53A04 (53C22 58F17)}, MRNUMBER = {0845704}, MRREVIEWER = {H. R. Gluck}, URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214440025}, ZBLNUMBER = {0592.53002}, }
[Al] $Go to document$ W. K. Allard, "On the first variation of a varifold," Ann. of Math., vol. 95, pp. 417-491, 1972.

Show bibtex

@article {Al, MRKEY = {0307015}, AUTHOR = {Allard, William K.}, TITLE = {On the first variation of a varifold}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {95}, YEAR = {1972}, PAGES = {417--491}, ISSN = {0003-486X}, MRCLASS = {49F20}, MRNUMBER = {0307015}, MRREVIEWER = {M. Klingmann}, ZBLNUMBER = {0252.49028}, DOI = {doi:10.2307/1970868}, }
[AAG] $Go to document$ S. Altschuler, S. B. Angenent, and Y. Giga, "Mean curvature flow through singularities for surfaces of rotation," J. Geom. Anal., vol. 5, iss. 3, pp. 293-358, 1995.

Show bibtex

@article {AAG, MRKEY = {1360824}, AUTHOR = {Altschuler, Steven and Angenent, Sigurd B. and Giga, Yoshikazu}, TITLE = {Mean curvature flow through singularities for surfaces of rotation}, JOURNAL = {J. Geom. Anal.}, FJOURNAL = {The Journal of Geometric Analysis}, VOLUME = {5}, YEAR = {1995}, NUMBER = {3}, PAGES = {293--358}, ISSN = {1050-6926}, MRCLASS = {58E15 (35K55 53A07)}, MRNUMBER = {1360824}, ZBLNUMBER = {0847.58072}, DOI = {10.1007/BF02921800}, }
[An] $Go to document$ B. Andrews, "Classification of limiting shapes for isotropic curve flows," J. Amer. Math. Soc., vol. 16, iss. 2, pp. 443-459, 2003.

Show bibtex

@article {An, MRKEY = {1949167}, AUTHOR = {Andrews, Ben}, TITLE = {Classification of limiting shapes for isotropic curve flows}, JOURNAL = {J. Amer. Math. Soc.}, FJOURNAL = {Journal of the American Mathematical Society}, VOLUME = {16}, YEAR = {2003}, NUMBER = {2}, PAGES = {443--459}, ISSN = {0894-0347}, MRCLASS = {53C44 (35K55 53A04)}, MRNUMBER = {1949167}, MRREVIEWER = {Kai Seng Chou}, DOI = {10.1090/S0894-0347-02-00415-0}, ZBLNUMBER = {1023.53051}, }
[A] S. B. Angenent, "Shrinking doughnuts," in Nonlinear Diffusion Equations and their Equilibrium States, Boston, MA: Birkhäuser, 1992, vol. 7, pp. 21-38.

Show bibtex

@incollection {A, MRKEY = {1167827}, AUTHOR = {Angenent, Sigurd B.}, TITLE = {Shrinking doughnuts}, BOOKTITLE = {Nonlinear Diffusion Equations and their Equilibrium States}, VENUE={{G}regynog, 1989}, SERIES = {Progr. Nonlinear Differential Equations Appl.}, VOLUME = {7}, PAGES = {21--38}, PUBLISHER = {Birkhäuser}, ADDRESS = {Boston, MA}, YEAR = {1992}, MRCLASS = {58E12 (35B99 53A10)}, MRNUMBER = {1167827}, MRREVIEWER = {Anders Linn{é}r}, ZBLNUMBER = {0762.53028}, }
[AChI] $Go to document$ S. B. Angenent, T. Ilmanen, and D. L. Chopp, "A computed example of nonuniqueness of mean curvature flow in $\Bbb R^3$," Comm. Partial Differential Equations, vol. 20, iss. 11-12, pp. 1937-1958, 1995.

Show bibtex

@article {AChI, MRKEY = {1361726}, AUTHOR = {Angenent, Sigurd B. and Ilmanen, T. and Chopp, D. L.}, TITLE = {A computed example of nonuniqueness of mean curvature flow in {$\bold R\sp 3$}}, JOURNAL = {Comm. Partial Differential Equations}, FJOURNAL = {Communications in Partial Differential Equations}, VOLUME = {20}, YEAR = {1995}, NUMBER = {11-12}, PAGES = {1937--1958}, ISSN = {0360-5302}, CODEN = {CPDIDZ}, MRCLASS = {58E12 (35K55 53A05)}, MRNUMBER = {1361726}, MRREVIEWER = {Emmanuel Hebey}, DOI = {10.1080/03605309508821158}, }
[Ba] J. Barta, "Sur la vibration fundamentale dúne membrane," C.R. Acad. Sci., vol. 204, pp. 472-473, 1937.

Show bibtex

@article{Ba, author={Barta, J.}, TITLE={Sur la vibration fundamentale dúne membrane}, JOURNAL={C.R. Acad. Sci.}, VOLUME={204}, YEAR={1937}, PAGES={472--473}, }
[B] K. A. Brakke, The Motion of a Surface by its Mean Curvature, Princeton, N.J.: Princeton Univ. Press, 1978, vol. 20.

Show bibtex

@book {B, MRKEY = {0485012}, AUTHOR = {Brakke, Kenneth A.}, TITLE = {The Motion of a Surface by its Mean Curvature}, SERIES = {Math. Notes}, VOLUME = {20}, PUBLISHER = {Princeton Univ. Press}, ADDRESS = {Princeton, N.J.}, YEAR = {1978}, PAGES = {i+252}, ISBN = {0-691-08204-9}, MRCLASS = {49F22 (35K99 49F20 58D25)}, MRNUMBER = {0485012}, MRREVIEWER = {Jean E. Taylor}, ZBLNUMBER = {0386.53047}, }
[CaIH] H-D. Cao, R. S. Hamilton, and T. Ilmanen, Gaussian densities and stability for some Ricci solitons, 2004.

Show bibtex

@misc{CaIH, author={Cao, H-D. and Hamilton, R. S. and Ilmanen, T.}, TITLE={Gaussian densities and stability for some {R}icci solitons}, NOTE={preprint}, YEAR={2004}, }
[Ca] I. Chavel, Eigenvalues in Riemannian Geometry, Orlando, FL: Academic Press, 1984, vol. 115.

