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Partial Regularity Theory of non - collapsed limits of RFS
Goat : Understand singularity formation in RF.
Blow -up analysis (M , lgelteco.TT
Fix Hi, ti) E M x Co,-13
,ti p T
T- ---
- - - -- -
-
-- -
.
.
*Hats)
Xi -Dcs gets ITII
subseq . c.xi.tn
numb
•
1-1M
??(Mi Ai SA? tttitef ti,o] ,Hi
,O)) (Mcs
,(gas,e)⇐ o ,
has,O))
-
blow-
up l singularity model[• Hamilton -
convergence :
require trans CXE near Gi,ti)
=D limit is smooth• F-
convergenceno requirements=D limit is metric flow
singularity formation -is singularity model
Gradieutshriukiugsoh.to#s ( GSS) (Meg ,f )
RiceTff - at g - O
MDGt = - t fig ,
t so RF
[flow of - t Pf
examples : ② Gaussian shrinker ( IR", goad , f = #xp )
② Einstein metrics ( Ric - tag , f- = const)
③ Cylinder ( 5×112,2 gsztgp ,f- =L 5)
Conjecture For any RF (M, Cge)ee↳⇒) , too,
"
most"
singularitymodels are GSS
.
[possibly singular
Example (Appleton)4DRFs sit. for tat
,feet
i¥÷€¥÷-
= Za (GSS)
-
neBryant soliton 1212 ③-
= RP3xR ( Gss) ③
Actual Goal : Find a compactness + partial regularity theory for RF-
2-
Recall : (+PR for Einstein metricsmummy : ( Gromov
, Golding , Cheeger, Tian,Nader)
(Mi, gi , Xi ) complete, Rig , ⇒ igi , Hill
Compactness : F subsequence st .
(Mi, gi , xi)
GH
qi'→ is
D (X , d,as )
dgi
PR : Assume non - collapsing : 1B Gi,r) ) Z v rn
Then X = R j g S.t. :Open dosed
② There is a smooth structure + Einstein metric go on R sit.
dlp = dogs (⇒ (Xd) = CRdg.TT#Yfon )② dingus in-4
③ Every tangent cone at any xex is a metric core.
④ 5<5 'a . . - - CS
"
-4=5 st.
• dim# S
"Ek
•
any x c- SIS" has a tangent cone that splits off IR" -factorexample ( MEH , 12gEH , x ) 1124/212
Ric EO Got
New CtPR for Rts
Given Rts (Mi, Cgi
, t)feet
,oy , Gi , o)) , Ti→Is 70
Compactness ( fall) F subsequence
(Mi , ( gist) , hi , o) )F
D ( Tf , (Kas ; t ) te C-Too,OT )its co
ofmetric flow• time- slices (At , At)•
conjugate heat kernels : x eat,set
Vx; s E Pots)
PR : Assume non - collapsing : Nx, oLto) Z -Yo
[ptd Nash entropy
Then I = R u s sit.
:
open closed
② R has a smooth
Ricciflowspa-etimei.EE:*!÷÷¥¥i!÷. IIT
and E is uniquely determined by this structure .② dimwit S f (ht2) - 4
Tusing P*_ parabolic ubhds
③ All tangent flows are (singular) GSS
④ 5°C Slc - - - -S"
-2=5 st.
:
• dim# Sk f k
•
any xe S''-S" has a tangent flow that
• splits off IRT- factor OR
• splits off Rk-2 - factor + static (Ric -=o) ( k > 2)
examples ② (MEH , ( HSE# 7*0 ,X) Static flow on 1124/22
S = Ex IR dim5=2T
the= (4+2)-4
② SIT? gt =- 2Tget gtz
S = * x TZ x { of dim 5--2=(4+2)-4
Main Ingredients of PR -theory in Einstein case Rico to besure
monotone quantity : Vfx,D= BGr f
(Bishop - Gromov)
almost cone- rigidity : FSG) >o : VG ,F) 3 Vfx , Sr) - S( Cheeger - Coldly)
x.÷g⇒ :":*::S:c:O:*:
come-splitting : If CZ, dz , Zf),i -42
,are metric cones
,2-it-22
,
vertex
then 2- ⇐ Z' xR.
almost cone- splitting : If (M , g.xi) ,i - 1,2,8k)- close to metric core
lfdlxyxz) SC,then (Meg , Xi) e- close Z' XR
E- regularity : H ( M,r -2g , x) is en - close to some Extra
,then
112mL f Cr?
