Regularity Theory for Mean Curvature Flow
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.
Devoted to the motion of surfaces for which the normal velocity at every point is given by the mean curvature at that point; this geometric heat flow process is called mean curvature flowsMean curvature flow and related geometric evolution equations are important tools in mathematics and mathematical physicsPresents techniques for the study of singularities of mean curvature flow and is largely based on the work of K. Brakke, although more recent developments are incorporated