Show bibtex

@book {Ca, MRKEY = {0768584}, AUTHOR = {Chavel, Isaac}, TITLE = {Eigenvalues in {R}iemannian Geometry}, SERIES = {Pure Appl. Math.}, VOLUME = {115}, PUBLISHER = {Academic Press}, ADDRESS = {Orlando, FL}, YEAR = {1984}, PAGES = {xiv+362}, ISBN = {0-12-170640-0}, MRCLASS = {58G25 (35P99 53C20)}, MRNUMBER = {0768584}, MRREVIEWER = {G{é}rard Besson}, ZBLNUMBER = {0551.53001}, }
[CGG] $Go to document$ Y. G. Chen, Y. Giga, and S. Goto, "Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations," J. Differential Geom., vol. 33, iss. 3, pp. 749-786, 1991.

Show bibtex

@article {CGG, MRKEY = {1100211}, AUTHOR = {Chen, Yun Gang and Giga, Yoshikazu and Goto, Shun'ichi}, TITLE = {Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations}, JOURNAL = {J. Differential Geom.}, FJOURNAL = {Journal of Differential Geometry}, VOLUME = {33}, YEAR = {1991}, NUMBER = {3}, PAGES = {749--786}, ISSN = {0022-040X}, CODEN = {JDGEAS}, MRCLASS = {35K65 (35D05 53C21 58E12)}, MRNUMBER = {1100211}, MRREVIEWER = {Ioan I. Vrabie}, URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214446564}, ZBLNUMBER = {0696.35087}, }
[Ch] $Go to document$ D. L. Chopp, "Computation of self-similar solutions for mean curvature flow," Experiment. Math., vol. 3, iss. 1, pp. 1-15, 1994.

Show bibtex

@article {Ch, MRKEY = {1302814}, AUTHOR = {Chopp, David L.}, TITLE = {Computation of self-similar solutions for mean curvature flow}, JOURNAL = {Experiment. Math.}, FJOURNAL = {Experimental Mathematics}, VOLUME = {3}, YEAR = {1994}, NUMBER = {1}, PAGES = {1--15}, ISSN = {1058-6458}, MRCLASS = {53A05 (58E12 65C99 65Y25)}, MRNUMBER = {1302814}, MRREVIEWER = {Guido Haak}, URL = {http://projecteuclid.org/getRecord?id=euclid.em/1062620999}, ZBLNUMBER = {0811.53011}, }
[CM1] T. H. Colding and W. P. Minicozzi II, Generic mean curvature flow II; dynamics of a closed smooth singularity.

Show bibtex

@misc{CM1, author = {Colding, Tobias H. and Minicozzi, II, William P.}, TITLE={Generic mean curvature flow {II}; dynamics of a closed smooth singularity}, NOTE={in preparation}, SORTYEAR={2012}, }
[CM2] T. H. Colding and W. P. Minicozzi II, Smooth compactness of self-shrinkers.

Show bibtex

@misc{CM2, author = {Colding, Tobias H. and Minicozzi, II, William P.}, TITLE={Smooth compactness of self-shrinkers}, NOTE={to appear in {\it Comm. Math. Helv.} {\bf 87} (2012)}, ARXIV={0907.2594}, SORTYEAR={2012}, }
[CM3] T. H. Colding and W. P. Minicozzi II, Minimal Surfaces, New York: New York University Courant Institute of Mathematical Sciences, 1999, vol. 4.

Show bibtex

@book {CM3, MRKEY = {1683966}, AUTHOR = {Colding, Tobias H. and Minicozzi, II, William P.}, TITLE = {Minimal Surfaces}, SERIES = {Courant Lect. Notes Math.}, VOLUME = {4}, PUBLISHER = {New York University Courant Institute of Mathematical Sciences}, ADDRESS = {New York}, YEAR = {1999}, PAGES = {viii+124}, ISBN = {0-9658703-3-2}, MRCLASS = {49Q05 (49-02 53A10 58E12)}, MRNUMBER = {1683966}, ZBLNUMBER = {0987.49025}, }
[CM4] $Go to document$ T. H. Colding and W. P. Minicozzi II, "Shapes of embedded minimal surfaces," Proc. Natl. Acad. Sci. USA, vol. 103, iss. 30, pp. 11106-11111, 2006.

Show bibtex

@article {CM4, MRKEY = {2242650}, AUTHOR = {Colding, Tobias H. and Minicozzi, II, William P.}, TITLE = {Shapes of embedded minimal surfaces}, JOURNAL = {Proc. Natl. Acad. Sci. USA}, FJOURNAL = {Proceedings of the National Academy of Sciences of the United States of America}, VOLUME = {103}, YEAR = {2006}, NUMBER = {30}, PAGES = {11106--11111}, ISSN = {1091-6490}, CODEN = {PNASFB}, MRCLASS = {53A10}, MRNUMBER = {2242650}, MRREVIEWER = {Magdalena Daniela Toda}, DOI = {10.1073/pnas.0510379103}, ZBLNUMBER = {1175.53008}, }
[CM5] $Go to document$ T. H. Colding and W. P. Minicozzi II, "The space of embedded minimal surfaces of fixed genus in a 3-manifold. I. Estimates off the axis for disks," Ann. of Math., vol. 160, iss. 1, pp. 27-68, 2004.

Show bibtex

@article {CM5, MRKEY = {2119717}, AUTHOR = {Colding, Tobias H. and Minicozzi, II, William P.}, TITLE = {The space of embedded minimal surfaces of fixed genus in a 3-manifold. {I}. {E}stimates off the axis for disks}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {160}, YEAR = {2004}, NUMBER = {1}, PAGES = {27--68}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {53A10}, MRNUMBER = {2119717}, MRREVIEWER = {Oliver C. Schn{ü}rer}, DOI = {10.4007/annals.2004.160.27}, ZBLNUMBER = {1070.53031}, }
[CM6] $Go to document$ T. H. Colding and W. P. Minicozzi II, "The space of embedded minimal surfaces of fixed genus in a 3-manifold. II. Multi-valued graphs in disks," Ann. of Math., vol. 160, iss. 1, pp. 69-92, 2004.

Show bibtex

@article {CM6, MRKEY = {2119718}, AUTHOR = {Colding, Tobias H. and Minicozzi, II, William P.}, TITLE = {The space of embedded minimal surfaces of fixed genus in a 3-manifold. {II}. {M}ulti-valued graphs in disks}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {160}, YEAR = {2004}, NUMBER = {1}, PAGES = {69--92}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {53A10}, MRNUMBER = {2119718}, MRREVIEWER = {Oliver C. Schn{ü}rer}, DOI = {10.4007/annals.2004.160.69}, ZBLNUMBER = {1070.53032}, }
[CM7] $Go to document$ T. H. Colding and W. P. Minicozzi II, "The space of embedded minimal surfaces of fixed genus in a 3-manifold. III. Planar domains," Ann. of Math., vol. 160, iss. 2, pp. 523-572, 2004.

