关于复Monge-Ampère方程的若干研究-豆柴文库

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关于复Monge-Ampère方程的若干研究
Abstract
TheMonge-Ampèreequationisatypeofnonlinearpartialdifferentialequationthatarisesinvariousmathematicaldisciplines,includingdifferentialgeometry,potentialtheory,andoptimaltransportation.Inrecentyears,therehasbeensignificantresearchonthemoregeneralclassofequationsknownasthecomplexMonge-Ampèreequations.ThispaperaimstoprovideanoverviewofsomerecentdevelopmentsinthestudyofcomplexMonge-Ampèreequationsandtheirapplications.
Introduction
TheMonge-Ampèreequation,namedafterGaspardMongeandAndréMarieAmpère,isafundamentalequationingeometrythatdescribesthedeformationofavolume-preservingmapinEuclideanspace.Theequationtakestheformdet(D²u)=f(x),whereuistheunknownfunction,f(x)isagivensourceterm,andD²udenotestheHessianmatrixofu.TheMonge-Ampèreequationhasbeenextensivelystudiedandhasimportantapplicationsingeometricoptics,optimaltransportationtheory,andconvexanalysis.
ThecomplexMonge-AmpèreequationisanaturalgeneralizationoftheMonge-Ampèreequationtothecomplexsetting.Itinvolvesaplurisubharmonic(psh)functionuandisgivenbydet(∂²u)=ω(x),where∂²uisthecomplexHessianofuandω(x)isagivenpositivemeasure.ThestudyofcomplexMonge-Ampèreequationsprovidesarichsourceofmathematicalproblems,connectingdiverseareassuchascomplexanalysis,geometricanalysis,andalgebraicgeometry.
RecentDevelopments
Inrecentyears,therehavebeensignificantadvancesinthestudyofcomplexMonge-Ampèreequations.Oneimportantdevelopmentistheregularitytheoryforsolutionsoftheequation,whichwasestablishedbyCegrell,Caffarelli,andtheCalabiconjecture.Thistheoryprovidesadeepunderstandingoftheregularitypropertiesofplurisubharmonicfunctionsandtheirconnectionstocomplexgeometry.
AnotherareaofactiveresearchistheexistenceanduniquenessofsolutionstothecomplexMonge-Ampèreequation.Thisproblemishighlynonlinearandchallenging,butprogresshasbeenmadebyusingtechniquesfrompluripotentialtheoryandnonlinearanalysis.Inparticular,theconceptofKähler-EinsteinmetricsandtheirrelationtothecomplexMonge-Ampèreequationhasbeenafruitfulavenueofinvestigation.