for gcemetric variational problems · rigidity of first variation in 150 perimetry alexandrovthm if...
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STABILITY THEOREMS
FOR GCEMETRIC VARIATIONAL PROBLEMS
FRANCESCO MAGGI
INTERNATIONAL CENTRE
FOR THEORETICAL PHYSICS TRIESTE
MATHEMATICS COLLOQUIUM
COURANT INSTITUTE NYU
2/2/2017
THE EUCLIDEAN ISOPERIMETRIC INQ
xiIF EEIRYO(lE÷ -
Hen PCE)znwFiEi¥XiIF = HOLDS THEN E = Brlx ) FOR Some NER "
r >oXIF E 15 NOT A BALL,
WHAT INFORMATION 15 ENCODED
INTO THE SMALNESS OF
SCE ) =MEIWIHIEFIT
- 1.
ALMOST ISOPCRIMETRIC SETS
SCE )=Pn!hE÷eF←In -1£80*1
Diana. Tentacles
ROUGH BOUNDARY g-'
"" . /! ,
↳ :{ ÷ =.
a÷÷¥€÷%I SMALL COMPONENTSSMALL HolesIn
, . . pm .
SHARP SPIKES
THM ( FUSOM.
PRATELLIANNMATH 08 ) 7 Cln ) > o S.
T.
IF EEIRYOCIEI< A THEN FNEIR ?rso
st. MetznwFiEi¥{stein , (
EOBRKTFZ}
IEI
OR,
EQUIVALENTLY,
8 (E) 2 an ) a ( EP
lsoperlmemlc→ SCE ) = nP¥g÷eFnIn - 1
DEFICIT
Asymmetry → LCEI
=inf{#9BgKT: NEIR ?rso }
THMCFUSOM .PRATELLIANNMATH 08 ) 8( E) 2 an ) & ( E )2
*EXPONENT 2 IS SHARP
SOBOLEV INEQUALITYXCONSTANT Cln ) NOT EXPLICIT
'× PROOF BY QUANTITATIVE symmetry , zqy ,o ,
) GAUSSIAN ISOPERIM .
FRACIIONAL ISOPCRIM .X.IDEA SYMMETRIZE E→E*
THEN or (E)
>_ggEµbTof| RKSZ INEQUALITY
✓UNNATURAL BY VARIOUS AUTHORS
LCE ) 2 &lE* ) >_ ccn )a( E)xIDEA SYMMETRIZC VERY GRADUALLY
1- CHOOSE RIGHT SYMMETRIES t DIMENSION INDUCTION
THE WULFF INEQUALITY
xIF EEIRYO(lE÷ -
HENBCE)znlk'TlEl¥BCE )=§g4w£)dH"
y :$ " -46 ,a ) WITH convex 1- tom.
EXTENSION
K=M{ ni NNEYIV ) } =Ky OPEN BOUNDED convex set
VE 8 " ixIF = HOLDS THEN
E=xtrkFOR Some NER "
r >oxcrystalline case
¥66196'[¥⇒ x isotropic case(tBz¥€€h€E¥*
THMCFIGALHM .PRATELLI WVMAT#11 )
8y(E) > an )2( E ,K )
'
where 8ytI=*¥- r & alektzn.fr IEOKTRKIn1khIEl¥ IEI
.XAGAIN EXPONENT 2 15 SHARP x.Clh ) EXPLICIT
,POLYNOMIAL IN n
×OPTIMAL MASS TRANSPORT FlfCONVEX S.
T.TV/e#tIg==.1gBReN1eR- MCCANN
Tio±
THM ( FIGALHM . PRATELLIWVMATELI )
fy(E) > an ) 2 ( E ,k )
?
where attn,¥yEpg,¥ - r & Netting! lEo,k¥ktX.AGAIN EXPONENT 2 15 SHARP X.Clh ) EXPLICIT,
POLYNOMIAL IN nx.STITH
.TT#wrnFEortF4convexstty*tE==Yy
,
Eq
K Y⇐¥¥*a###""
IIIIEIIII
THMCFIGALHM .PRATELLI WVMAT#11 )
fy(E) > an )2( E ,k )
"
XiOPTIMAL MASS TRANSPORT
FlfCoNVeXS.T.detD3f-lkHEgBReN1eR-MCCANNTl1MiXKNOTHe-GROM0VARcrUmenTnlklHeFtinfddettyMefoy-feuy.ueEqkEfgyludqlJyjEIlEIE@FEHTiNFtIIasx.fexy.aeHek.xTHus8ylE12f0n1.1dettiyFxf.t
iy - Idp ( IKHIED
THM ( FIGALHM .PRATELLI WVMAT#11 )fy( E) > an ) 2 ( E ,k )
"
X.OPTIMAL MASS TRANSPORT FlfCONVEX S .T . detDZ= IKHE ,
BREMER - MCCANN TIM
ixoyktz§ lot -Idiz going.to/5elapialpzlEoH2
C l E ) =0 f÷ C E)zccn )Poincare Poincare
t.DE#do:tsE:0EiE:TRIMMING DOWN
LEMMA
SE
A DETOUR : PLATEAU PROBLEM
€614,meNn's
am
xiNYMKHIMT whenever an.org#tM&X.Does HYK ) - NYM ) control IEI where 2E=M - Mt ?xNONUNIQUENESS j€+ §Mz€
THMCDEPHIUPPISM . JDG 14 ) M SMOOTH & UNQUCLYAREAMIMMIZ .
( AMONG CURRENTS )
Ttit HYMT - HTM )zk(m)min{ IEI ? IEI "htY tdM=FF OE=M - Av
0
F&onLYF AttuMciiskm)=inf{fnheifiyifyytn ,y=ooM } >o
X SHARP EXPONCNTS 2 & n÷z ( ISOPERIMETRIC REGIME )
X CONJECTURE : IF NMKO THEN 2→k,
KEN
× WHAT ABOUT SINGULARITIES ?