Show bibtex

@article {CM7, MRKEY = {2123932}, AUTHOR = {Colding, Tobias H. and Minicozzi, II, William P.}, TITLE = {The space of embedded minimal surfaces of fixed genus in a 3-manifold. {III}. {P}lanar domains}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {160}, YEAR = {2004}, NUMBER = {2}, PAGES = {523--572}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {53A10 (53C42)}, MRNUMBER = {2123932}, MRREVIEWER = {Fei-Tsen Liang}, DOI = {10.4007/annals.2004.160.523}, ZBLNUMBER = {1076.53068}, }
[CM8] $Go to document$ T. H. Colding and W. P. Minicozzi II, "The space of embedded minimal surfaces of fixed genus in a 3-manifold. IV. Locally simply connected," Ann. of Math., vol. 160, iss. 2, pp. 573-615, 2004.

Show bibtex

@article {CM8, MRKEY = {2123933}, AUTHOR = {Colding, Tobias H. and Minicozzi, II, William P.}, TITLE = {The space of embedded minimal surfaces of fixed genus in a 3-manifold. {IV}. {L}ocally simply connected}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {160}, YEAR = {2004}, NUMBER = {2}, PAGES = {573--615}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {53A10 (53C42)}, MRNUMBER = {2123933}, MRREVIEWER = {Fei-Tsen Liang}, DOI = {10.4007/annals.2004.160.573}, ZBLNUMBER = {1076.53069}, }
[CM9] T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold V; Fixed genus.

Show bibtex

@misc{CM9, author = {Colding, Tobias H. and Minicozzi, II, William P.}, TITLE = {The space of embedded minimal surfaces of fixed genus in a 3-manifold {V}; Fixed genus}, SORTYEAR={2013}, ARXIV={math.DG/0509647}, NOTE={preprint}, }
[CM10] T. H. Colding and W. P. Minicozzi II, A course on the heat equation and mean curvature flow.

Show bibtex

@misc{CM10, author = {Colding, Tobias H. and Minicozzi, II, William P.}, TITLE = {A course on the heat equation and mean curvature flow}, NOTE={in preparation}, SORTYEAR={2020}, }
[CM11] T. H. Colding and W. P. Minicozzi II, A Course in Minimal Surfaces, Providence, RI: Amer. Math. Soc., 2011, vol. 121.

Show bibtex

@book {CM11, MRKEY = {2780140}, AUTHOR = {Colding, Tobias H. and Minicozzi, II, William P.}, TITLE = {A Course in Minimal Surfaces}, SERIES = {Grad. Stud. Math.}, VOLUME = {121}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {2011}, PAGES = {xii+313}, ISBN = {978-0-8218-5323-8}, MRCLASS = {53A10 (35J93 49Qxx)}, MRNUMBER = {2780140}, ZBLNUMBER = {05881490}, }
[D] R. J. Duffin, "Lower bounds for eigenvalues," Physical Rev., vol. 71, pp. 827-828, 1947.

Show bibtex

@article {D, MRKEY = {0020698}, AUTHOR = {Duffin, R. J.}, TITLE = {Lower bounds for eigenvalues}, JOURNAL = {Physical Rev.}, VOLUME = {71}, YEAR = {1947}, PAGES = {827--828}, MRCLASS = {36.0X}, MRNUMBER = {0020698}, MRREVIEWER = {R. E. Langer}, }
[E1] $Go to document$ K. Ecker, "Local monotonicity formulas for some nonlinear diffusion equations," Calc. Var. Partial Differential Equations, vol. 23, iss. 1, pp. 67-81, 2005.

Show bibtex

@article {E1, MRKEY = {2133662}, AUTHOR = {Ecker, Klaus}, TITLE = {Local monotonicity formulas for some nonlinear diffusion equations}, JOURNAL = {Calc. Var. Partial Differential Equations}, FJOURNAL = {Calculus of Variations and Partial Differential Equations}, VOLUME = {23}, YEAR = {2005}, NUMBER = {1}, PAGES = {67--81}, ISSN = {0944-2669}, MRCLASS = {35K55 (35K57)}, MRNUMBER = {2133662}, MRREVIEWER = {Maura Ughi}, DOI = {10.1007/s00526-004-0290-9}, ZBLNUMBER = {1119.35026}, }
[E2] K. Ecker, Regularity Theory for Mean Curvature Flow, Boston, MA: Birkhäuser, 2004, vol. 57.

Show bibtex

@book {E2, MRKEY = {2024995}, AUTHOR = {Ecker, Klaus}, TITLE = {Regularity Theory for Mean Curvature Flow}, SERIES = {Progr. Nonlinear Differential Equations Appl.}, VOLUME={57}, PUBLISHER = {Birkhäuser}, ADDRESS = {Boston, MA}, YEAR = {2004}, PAGES = {xiv+165}, ISBN = {0-8176-3243-3}, MRCLASS = {53C44 (35-02 35B65 35K55)}, MRNUMBER = {2024995}, MRREVIEWER = {Xi Ping Zhu}, ZBLNUMBER = {1058.53054}, }
[E3] $Go to document$ K. Ecker, "A formula relating entropy monotonicity to Harnack inequalities," Comm. Anal. Geom., vol. 15, iss. 5, pp. 1025-1061, 2007.

Show bibtex

@article {E3, MRKEY = {2403194}, AUTHOR = {Ecker, Klaus}, TITLE = {A formula relating entropy monotonicity to {H}arnack inequalities}, JOURNAL = {Comm. Anal. Geom.}, FJOURNAL = {Communications in Analysis and Geometry}, VOLUME = {15}, YEAR = {2007}, NUMBER = {5}, PAGES = {1025--1061}, ISSN = {1019-8385}, MRCLASS = {53C44}, MRNUMBER = {2403194}, MRREVIEWER = {John Urbas}, URL = {http://intlpress.com/CAG/CAG-v15.php#v15n5}, ZBLNUMBER = {1167.53055}, }
[E4] K. Ecker, "Heat equations in geometry and topology," Jahresber. Deutsch. Math.-Verein., vol. 110, iss. 3, pp. 117-141, 2008.

