PEIpogor - math.ucsd.edu
Transcript of PEIpogor - math.ucsd.edu
Regularity of dutiful f in r
fIsnterion
regularity
elatedtotheLU zr G boundary value
is very tricky Seeexample in Exercise 31 This is drastically
Regularity is dual toLiouville type theorem different.fm
Bernstein type theoremthe linear case
Theorem to'rgeus Calabi pogorelon
U ECT IR dit 17541 1 TTU o
ther U is a quadratic polynomial
I shall present a proof usingJohn's Lemma Ev 1982
interior C estimate The key is a d estimate
on calabi.sc shya k 1958of Pogo rein tused ag.me
icarguemeattifIIPPEIpogorelou's originalproof of TCP theorembe
Thema Pomelo U C Ehr nc I
is a convex solution of dit 4 9 0 in r
nineThen Hillel Fff Helo Halo 1171net 7l4n5lo dx.smMaxlul
l
f f a 121 1 direction consider W toilet UK o on 2hi say co o
It attains its maximum somewhere Xo Weestimate who
gatXo o
first pick a framerttu U U.j Uiifij at Ko i a
unit'h o I t Y o
0 17login t_ fbad
no TH logue Y Ha aauii iuitkik.it Yi
Imind
Using it U
I't Es't E Tfi
Now we use n'in.j He µiiidish
uii Closet
n Eu4 i Ese EY uit unenclosedis
yBad
Hui E.euiii soBad
E Eui E Ii o
w i't
Uit n a Undoge 1 dose I fo
TMultiplying n'e we Aw Bso A ne
A B depends on u U doggy doggytudughue
Hence weAt It D UHogy e
T LU 2e42
Namely at themaximum who E C
ul et unlocole C
une et Iulxxiec t.ec
simplyusingthecomexityHukxxkdiThis is similar to Aubin Yan's C estimate on
a compact manifold which was provedlater fpsahpe.mgr2aoYfpIasJeMpwog
hem Estimatesfor tr Can always be doneby affinetransformation
Assume Bf C R C Bhomex via John's Lemma
Then c Cz o Ci Ciadet72u i inn i
tube h constant IUtd 1 0 fr some doC r
fUe E h in Bath applyCaleb
x C r l Ub t liiilc.I pusc.ee
t.tn Ic i E.nThe above is the key ingredient 47.4 of Gilbey Trudingertobe proved in next Leiture
Undertheassumptionof ICP if C I he c I
dsj U.joxidxJ is a complete metric on IR
Calabi's result 1958 paper Corollary on P 108
his quadrateor is Kuba
It x y E B
Thnx duh I
t.geI pin Hussle CH o as
R R o
7in is constant
We shall first show how to use Them to jcp situation
Then we prove Them
By translation.ro yikiiiEEiiiEIiauoTheCole I U
Now we consider R x Ubc Ea
which is a comex body in IR by John's theorem
7 E centered at x er do is the center of gravityoff
E f x l life E Crc ENK
This is due to that
Iet 174 I does not change unden an
affine transformation T K A etc will let A
Namely if 4 Cx U Aetc
3 Ena 3gyeafa
Cake Iff af Ii CA5 air A
delta detCA dutCA detain
This has been used when we assume Tin lok id
clearly using orthogonaltransformation E can be put into
the firm we claimed But it does not changeInfo I
Now let Y fci biyi.sci B
E B badr ri
Ent B smaller ball
hk
Tiff Xu By 1 do
172 13 d 7in B detail ib d
i bij
Nour a vbi t.LIThi is the h in Thm a stake
c a eds Cza
on Siya vs he c due he
c Iue E
I us GI lc.xtebi.ec fa is iM
In particular c Ift B B f e I
at K oor To fur
C E X bi E C
a'Ie Tu e Ca't in Ius ElH
S y also contains a smaller ball aZ
with radius Ea
This put us intothe situation of applying Calabi's
theorem
precisely d shows that E contains
z E
B Glo lxi.xi.llxi.sc 1ara E b.ie
E I
Also since Be.GE C SE B lo
na
fully Else C SE c sa
fxllisiyet.SC SE tin n'Ent
of aY Bbe I
E Byte Beo C S any
I B do a
Namely Then provide the normalized Situtation to
Hain the needed estimates for JCP
Now we focus on the proof of Thin
Proof of ThmU the one in Ttm not in Jcp
We use the comparison principle 2 the Alexandrov maximum4 Cpg principle
pogorelorestimate
Ford Apply Cpa to Ichi 4th d u
U
on Ir U E u Yin Lu Ui o
dutiful IU Ui 2 n n
ditchUbc z iz bei hth
UGH he Ici batte T
YApply Cpr to txt th n
TUz
4 In 341 7 u u in r2h
I h Uco on't
no
Fort We need thegradient estimate fr convex functions
9 91,5yay Z 41Mt 71964 Y n
unmeetIf we pick y xtYp9gY
lD9k Gf
As long as such y E N
by taking y tothe limit
tR of such possibility
The key is still A maximum principle
Eun
the
theine C da.mku
C desc arDfl
Hence Exe Sy f Shm
H eda SETIdist Sk 254 3 Ct hz is ccn
Similarlydist Shg 25 3 Can
Note that xoESn
BK.sc f Bapu isaway from 22
In fact Bc du will not tonnch the boundaryof 25hLhence utockth
If want to be sure
lunate
To get Hit the Hessianestimate We apply to
ughh
Then 17M ftp.fdistchsIasaT E C 9
cents distCSzh SaNow Pogorehr Estimate
Snp Ih u U.IE C
In particular is BaGlo E S y2
sup un E zE C
4
Since Ui I the minimum one Upp 7 C
app I
This proves the uniformestimate in
The Hidden estimate how followsfrom Evans estimate
G T 17.4
Next We show how to prove Evansestimate The key is
a Harnack estimate of Krylov Safanor Only need the
weaker form