Einstein case RF case
metricspaces metric flows
GH - convergence F- convergence✓ (x , r) > V r
"
Nx, o(to) Z -Yo
dimm , drink dinah , dimatangent cone tangent flowsmetric cones (metrics gradient shrinking solitons
monotone quantity VG,m Nyo (t)
almost one rigidity almost monotonicity of N=D E - selfsimilarity
( integral condition)
come splitting no one splitting for metric GSS N/A Caprivi)
almost one splitting E- self-similar at nearby pts⇒ F weak e-splitting map
OR e-static
T tintegral conditions
E- regularity weak e - splitting=D strong e- splitting=D curvature bound
no strict localizations ( cutoff fatsno distance expansion boards
no lower HK bounds
Recall GSS equations (M , Cgttso,f) , t= -t
Riege -1172ft - ¥gt=O If = 117ft-
Consequencesmum
:
Rt If - 2¥ = O
- EUR+11712) tf = WEIR-de-DTR
- off = Df - lPfktR-Lz SED 1I¥l4ae5% e-t) = o
conventionlf-f-dmi.fm Hat)-42 e-tdgt =LFurther consequencesrunny
!
c- (- ITAI'tDf) + f - Iz -W = o ( W - ft - - -K¥5"e-tdgr-fflkxe-TZ-fdgf-I-wcgt.EEc- ( 2 of -11712+12)tf- new (W- ft - -That-"e-tdgt-WLge.fe.TT)-
=: w Perelman's Harnack qty
DEH + 2- +W=O
Improved Integral Bounds ( M , Cgt)tef2, og),Xo EM
,RF - I
conjugate HK : dot =d%,o;t = (4te5ke-ttdgt-kcxo.co ; ; t) dyeat exo,o) tee
- t
NE) Do,ok)= ) ft dot - Z fo
WH :=W[gt , ftp.IE-J#CRtlPfI7tf-n)dutfOWEI = ENG)) '
W'E) = - 2tJmlRictDf - ¥gTd¥ TO
D T
• WET
=D W G) f NLT)TNE) • INE)
0
Suppose W (2) z -Y =D Wh) > -2T
D K (xo,o ;
.
,t) f ft e
-N-t)f9
=D f z - CCT)
Imf diet = NII t Z F ZE O
=p J Ifl dot f CCT)
t Jm( 11712 try,
due = WE) - WH thz f 'f th- 2
f O
d¥ ImR dot = ) DR dot t ) R D* (Hattiee-f) dgt = 2) IthickdotM¥2Rick
M To M
fi"
Tm kiddedt t fylde ,F CH)
Like Jm ltac-DF-fgkdqdt-W.lk) -WLD FLY2-
-W' (t)
{"
Im 117712 dadt + Jm ITAI' die , TCH)
Jm ITAM dot = Im ftp.TACKD-ke-tdgt
= Sm dix (Nfl' Df) dgt f CfmlDFI - 117ft diet
f c)MID 'S Pdq + If 1171 " dotM
=p SMITHY dot f C Im 117712 diet
{"
Im Nfl " dqndt § can
Set a text)-
Ke-f (D*u=o)
W'
-
= that - ITAKTR) tf - n Perelman's Harnack
Jw dot = WE)M
Perelman 1 section 9 :
D*(w u) = - 24RiotDF -LtgKu
,we 0
2+1 Dfl =/ wt that-TR -ftnlf -wit IDFlit IRI t Hin
=D J l Dfl die ,F Ca)
M
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