Show bibtex

@article {E4, MRKEY = {2456499}, AUTHOR = {Ecker, Klaus}, TITLE = {Heat equations in geometry and topology}, JOURNAL = {Jahresber. Deutsch. Math.-Verein.}, FJOURNAL = {Jahresbericht der Deutschen Mathematiker-Vereinigung}, VOLUME = {110}, YEAR = {2008}, NUMBER = {3}, PAGES = {117--141}, ISSN = {0012-0456}, CODEN = {JDMVA7}, MRCLASS = {58J35 (35K55 53C44)}, MRNUMBER = {2456499}, MRREVIEWER = {Esther Cabezas Rivas}, ZBLNUMBER = {1151.53061}, }
[EpG] C. L. Epstein and M. Gage, "The curve shortening flow," in Wave Motion: Theory, Modelling, and Computation, New York: Springer-Verlag, 1987, vol. 7, pp. 15-59.

Show bibtex

@incollection {EpG, MRKEY = {0920831}, AUTHOR = {Epstein, C. L. and Gage, Michael}, TITLE = {The curve shortening flow}, BOOKTITLE = {Wave Motion: Theory, Modelling, and Computation}, VENUE={{B}erkeley, {C}alif., 1986}, SERIES = {Math. Sci. Res. Inst. Publ.}, VOLUME = {7}, PAGES = {15--59}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1987}, MRCLASS = {58G11 (53A04)}, MRNUMBER = {0920831}, MRREVIEWER = {R. Osserman}, ZBLNUMBER = {0645.53028}, }
[EpW] $Go to document$ C. L. Epstein and M. I. Weinstein, "A stable manifold theorem for the curve shortening equation," Comm. Pure Appl. Math., vol. 40, iss. 1, pp. 119-139, 1987.

Show bibtex

@article {EpW, MRKEY = {0865360}, AUTHOR = {Epstein, C. L. and Weinstein, M. I.}, TITLE = {A stable manifold theorem for the curve shortening equation}, JOURNAL = {Comm. Pure Appl. Math.}, FJOURNAL = {Communications on Pure and Applied Mathematics}, VOLUME = {40}, YEAR = {1987}, NUMBER = {1}, PAGES = {119--139}, ISSN = {0010-3640}, CODEN = {CPAMA}, MRCLASS = {53A04 (35K22 58F10)}, MRNUMBER = {0865360}, MRREVIEWER = {R. Osserman}, DOI = {10.1002/cpa.3160400106}, ZBLNUMBER = {0602.34026}, }
[Ev] L. C. Evans, Partial Differential Equations, Providence, RI: Amer. Math. Soc., 1998, vol. 19.

Show bibtex

@book {Ev, MRKEY = {1625845}, AUTHOR = {Evans, Lawrence C.}, TITLE = {Partial Differential Equations}, SERIES = {Grad. Stud. Math.}, VOLUME = {19}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {1998}, PAGES = {xviii+662}, ISBN = {0-8218-0772-2}, MRCLASS = {35-01}, MRNUMBER = {1625845}, MRREVIEWER = {Luigi Rodino}, ZBLNUMBER = {0902.35002}, }
[EvSp] $Go to document$ L. C. Evans and J. Spruck, "Motion of level sets by mean curvature. I," J. Differential Geom., vol. 33, iss. 3, pp. 635-681, 1991.

Show bibtex

@article {EvSp, MRKEY = {1100206}, AUTHOR = {Evans, Lawrence C. and Spruck, J.}, TITLE = {Motion of level sets by mean curvature. {I}}, JOURNAL = {J. Differential Geom.}, FJOURNAL = {Journal of Differential Geometry}, VOLUME = {33}, YEAR = {1991}, NUMBER = {3}, PAGES = {635--681}, ISSN = {0022-040X}, CODEN = {JDGEAS}, MRCLASS = {35K55 (35B05 35D05 53A10)}, MRNUMBER = {1100206}, MRREVIEWER = {Friedrich Sauvigny}, URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214446559}, ZBLNUMBER = {0726.53029}, }
[GH] $Go to document$ M. Gage and R. S. Hamilton, "The heat equation shrinking convex plane curves," J. Differential Geom., vol. 23, iss. 1, pp. 69-96, 1986.

Show bibtex

@article {GH, MRKEY = {0840401}, AUTHOR = {Gage, M. and Hamilton, R. S.}, TITLE = {The heat equation shrinking convex plane curves}, JOURNAL = {J. Differential Geom.}, FJOURNAL = {Journal of Differential Geometry}, VOLUME = {23}, YEAR = {1986}, NUMBER = {1}, PAGES = {69--96}, ISSN = {0022-040X}, CODEN = {JDGEAS}, MRCLASS = {53A04 (35K05 52A40 58E99 58G11)}, MRNUMBER = {0840401}, MRREVIEWER = {R. Osserman}, URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214439902}, ZBLNUMBER = {0621.53001}, }
[G1] $Go to document$ M. A. Grayson, "The heat equation shrinks embedded plane curves to round points," J. Differential Geom., vol. 26, iss. 2, pp. 285-314, 1987.

Show bibtex

@article {G1, MRKEY = {0906392}, AUTHOR = {Grayson, Matthew A.}, TITLE = {The heat equation shrinks embedded plane curves to round points}, JOURNAL = {J. Differential Geom.}, FJOURNAL = {Journal of Differential Geometry}, VOLUME = {26}, YEAR = {1987}, NUMBER = {2}, PAGES = {285--314}, ISSN = {0022-040X}, CODEN = {JDGEAS}, MRCLASS = {53A04 (35K99 58G11)}, MRNUMBER = {0906392}, MRREVIEWER = {R. Osserman}, URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214441371}, ZBLNUMBER = {0667.53001}, }
[G2] $Go to document$ M. A. Grayson, "A short note on the evolution of a surface by its mean curvature," Duke Math. J., vol. 58, iss. 3, pp. 555-558, 1989.

Show bibtex

@article {G2, MRKEY = {1016434}, AUTHOR = {Grayson, Matthew A.}, TITLE = {A short note on the evolution of a surface by its mean curvature}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {58}, YEAR = {1989}, NUMBER = {3}, PAGES = {555--558}, ISSN = {0012-7094}, CODEN = {DUMJAO}, MRCLASS = {53A10 (35K55)}, MRNUMBER = {1016434}, MRREVIEWER = {Gerhard Huisken}, DOI = {10.1215/S0012-7094-89-05825-0}, ZBLNUMBER = {0677.53059}, }
[G3] $Go to document$ M. A. Grayson, "Shortening embedded curves," Ann. of Math., vol. 129, iss. 1, pp. 71-111, 1989.