LAWSON Cones Mke={ ( n ,y)ElRtxlRhiln¥=lY€ } ZEKEL
IF kth 29 OR (K ,h)=( 4,4),(3,5 ) THEN UNIQUELY AREA MINIMIZING
THMCDEPHIUPPISM . JDG 14 ) IF M~nBr=M£hnBp & OE=M•a - M
THEN HTTMBP ) - NYMrenBr)zC( KHR "(k¥+ . )"
UNLESS k=2 7ehE11 OR k=3 h=5
13/8
car ,a=¥tT¥I "
⇒x( manner )zzzHey%hM(h -11312 hkuwwh
THE STABILITY OF SIMONS ONES k=h INCREASES AS h→a
RIGIDITY OF FIRST VARIATION IN 150 PERIMETRY
ALEXANDROVTHM IF E OPEN BOUNDED SET WITH SMOOTH BOUNDARY
SUCH THATHeIS CONSTANT THEN £ = BALL.*He= MEAN CURVATURE ( WRT OUTER UNIT NORMALU£)
CAPILLARITY THEORY
'X Mean CURVATURE Flow ⇒ E WITHHETCONSTANT
CMC FOLIATIONS FORGEDx.CMC DEFICIT
QMDEKHTIEE- Ylcqo ,
where
H°=nPl±£ E
£( nt1 )
IEIX.REMARK IF H = CONST . THEN CONST . =
tiC
SLIGHTLY PERTURBED UNDULOIDS
mm.
E H=n=HE Be
=.
MiixUNDULOIDS ARE UNBOUNDED AXIALLY SYMM . CMC SURFACESXUNDULOIDS
CONVERGETO AN ARRAY of SPHERES WITH SAME RADI )X.A TRUNCATED UNDOLOID WITH Necks = 018 ) CAN Be
"Closed
"
WITH ERROR 8cµc=O( 8)Hunton,
←n#D: loglkr )
THM CIRAOLO - MCPAM 17 H£=hPl¥=n BY SCALING( html El
E Open BOUNDED C }eT ST . { PIE ) e- ( Ltn - a )P( B ,) WHERE LEN
EE ) e- Joan ,L ,a )
At 6.1 )
THEN 7G=¥, Bzlzj ) DISJOINT UNIT BALLS NEL s .T .
x.IEOGHIPIE ) -
NPCBd1ECln1dcmlFP.2eOCri1jxhdloE.aG1eundmlF5.x-0lriY@EzfxVZjFzbS.T
Ilzj- Zwl - ZIECK ) dmlFP,
2=0154
INGREDIENTS ! TORSION POTENTIAL HINTZE - KARCHER INQ POHOZAEU ID£
GLOBAL ELLIPTIC ESTIMATES ALLARD CLOSURE THM
THM CIRAOLO - M CPAM 17 Heo=hPl¥=n BY SCALING( html El
E Open BOUNDED d SET 5.T . { PIE ) e- ( Ltn - a )P( Be) WHERE LEN
EE ) e- Joan , L,
a )a €6,1 )
Moreover OE Is FOR THE MOST PART A C' 't GRAPH Over [
WHERE [ = OG \ SPHERICAL CAPS WITH DIAMETERE dcmlF)? a- 0152 )
& ue CMCE) st.
OE 2 (
idtuva) ( E )
;gnm ' '
COROLLARY ALEXANDROUTHM IS MORALLY WRONG...
COROLLARY LOCAL MINIMIZERS OF PIE )+§g ( GAUSSCAPILLARIT 's )
WITH IEI = m SMALL Are Close To SINGLE SPHERES
- ..
i
. .
REMARK GLOBALMINIMIZERS✓B✓(a)
so
ON IR "→ FIGALLI M ARMAN ( ALSO CRYSTALS ) ~ '
'
'
i.
CONTAINER → M MIHAILA CALC VAR PDE 16-
↳ De PHILIPPIS M. ARMA 14
CICALESE LEONARDI M INDIANA UMJ 17
QUEST FOR SHARPNESS : SINGLE SPHERE
THMKRUMMCL . M.
IF Hf÷n , 0<24 THEN dE=KdtuUBy( FBIPCE )eH+c)P( B , ) Hullo ,×EC1n2)8
,!nF)
9k¥10 .in , -4
IN ADDITION
sosiitiouieunufatteni
RELATED TO ALMGREN ISOPERIMETRIC PRINCIPLE
ALMGREN ISOPERIMETRIC PRINCIPLE
( CODIMENSION 1 VERSION )
IF Hein THEN PCE )2P( Bs )
ALMGREN ISOPERIMETRIC PRINCIPLE
IF Hein THEN PCE )2P( Bs )
PCB,)=M"Copy )=f ldettyl
oa€ =s¥oat÷n%
€Ia %Ie¥t¥nH⇒etiloanot ){ MYOEKPIE )
RMKI EQUALITY HOLDS RMKZ Yes ! IT REMINDS
E⇒ E=Bdn) A Lot ABP !
THM Krummel . m IF HEIN & dt )=PlE ) - PC Bs ) I ddn ) SMALLA
E Then 2E=
OR CONNECTED
@ fixFR*2R WITH
nzHrxiQhn@I.xlrTr1tHYortidNeciniffEIx2rtilidtuiupnlloBdwHeRen-1nuiwntnutncoeumffEtollnHfclmfAfggEfgl.a
:o) rn=z
# TRUNCATING MEAN curvatureIF A 23
BY Free BOUNDARY THEORY