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