Show bibtex

@article {G3, MRKEY = {0979601}, AUTHOR = {Grayson, Matthew A.}, TITLE = {Shortening embedded curves}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {129}, YEAR = {1989}, NUMBER = {1}, PAGES = {71--111}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {53C22 (58E10)}, MRNUMBER = {0979601}, MRREVIEWER = {Gudlaugur Thorbergsson}, DOI = {10.2307/1971486}, ZBLNUMBER = {0686.53036}, }
[Ha1] R. S. Hamilton, "Isoperimetric estimates for the curve shrinking flow in the plane," in Modern Methods in Complex Analysis, Princeton, NJ: Princeton Univ. Press, 1995, vol. 137, pp. 201-222.

Show bibtex

@incollection {Ha1, MRKEY = {1369140}, AUTHOR = {Hamilton, Richard S.}, TITLE = {Isoperimetric estimates for the curve shrinking flow in the plane}, BOOKTITLE = {Modern Methods in Complex Analysis}, VENUE={{P}rinceton, {NJ}, 1992}, SERIES = {Ann. of Math. Stud.}, VOLUME = {137}, PAGES = {201--222}, PUBLISHER = {Princeton Univ. Press}, ADDRESS = {Princeton, NJ}, YEAR = {1995}, MRCLASS = {58E10 (53A04)}, MRNUMBER = {1369140}, MRREVIEWER = {Anders Linn{é}r}, ZBLNUMBER = {0846.51010}, }
[Ha2] $Go to document$ R. S. Hamilton, "Harnack estimate for the mean curvature flow," J. Differential Geom., vol. 41, iss. 1, pp. 215-226, 1995.

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@article {Ha2, MRKEY = {1316556}, AUTHOR = {Hamilton, Richard S.}, TITLE = {Harnack estimate for the mean curvature flow}, JOURNAL = {J. Differential Geom.}, FJOURNAL = {Journal of Differential Geometry}, VOLUME = {41}, YEAR = {1995}, NUMBER = {1}, PAGES = {215--226}, ISSN = {0022-040X}, CODEN = {JDGEAS}, MRCLASS = {53C21 (53A07 58G30)}, MRNUMBER = {1316556}, MRREVIEWER = {Emmanuel Hebey}, URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214456010}, ZBLNUMBER = {0827.53006}, }
[H1] $Go to document$ G. Huisken, "Flow by mean curvature of convex surfaces into spheres," J. Differential Geom., vol. 20, iss. 1, pp. 237-266, 1984.

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@article {H1, MRKEY = {0772132}, AUTHOR = {Huisken, Gerhard}, TITLE = {Flow by mean curvature of convex surfaces into spheres}, JOURNAL = {J. Differential Geom.}, FJOURNAL = {Journal of Differential Geometry}, VOLUME = {20}, YEAR = {1984}, NUMBER = {1}, PAGES = {237--266}, ISSN = {0022-040X}, CODEN = {JDGEAS}, MRCLASS = {53C45 (49F05 58F17)}, MRNUMBER = {0772132}, MRREVIEWER = {R. Schneider}, URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214438998}, ZBLNUMBER = {0556.53001}, }
[H2] G. Huisken, "Local and global behaviour of hypersurfaces moving by mean curvature," Proc. CMA, ANU, vol. 26, 1991.

Show bibtex

@article{H2, author= {Huisken, Gerhard}, TITLE = {Local and global behaviour of hypersurfaces moving by mean curvature}, JOURNAL={Proc. CMA, ANU}, VOLUME={26}, YEAR={1991}, }
[H3] $Go to document$ G. Huisken, "Asymptotic behavior for singularities of the mean curvature flow," J. Differential Geom., vol. 31, iss. 1, pp. 285-299, 1990.

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@article {H3, MRKEY = {1030675}, AUTHOR = {Huisken, Gerhard}, TITLE = {Asymptotic behavior for singularities of the mean curvature flow}, JOURNAL = {J. Differential Geom.}, FJOURNAL = {Journal of Differential Geometry}, VOLUME = {31}, YEAR = {1990}, NUMBER = {1}, PAGES = {285--299}, ISSN = {0022-040X}, CODEN = {JDGEAS}, MRCLASS = {53A10 (35B99 53C45 58G11)}, MRNUMBER = {1030675}, MRREVIEWER = {Dennis M. DeTurck}, URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214444099}, ZBLNUMBER = {0694.53005}, }
[H4] G. Huisken, "Local and global behaviour of hypersurfaces moving by mean curvature," in Differential Geometry: Partial Differential Equations on Manifolds, Providence, RI: Amer. Math. Soc., 1993, vol. 54, pp. 175-191.

Show bibtex

@incollection {H4, MRKEY = {1216584}, AUTHOR = {Huisken, Gerhard}, TITLE = {Local and global behaviour of hypersurfaces moving by mean curvature}, BOOKTITLE = {Differential Geometry: Partial Differential Equations on Manifolds}, VENUE={{L}os {A}ngeles, {CA}, 1990}, SERIES = {Proc. Sympos. Pure Math.}, VOLUME = {54}, PAGES = {175--191}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {1993}, MRCLASS = {58E15 (53A10 58G11)}, MRNUMBER = {1216584}, MRREVIEWER = {Li Ma}, ZBLNUMBER = {0791.58090}, }
[H5] $Go to document$ G. Huisken, "A distance comparison principle for evolving curves," Asian J. Math., vol. 2, iss. 1, pp. 127-133, 1998.

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@article {H5, MRKEY = {1656553}, AUTHOR = {Huisken, Gerhard}, TITLE = {A distance comparison principle for evolving curves}, JOURNAL = {Asian J. Math.}, FJOURNAL = {The Asian Journal of Mathematics}, VOLUME = {2}, YEAR = {1998}, NUMBER = {1}, PAGES = {127--133}, ISSN = {1093-6106}, MRCLASS = {58E10 (53A04)}, MRNUMBER = {1656553}, MRREVIEWER = {Anders Linn{é}r}, ZBLNUMBER = {0931.53032}, URL = {http://intlpress.com/CAG/CAG-v15.php#v15n5}, }
[HP] $Go to document$ G. Huisken and A. Polden, "Geometric evolution equations for hypersurfaces," in Calculus of Variations and Geometric Evolution Problems, New York: Springer-Verlag, 1999, vol. 1713, pp. 45-84.

Show bibtex

@incollection {HP, MRKEY = {1731639}, AUTHOR = {Huisken, Gerhard and Polden, Alexander}, TITLE = {Geometric evolution equations for hypersurfaces}, BOOKTITLE = {Calculus of Variations and Geometric Evolution Problems}, VENUE={{C}etraro, 1996}, SERIES = {Lecture Notes in Math.}, VOLUME = {1713}, PAGES = {45--84}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1999}, MRCLASS = {53C44 (35K55 49Q05)}, MRNUMBER = {1731639}, MRREVIEWER = {John Urbas}, ZBLNUMBER = {0942.35047}, DOI = {10.1007/BFb0092669}, }
[HS1] $Go to document$ G. Huisken and C. Sinestrari, "Convexity estimates for mean curvature flow and singularities of mean convex surfaces," Acta Math., vol. 183, iss. 1, pp. 45-70, 1999.

