Moduliof Einsteinmfds part I C

Hj Hein WWUMinstersome review somejointwork w 0 Biquard

part 2 Thu 8am byT Auch

Anderson Bando Cheeger Kasue Nakajima Tian

Kronheimer'sgravitational instantons

Renormativedvolumeof ALE spaces

then Let Ming7 be a sequenceofclosed Einstein4 infds such that

Ridgi digi Hilesdiam Ming ED M

voecni.gov o

Osx Mike GHEinstein

SmithiemlgidVgi Gamp Bonnet

2 Then a subsequencehas a GH limit Magoowhich is a compact 4 dime Einsteinorbifold with onlyisolated singularitiesmodeled an RMP Pc8014 actingfreelyon 5 Convergence is in CF over theregularpartof Ma ga

hr s 1990

i Gromov'saptness thin2 E regularity if S 1RmPdVs E D V

Br p

thensup lRmIE CCD V f f lRm1211BEp Br p

Sobolev ineq dependson Rio's 3g

upperdiameter Dlowervolume V

limit Ma ga has atworst finitesingularset

with at worst of curative blowupat each singular point

3 f lAml IV isscale invariant in dim4

remains to establish orbifold structure at sing pointsslightlymodernized version

can passtotangent cones

repeattheory on unit ball in tangent onetangent cones havefinitesingularsetand o F curvature blowup as well

tangent cones must be CC IT RMTalsodirectlyfrom 4 dime Ricci flat w isolatedsings

Cheeger Tian generalanalysisof Ric 0on Riemannian cones 5 orbifold regularity

Meer bubbletreestructure

o U

Siret

bubbles iteratedbubbles are ALEspac.es

complete Ricci flat orbifoldMfg with finitesingular set such that I differ 4124lb toDIP Mlks th oIg t Off

4as r a

gU

EatinEan Tanostahomtheimentiaraergertifan

5 Bgh dirgh thdtrgh g mopanwadfr

usefulfor gangefixing on Einsteinmfds 4

Bg g DgDgX t BigXeasytosolvefor X

h hot higher order

Dgoho O lBgoho O THE

I

f hiifeaf.anardratifGreerisfdrlHogdgfjni.s2ndder ofGreen's

generalideai on 11241T havespecial vectorfields

scaling Xo _r2r conformal hilligfieldisometries XoC so killingfields

can we extend these to conformal

killingfields on Mtgcan extend harmonically D DgX O X Xcan trytointegratebyparts in 1 7

Too naive Therehaveto be obstructions 6C

forconf Killingfield if Xexists thenthobeiometric to RHP go

forthillingfieldi Eguchi Hanson P Id

isometrysoonp is U2 E SO 4

directionsXoC so 4 tubmust beobstructed eventhoughthey are indeed T invariant

BHi forconthikingfields X Xo Fdr on 11248calculateobstructions toextensioninterpret geometrically

keypoint precisechoiceofdiffeo IT

A construct a function u M IRStu Agu 8 and oI n r't lowerorder

Igor 8HOP

Etfbudging fixing

A q g r 8 2trg hot gohoXr2rD

to G s

I

8assuming in additionthat h o tar idr O

upshot f trgoho O.Bgoho O.to 2ridrITthen b 30 and b O iffMYg E RYP go

c geometric interpretation

8hrs1g fyagudVg hintbyparts

8 BsbYp go b t 0dg'tE

assumingagainb tenormalizedordn me

i sofar not clear if Rs depends on E hasgeommeaningNotice also under mean wratwe of22g f It D

one orderbetterthanexpected