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LI [ Existence of conical functions ] Rc Z - In - D8 on ( M " . g) uol.it#shaBiIi.QVIslr7---/Brtpil in Mts Ilan) o Ig Ir ) = I 213.4511 when s o Then I ¥1814 - conical function h on B , . # ( sin , Cpl . . We say h :B R is an Ewm'cN it f 15h - 2g I ' s e Brlpl f l Ion ' - 4h I a er ' Brlp ) Z f I 01h - dis ) a er , dplxl - dip . x ) . Brlp ) sup Ih - dit a er ' . Brlpl

Z In D8 on M g) uol.it#shaBiIi.QVIslr7---/Brtpilbenchow/lcct/CCLecture14hw.pdfLI [Existence of conical functions] Rc Z-In-D8 on (Mg) uol.it#shaBiIi.QVIslr7---/Brtpil in Mts Ilan) →

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. g)
.
f 15h - 2g I ' s e
Brlpl
Brlp)
Z
f I 01h - dis ) a er , dplxl - dip . x ) .
Brlp)
.
h = d ,5= I on 213,1ps .
Maximum principle ⇒ sup tht E Cin) .
B#
Try to estimate f loch - disk = - f ch - dj ) - och - d;)
B. Ip) B. Ip) - £ supthltdp ) . flolh - did
od; = zdpodp + 2104.12 B = 2in -Ddp - nfswthlrdp) +2
= Zn t 21ms { coshlrsdp) - s?hf÷q, - I } E Zn t Ccn) S on B
, Ip ) .
e utlslnlllipl -
Isla) at B'll" s t -- Hsin) - -
ht -10181 I -1018 )
O 10h12 = z 15h12 t 212cL oh , oh )
Z - 2 In -D8 10h12 .
Elliptic estimates : F I , = If Slu ) s > I sit .
sup 10h12 B , - yep,
'
Bslp)
ha dp ' oh a zdp - Pdp
f ion ' - ahl e f ion ' - lodi l 't BslpsBs 'M t f Ipd;T - 4hL
Bslp )
+ I ,
< I , .
f Iah - zgl '
Bl - 2¥ , Cp)
Lee Rc z - In -178 , a S - conical h on B. Ipl .
Then u x. Yi , y. c- As , , Cpl = B. 4) I Belpt .
" I lpy.lt/y.yz1-lpyil - S .
Y , V

f f a Elan ) .
c f t cut 1×1×21
Fflxiixzl = int f fins , ) Is .
1354"
" LETS ] = lls ) .
1×51 x ly , if I t ly. It s ut
Cbl 7 ! minimal r : fo . I ] → M sit -
T T
J ? f- Iris , ) Is a E
.
Id ) J 't ds J !" flesh , )dt = Fogle
. . > IT 'T) s t -
/ s I .
= ) ! ( hors "
⇐ / ! I Chori'm - 2 I do - E .
T 5h18.ir )
dykes . ) djlyT )
H1nnm-n%n%w.,EY"÷-"III!
.
• Ts les) = I Iris , I = LETS] .
P a K
cos Iris> pI =
| us Erisipx - / e Els ' .
'
x
I? " oh - zcbts) regs , 12 1h04
' ' sad!.is)
( hor) '
;%5.
-
"
"
15h - zglltscti ) dt + I .
J ! I das lbtsYa;¥;l / e 53 /! I chest .- e ' - a ' - uhesiee't
⇐ Ifs' t 53 J! lls) ds / !" I oh - zglleslti ) dt
E Ils ' .
Za( bts)