Show bibtex

@article {HS1, MRKEY = {1719551}, AUTHOR = {Huisken, Gerhard and Sinestrari, Carlo}, TITLE = {Convexity estimates for mean curvature flow and singularities of mean convex surfaces}, JOURNAL = {Acta Math.}, FJOURNAL = {Acta Mathematica}, VOLUME = {183}, YEAR = {1999}, NUMBER = {1}, PAGES = {45--70}, ISSN = {0001-5962}, CODEN = {ACMAA8}, MRCLASS = {53C44 (35K55)}, MRNUMBER = {1719551}, MRREVIEWER = {Ben Andrews}, DOI = {10.1007/BF02392946}, ZBLNUMBER = {0992.53051}, }
[HS2] $Go to document$ G. Huisken and C. Sinestrari, "Mean curvature flow singularities for mean convex surfaces," Calc. Var. Partial Differential Equations, vol. 8, iss. 1, pp. 1-14, 1999.

Show bibtex

@article {HS2, MRKEY = {1666878}, AUTHOR = {Huisken, Gerhard and Sinestrari, Carlo}, TITLE = {Mean curvature flow singularities for mean convex surfaces}, JOURNAL = {Calc. Var. Partial Differential Equations}, FJOURNAL = {Calculus of Variations and Partial Differential Equations}, VOLUME = {8}, YEAR = {1999}, NUMBER = {1}, PAGES = {1--14}, ISSN = {0944-2669}, MRCLASS = {58E12 (35K55 53A10)}, MRNUMBER = {1666878}, MRREVIEWER = {John Urbas}, DOI = {10.1007/s005260050113}, ZBLNUMBER = {0992.53052}, }
[HS3] $Go to document$ G. Huisken and C. Sinestrari, "Mean curvature flow with surgeries of two-convex hypersurfaces," Invent. Math., vol. 175, iss. 1, pp. 137-221, 2009.

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@article {HS3, MRKEY = {2461428}, AUTHOR = {Huisken, Gerhard and Sinestrari, Carlo}, TITLE = {Mean curvature flow with surgeries of two-convex hypersurfaces}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {175}, YEAR = {2009}, NUMBER = {1}, PAGES = {137--221}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {53C44 (35K93 53C21)}, MRNUMBER = {2461428}, MRREVIEWER = {John Urbas}, DOI = {10.1007/s00222-008-0148-4}, ZBLNUMBER = {1170.53042}, }
[I1] $Go to document$ T. Ilmanen, Singularities of Mean Curvature Flow of Surfaces, 1995.

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@misc{I1, author={Ilmanen, Tom}, TITLE = {Singularities of Mean Curvature Flow of Surfaces}, YEAR={1995}, NOTE={preprint}, URL = {http://www.math.ethz.ch/~ilmanen/papers/pub.html}, }
[I2] T. Ilmanen, "Elliptic regularization and partial regularity for motion by mean curvature," Mem. Amer. Math. Soc., vol. 108, iss. 520, p. x, 1994.

Show bibtex

@article {I2, MRKEY = {1196160}, AUTHOR = {Ilmanen, Tom}, TITLE = {Elliptic regularization and partial regularity for motion by mean curvature}, JOURNAL = {Mem. Amer. Math. Soc.}, FJOURNAL = {Memoirs of the American Mathematical Society}, VOLUME = {108}, YEAR = {1994}, NUMBER = {520}, PAGES = {x+90}, ISSN = {0065-9266}, CODEN = {MAMCAU}, MRCLASS = {49Q20 (53A10 58E12)}, MRNUMBER = {1196160}, MRREVIEWER = {Wei Yue Ding}, ZBLNUMBER = {0798.35066}, }
[I3] $Go to document$ T. Ilmanen, Lectures on Mean Curvature Flow and Related Equations (Trieste Notes), 1995.

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@misc{I3, author = {Ilmanen, Tom}, TITLE = {Lectures on Mean Curvature Flow and Related Equations ({T}rieste Notes)}, YEAR={1995}, URL = {http://www.math.ethz.ch/~ilmanen/papers/pub.html}, }
[L] $Go to document$ B. H. Lawson Jr., "Local rigidity theorems for minimal hypersurfaces," Ann. of Math., vol. 89, pp. 187-197, 1969.

Show bibtex

@article {L, MRKEY = {0238229}, AUTHOR = {Lawson, Jr., H. Blaine}, TITLE = {Local rigidity theorems for minimal hypersurfaces}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {89}, YEAR = {1969}, PAGES = {187--197}, ISSN = {0003-486X}, MRCLASS = {53.74}, MRNUMBER = {0238229}, MRREVIEWER = {K. Nomizu}, ZBLNUMBER = {0174.24901}, DOI = {10.2307/1970816}, }
[MM] C. Mantegazza and A. Magni, "Some Remarks on Huisken’s Monotonicity Formula for Mean Curvature Flow," in Singularities in Nonlinear Evolution Phenomena and Applications, Novaga, M. and Orlandi, G., Eds., , 2009, vol. 9, pp. 157-169.

Show bibtex

@incollection{MM, author={Mantegazza, C. and Magni, A.}, TITLE={Some Remarks on {H}uisken's Monotonicity Formula for Mean Curvature Flow}, BOOKTITLE={Singularities in Nonlinear Evolution Phenomena and Applications}, EDITOR={Novaga, M. and Orlandi, G.}, SERIES={CRM Ser. Center ``Ennio De Giorgi'', Pisa}, YEAR={2009}, MRNUMBER = {2528703}, VOLUME = {9}, PAGES = {157--169}, }
[MiSi] $Go to document$ J. H. Michael and L. M. Simon, "Sobolev and mean-value inequalities on generalized submanifolds of $R^{n}$," Comm. Pure Appl. Math., vol. 26, pp. 361-379, 1973.

