Details

Autor(en) / Beteiligte

• Ecker, Klaus

Titel

Regularity Theory for Mean Curvature Flow [Elektronische Ressource]

Ist Teil von

• Progress in Nonlinear Differential Equations and Their Applications : 57

Ort / Verlag

Boston, MA : Birkhäuser Boston

Erscheinungsjahr

2004

Link zum Volltext

• Volltext

Link zu anderen Inhalten

• Volltext

Beschreibungen/Notizen

• ThisbookstartedasasetofinformalnotesonBrakke’sregularitytheoryfor meancurvature?ow([B]). Thesenotesfocussedonthespecialcasewhere smooth solutions of mean curvature ?ow develop singularities for the ?rst time,thusexpressingtheunderlyingideasalmostentirelyinthelanguageof differentialgeometryandpartialdifferentialequations.Inparticular,notation fromgeometricmeasuretheorywaskepttoaminimum. After I gave lectures on Brakke’s work during 1994 in the Mathematics DepartmentsatStanfordUniversityandtheUniversityofTubingen ¨ andata workshop on Motion by Mean Curvature in Trento, I was encouraged by a numberofcolleaguestopublishmynotes. Since that time, but particularly since 1978, when Brakke’s work ?rst appeared, there have been many new developments in mean curvature ?ow startingwithHuisken’sworkin1984([Hu1]).Someofthesehaveresultedin signi?cantsimpli?cationsofBrakke’soriginalargumentsaswellasimpro- mentsofhisresultsinspecialsituations.^
• Thisincludesparticularlythecase ofevolvinghypersurfaceswithpositivemeancurvature.Remarkablythough, inthegeneralcasetheestimateofthesingularsetprovidedinBrakke’smain regularitytheorem([B],Theorem6.12)hasnotbeenimprovedupontodate. ThebulkofthematerialinthistextisbasedonlecturesIgaveinthe- partmentofMathematicsattheUniversitat ¨ Freiburg,Germany,fromNov- ber2000toFebruary2001andatMonashUniversity,Melbourne,Australia duringthe?rsthalfof2001. Thecentralthemeistheregularitytheoryformeancurvature?owleading up to a proof of Brakke’s main regularity theorem ([B], Theorem 6.12) for the special case where smooth solutions develop singularities. In this se- containedpresentation,IhavereplacedmanyofBrakke’soriginaltechniques by more recent methods wherever this led to a clear simpli?cation of his x Preface arguments. Someofhisoriginalideas,inonlyslightlymodi?edform,have beenincludedinanappendix.^
• Under additional assumptions such as symmetry conditions or dim- sionalrestrictionsonthesolutionorsignconditionsonthemeancurvature, improvedestimatesforthedimensionofthesingularsetorre?neddescr- tions of the behaviour of the solution near singularities can be established. I have, however, decided not to include a treatment of such results in this presentation.^
• Inparticular,thebookdoesnotcoverthefollowingimportant contributions:Altschuler,AngenentandGiga’sworkonisolatedsingularities of surfaces of revolution ([AAG]), Angenent and Velazquez’s construction of solutions exhibiting degenerate neckpinches ([AV]), Hamilton’s in?u- tialHarnackinequalityforconvexsolutions([Ha4]),Huisken’sclassi?cation of self-similar solutions with nonnegative mean curvature ([Hu3]), Huisken and Sinestrari’s and White’s asymptotic description of singularities in the mean convex case ([HS1], [W4]), Ilmanen’s results on smooth blow-ups in twodimensions([I1],[I2])aswellasWhite’sdimensionreductionargument ([W1]). Thelatterworkswithoutadditionalassumptionsonthesolutionbut so far implies improved (and optimal) estimates for the singular set only in themeanconvexcase([W1],[W2],[W4]).^
• Ialsohavechosennottoincludeimportantalternativeapproachestomean curvature?owsuchasthelevel-setapproachadoptedbyEvansandSpruck ([ES1]-[ES3])andbyChen,GigaandGoto([CGG],[GG1],[GG2])aswell astheworkofIlmanen([I1])whichestablishesalinkbetweenlevel-set?ow andBrakke’svarifoldsolutionframework