iins er berseminar

> ifferenial Geometry

n iy , Juno-

,2 2

Cp Convergence and e- Zeguariy he rems

fr ntropy and Scalar Curvature wer Bounds

' '

n-

w an- Chun ee and -

ar n aber

heme :

M"

. g) Riemannian manifold.

Put some constraints What structure does

on the curvature~

this impose on ( M. g) ?

to study :

lake sequence(Mi

, g;) satisfying constraint , look at limit-

Mi >

I 1-

What structure does have ?In what sense ?

F- ✗ Spaces with Ricci lower bounds

( Mi, g;) sat

.Ric ; 3

- IgiGH

( Mi , di ) → ( ✗ , d) Cheeger - Codding .

E - Regularity 1hm ( C '

ng'

97, Creager - C

'

ng'

97

tix E > O. there exists of = die

,n) > 0 such that if

( M. g) Riemannian manifold .

x c- M w/

Ricg 3 -

org volgl BC x. l)) 3 ( I - 8) Wn

'

henoc. " ( Bix,

D, 1310"

.

I ) ) s E.

His

In fact,GH can be

improved to biHolder

homeomorphic by iteratingon all scales .

to summarize :

Almost Euclidean nc apsing- Im s

-

Ricci cower boundt

uclidean volume

Geometrically close

to Euclidean

ur goal today :

E regularity assuming only scalar curvature lower bounds.

- mos- uclidean "l oncollapsing

"- Almost

scaar I wer s una Euclidean Perelman

entropy

Rg ? - d

"

Geometrically close < Not in any metric

0 uclidean" Space sense !

In"

dp sense."

Background Perelman Entropy µ g. t ) . T> 0

Optimal constant in family of log- Sobolev inequalities .

Def

(a) u g. I inf{ Ñlu) : /µ uidvolg = }

Ñlu = fm 4 rut + Rgu'

n' Iogu'

dvol.gr I log'll

if : og Sobolev on Euclidean space ⇐ ul gens , 1)= 0

.

fuilogu' s / 410nF dx n + log (4-11)-% if Ju' =L .

wi-h =L v2 = ( 4Ñ%e×p{ - ✗☐

2a }

t(b) vulg , L) = je T -1g ,

I

Some key perspective notes

M . g) complete with bounded curvature.

•• u( g. T ) ± for all T > 0

with equality < > M. g) = R

"

, gene

• Mlg ,T ) = - d for TE ,

I, 3g ? d

g Bglxir ? I - E) r "wn

re 0.1 ].

We will assume

µ / g. T 3 - d for TE 0,I

I w e 's turn to "

Geometrically close

understanding "

o uclidean

ok at limits of sequences ( Mi , g;) satisfying✓

Rg ;> '

i, m gi . L

) Z - Yi -

Lc- (O

. I

n what sense do they converge-

o IR? geuc?

A key point :

heir metric space structures

need not converge!

in: 3asic Idea :

' etric go on 2"

2" '

x 3

d×zEdx '

*A fibers

degenerateSaurer slowlyfrom one scale to

next

• ••.

• (IR" ", gene

r - I r = I

•

Metric is Euclidean

out here.

go

*

a ro

Can make this change

sow enough such that

IR

° Rage - de ) ' O, a

white lie : o

get scalar lower sound ,we need iery flat

• JU Ge , L ) - CCE ) s O cone me ric'

on 2h- '

n > 4.

T Te ( O,I

3.u n the pointed G - innit,centra line co apses to a point.

and

Voge Bge .

'

e : After rescaling chopping -action

, we ge

2 :

ake a flat ones I ? g. flat ①

2 btain ! gi by pasting

increasingly cense coaction f disjointostrips along parallel copies of S!

3 Can choose parameters such that

Rg ; > -'i µ gi . L 3 of t TECO

,I

3. ul

Gromov - Hausdorff ④G1

limit is 1" ! goat

intrinsic flat imi0 current

.

is zero Curren

' "

ake a flat oras in. go.at

÷÷÷÷÷i÷÷÷÷÷÷÷÷÷÷÷÷:÷¥E¥#¥E3 Can choose parameters such that

Rg ; > -'i ju gi . L 3 do F Te (O

,I

3. ul

c

: ¥¥¥¥#¥ ... .

#¥¥¥#¥ o ...

Up Sho : under scalar curva are and entropy cower bounds,

cis ance functions can

,

behave very poorly .

PER, p>

newnotion :

cp distance .

dlx.gl = sup { fix)- fly) : 1199-16<-1 }

Jef : M.gl Riemannian manifold. x.ge M, P > n

.

-

a) dplx . y) sup { Ifcx ) fly) / : f. left " dvolg ± I } .

b Bp × .r ) = { ye M

: dplx,y< r}

✓ FkFk → I a. e. fk → 1 in LP

.

£fk ↳ 1 in tf

some key perspective no-

es

• dolx.gl = dlx.gl and Iimp→ ,

dplxiy ) = dlx.glI

• dp reflects behavior of" P Sobolev space

• On euclidean space .- Mp

dplx.gl Spin ✗ - y .

• M, g) ,

(Nih) Cpt Riemannian manifolds .

If (M.dp.gr) , N, dp.in) isometric as metric spaces

then (M . g) ,( N

,h) isometric . as Riem . mflds .

1hm (-ee - Naber -N.

'

20)

tix n >_ 2, p ? n -11

,E > 0

.There exists D= dcn.p.ae) such that if

Mn, g complete with bounded curvature with

Ray Z - d, Mlg .

c) Z - J H Te 10.1-

then for any ✗ e-M,

E,•

CGH ( Bp.gl/.l,dp.g,Bp.geuc 0"

,I

, pieuc

V0 /g Bp.g ×

,r

• E S + E

Vgene(Bpigeuc 0 .

r )it recoil) .

1hm (-ee - Naber -N.

'

20)

Fix n > 2, oft 1

,E > 0

.There exists D= dcn

, ,e) such that if-

M"

, g closed with

Ray Z - d, vulg .

-43 - S H Te 10.1-

then

fml Rg 't

dvolg S E

Schoen Yau : if ( T? g) has Rg 30 ,then

g= gflat .

1hm (-ee - Naber -N.

'

20)

tix n >_ 2, p ? n -11

,> 0

.here exists D= dln ,p s

.

-

.if ?g ;)

satisfy

Rgi ? -'i, µ(gi.tl?-dV-- c- (0,1 ]

, Volgi(Mi s.

'

then up to subsequence,

Mi. gi

" ( T". gfiat in dp sense .

Limit spaces are \ metric spaces , they are

Rectifiable Ziemann ian spaces. g

I , m topological space with a Borel measure, equipped with

• Atlas of charts with bi Lipschitz transition

maps covering up to set of measure zero.

•• Possibly singular metric g defined in charts.

This is enough structure

to define W " ?

1hm (-ee - Naber -N.

'

20)

tix n > 2, p s n -11

.there exists or = den

, p) such that if { Mi, g ; , Xi

}

complete with bounded curvature with

Rg ;3 - d

, dig i. T) Z - J V te (0,1

Some subtlety tothen up to subsequence,

defining this

I i . gi , Xi" X

, g .x) in pointed dp sense a

where, g. x is a rectifiabe Riemannian space that is

• W" " - rectifiable complete

( W " "

space is"

big" and " well - behaved

"

• dp - complete

dp is metric generating same topology as

• is a smooth topological manifold .

u !