Show bibtex

@article {MiSi, MRKEY = {0344978}, AUTHOR = {Michael, J. H. and Simon, L. M.}, TITLE = {Sobolev and mean-value inequalities on generalized submanifolds of {$R\sp{n}$}}, JOURNAL = {Comm. Pure Appl. Math.}, FJOURNAL = {Communications on Pure and Applied Mathematics}, VOLUME = {26}, YEAR = {1973}, PAGES = {361--379}, ISSN = {0010-3640}, MRCLASS = {49F10 (46E35)}, MRNUMBER = {0344978}, MRREVIEWER = {David Kinderlehrer}, ZBLNUMBER = {0256.53006}, DOI = {10.1002/cpa.3160260305}, }
[N1] $Go to document$ X. H. Nguyen, "Construction of complete embedded self-similar surfaces under mean curvature flow. I," Trans. Amer. Math. Soc., vol. 361, iss. 4, pp. 1683-1701, 2009.

Show bibtex

@article {N1, MRKEY = {2465812}, AUTHOR = {Nguyen, Xuan Hien}, TITLE = {Construction of complete embedded self-similar surfaces under mean curvature flow. {I}}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {361}, YEAR = {2009}, NUMBER = {4}, PAGES = {1683--1701}, ISSN = {0002-9947}, CODEN = {TAMTAM}, MRCLASS = {53C44}, MRNUMBER = {2465812}, MRREVIEWER = {Henri Anciaux}, DOI = {10.1090/S0002-9947-08-04748-X}, ZBLNUMBER = {1166.53046}, }
[N2] X. H. Nguyen, "Construction of complete embedded self-similar surfaces under mean curvature flow. II," Adv. Differential Equations, vol. 15, iss. 5-6, pp. 503-530, 2010.

Show bibtex

@article{N2, author = {Nguyen, Xuan Hien}, TITLE = {Construction of complete embedded self-similar surfaces under mean curvature flow. {II}}, JOURNAL = {Adv. Differential Equations}, FJOURNAL = {Advances in Differential Equations}, VOLUME = {15}, YEAR = {2010}, NUMBER = {5-6}, PAGES = {503--530}, ISSN = {1079-9389}, MRCLASS = {53C44 (35C06 35J93)}, MRNUMBER = {2643233}, MRREVIEWER = {James McCoy}, }
[N3] $Go to document$ X. H. Nguyen, "Translating tridents," Comm. Partial Differential Equations, vol. 34, iss. 1-3, pp. 257-280, 2009.

Show bibtex

@article {N3, MRKEY = {2512861}, AUTHOR = {Nguyen, Xuan Hien}, TITLE = {Translating tridents}, JOURNAL = {Comm. Partial Differential Equations}, FJOURNAL = {Communications in Partial Differential Equations}, VOLUME = {34}, YEAR = {2009}, NUMBER = {1-3}, PAGES = {257--280}, ISSN = {0360-5302}, CODEN = {CPDIDZ}, MRCLASS = {53C44 (53A10 53C21)}, MRNUMBER = {2512861}, MRREVIEWER = {Shouwen Fang}, DOI = {10.1080/03605300902768685}, ZBLNUMBER = {1187.53071}, }
[OSe] $Go to document$ S. Osher and J. A. Sethian, "Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations," J. Comput. Phys., vol. 79, iss. 1, pp. 12-49, 1988.

Show bibtex

@article {OSe, MRKEY = {0965860}, AUTHOR = {Osher, Stanley and Sethian, James A.}, TITLE = {Fronts propagating with curvature-dependent speed: algorithms based on {H}amilton-{J}acobi formulations}, JOURNAL = {J. Comput. Phys.}, FJOURNAL = {Journal of Computational Physics}, VOLUME = {79}, YEAR = {1988}, NUMBER = {1}, PAGES = {12--49}, ISSN = {0021-9991}, CODEN = {JCTPAH}, MRCLASS = {80A25 (80-08)}, MRNUMBER = {0965860}, ZBLNUMBER = {0659.65132}, DOI = {10.1016/0021-9991(88)90002-2}, }
[ScSi] $Go to document$ R. Schoen and L. Simon, "Regularity of stable minimal hypersurfaces," Comm. Pure Appl. Math., vol. 34, iss. 6, pp. 741-797, 1981.

Show bibtex

@article {ScSi, MRKEY = {0634285}, AUTHOR = {Schoen, Richard and Simon, Leon}, TITLE = {Regularity of stable minimal hypersurfaces}, JOURNAL = {Comm. Pure Appl. Math.}, FJOURNAL = {Communications on Pure and Applied Mathematics}, VOLUME = {34}, YEAR = {1981}, NUMBER = {6}, PAGES = {741--797}, ISSN = {0010-3640}, CODEN = {CPAMAT}, MRCLASS = {49F22 (53C42 58E15)}, MRNUMBER = {0634285}, MRREVIEWER = {F. J. Almgren, Jr.}, DOI = {10.1002/cpa.3160340603}, ZBLNUMBER = {0497.49034}, }
[ScSiY] $Go to document$ R. Schoen, L. Simon, and S. T. Yau, "Curvature estimates for minimal hypersurfaces," Acta Math., vol. 134, iss. 3-4, pp. 275-288, 1975.

Show bibtex

@article {ScSiY, MRKEY = {0423263}, AUTHOR = {Schoen, R. and Simon, L. and Yau, S. T.}, TITLE = {Curvature estimates for minimal hypersurfaces}, JOURNAL = {Acta Math.}, FJOURNAL = {Acta Mathematica}, VOLUME = {134}, YEAR = {1975}, NUMBER = {3-4}, PAGES = {275--288}, ISSN = {0001-5962}, MRCLASS = {53C40 (49F10)}, MRNUMBER = {0423263}, MRREVIEWER = {Robert Gulliver}, ZBLNUMBER = {0323.53039}, DOI = {10.1007/BF02392104}, }
[Si] L. Simon, Lectures on geometric measure theory, Canberra: Australian National University Centre for Mathematical Analysis, 1983.

Show bibtex

@book {Si, MRKEY = {0756417}, AUTHOR = {Simon, Leon}, TITLE = {Lectures on geometric measure theory}, SERIES = {Proceedings of the Centre for Mathematical Analysis, Australian National University}, NUMBER = {3}, PUBLISHER = {Australian National University Centre for Mathematical Analysis}, ADDRESS = {Canberra}, YEAR = {1983}, PAGES = {vii+272}, ISBN = {0-86784-429-9}, MRCLASS = {49-01 (28A75 49F20)}, MRNUMBER = {0756417}, MRREVIEWER = {J. S. Joel}, ZBLNUMBER = {0546.49019}, }
[Sm] $Go to document$ K. Smoczyk, "Self-shrinkers of the mean curvature flow in arbitrary codimension," Int. Math. Res. Not., vol. 2005, iss. 48, pp. 2983-3004, 2005.

Show bibtex

@article {Sm, MRKEY = {2189784}, AUTHOR = {Smoczyk, Knut}, TITLE = {Self-shrinkers of the mean curvature flow in arbitrary codimension}, JOURNAL = {Int. Math. Res. Not.}, FJOURNAL = {International Mathematics Research Notices}, YEAR = {2005}, NUMBER = {48}, PAGES = {2983--3004}, ISSN = {1073-7928}, MRCLASS = {53C44}, MRNUMBER = {2189784}, MRREVIEWER = {Henri Anciaux}, DOI = {10.1155/IMRN.2005.2983}, VOLUME = {2005}, ZBLNUMBER = {1085.53059}, }
[St] $Go to document$ A. Stone, "A density function and the structure of singularities of the mean curvature flow," Calc. Var. Partial Differential Equations, vol. 2, iss. 4, pp. 443-480, 1994.

Show bibtex

@article {St, MRKEY = {1383918}, AUTHOR = {Stone, Andrew}, TITLE = {A density function and the structure of singularities of the mean curvature flow}, JOURNAL = {Calc. Var. Partial Differential Equations}, FJOURNAL = {Calculus of Variations and Partial Differential Equations}, VOLUME = {2}, YEAR = {1994}, NUMBER = {4}, PAGES = {443--480}, ISSN = {0944-2669}, MRCLASS = {58E15 (35K55 58G11)}, MRNUMBER = {1383918}, MRREVIEWER = {Thomas Ivey}, DOI = {10.1007/BF01192093}, ZBLNUMBER = {0833.35062}, }
[V] $Go to document$ J. J. L. Velázquez, "Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow," Ann. Scuola Norm. Sup. Pisa Cl. Sci., vol. 21, iss. 4, pp. 595-628, 1994.

Show bibtex

@article {V, MRKEY = {1318773}, AUTHOR = {Vel{á}zquez, J. J. L.}, TITLE = {Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow}, JOURNAL = {Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, FJOURNAL = {Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV}, VOLUME = {21}, YEAR = {1994}, NUMBER = {4}, PAGES = {595--628}, ISSN = {0391-173X}, CODEN = {PSNAAI}, MRCLASS = {53A10 (35K99 58E15 58G11)}, MRNUMBER = {1318773}, MRREVIEWER = {Ben Andrews}, URL = {http://www.numdam.org/item?id=ASNSP_1994_4_21_4_595_0}, ZBLNUMBER = {0926.35023}, }
[W1] B. White, "Evolution of curves and surfaces by mean curvature," in Proceedings of the International Congress of Mathematicians, Vol. I, Beijing, 2002, pp. 525-538.

Show bibtex

@inproceedings {W1, MRKEY = {1989203}, AUTHOR = {White, Brian}, TITLE = {Evolution of curves and surfaces by mean curvature}, BOOKTITLE = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. {I}}, VENUE={{B}eijing, 2002}, PAGES = {525--538}, PUBLISHER = {Higher Ed. Press}, ADDRESS = {Beijing}, YEAR = {2002}, MRCLASS = {53C44}, MRNUMBER = {1989203}, MRREVIEWER = {Tommaso Pacini}, ZBLNUMBER = {1036.53045}, }
[W2] $Go to document$ B. White, "The size of the singular set in mean curvature flow of mean-convex sets," J. Amer. Math. Soc., vol. 13, iss. 3, pp. 665-695, 2000.

Show bibtex

@article {W2, MRKEY = {1758759}, AUTHOR = {White, Brian}, TITLE = {The size of the singular set in mean curvature flow of mean-convex sets}, JOURNAL = {J. Amer. Math. Soc.}, FJOURNAL = {Journal of the American Mathematical Society}, VOLUME = {13}, YEAR = {2000}, NUMBER = {3}, PAGES = {665--695}, ISSN = {0894-0347}, MRCLASS = {53C44 (49Q20)}, MRNUMBER = {1758759}, MRREVIEWER = {Harold Parks}, DOI = {10.1090/S0894-0347-00-00338-6}, ZBLNUMBER = {0961.53039}, }
[W3] $Go to document$ B. White, "The nature of singularities in mean curvature flow of mean-convex sets," J. Amer. Math. Soc., vol. 16, iss. 1, pp. 123-138, 2003.

Show bibtex

@article {W3, MRKEY = {1937202}, AUTHOR = {White, Brian}, TITLE = {The nature of singularities in mean curvature flow of mean-convex sets}, JOURNAL = {J. Amer. Math. Soc.}, FJOURNAL = {Journal of the American Mathematical Society}, VOLUME = {16}, YEAR = {2003}, NUMBER = {1}, PAGES = {123--138}, ISSN = {0894-0347}, MRCLASS = {53C44 (49Q20)}, MRNUMBER = {1937202}, MRREVIEWER = {Shu-Yu Hsu}, DOI = {10.1090/S0894-0347-02-00406-X}, ZBLNUMBER = {1027.53078}, }
[W4] B. White, "Stratification of minimal surfaces, mean curvature flows, and harmonic maps," J. Reine Angew. Math., vol. 488, pp. 1-35, 1997.

Show bibtex

@article {W4, MRKEY = {1465365}, AUTHOR = {White, Brian}, TITLE = {Stratification of minimal surfaces, mean curvature flows, and harmonic maps}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal für die Reine und Angewandte Mathematik}, VOLUME = {488}, YEAR = {1997}, PAGES = {1--35}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {49Q05 (49Q20 53C42 58E20)}, MRNUMBER = {1465365}, MRREVIEWER = {Harold Parks}, ZBLNUMBER = {0874.58007}, }
[W5] $Go to document$ B. White, "A local regularity theorem for mean curvature flow," Ann. of Math., vol. 161, iss. 3, pp. 1487-1519, 2005.

Show bibtex

@article {W5, MRKEY = {2180405}, AUTHOR = {White, Brian}, TITLE = {A local regularity theorem for mean curvature flow}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {161}, YEAR = {2005}, NUMBER = {3}, PAGES = {1487--1519}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {53C44}, MRNUMBER = {2180405}, MRREVIEWER = {James McCoy}, DOI = {10.4007/annals.2005.161.1487}, ZBLNUMBER = {1091.53045}, }
[Wi] N. Wickramasekera, A general regularity theorem for stable codimension $1$ integral varifolds, 2009.

Show bibtex

@misc{Wi, author={Wickramasekera, N.}, TITLE={A general regularity theorem for stable codimension $1$ integral varifolds}, NOTE={preprint}, YEAR={2009}, ARXIV={0911.4883}, }

Generic mean curvature flow I; generic singularities

Abstract

Authors