WEBVTT
Kind: captions
Language: en
00:00:43.469 --> 00:00:52.379
Alright so let us continue with the proof
of Zalcman's Lemma, so basically what this
00:00:52.379 --> 00:01:06.690
Lemma is about, it is about characterising
a non-normality of a family okay. So I begin
00:01:06.690 --> 00:01:15.979
with a family script F of Meromorphic functions
okay and am assuming that the family is defined
00:01:15.979 --> 00:01:23.319
on this domain D alright and I assume that
the family is not normal okay then as the
00:01:23.319 --> 00:01:31.009
non-normality will manifest at some point
okay the family normally if it is not normal
00:01:31.009 --> 00:01:38.270
at at least 1 point in the domain and how
do you get that non-normal point that is exactly
00:01:38.270 --> 00:01:40.130
about Zalcman's Lemma is all about.
00:01:40.130 --> 00:01:45.039
So you see what it says is that you can find
a sequence of points in the domain converging
00:01:45.039 --> 00:01:52.250
with this point Z not which is the so-called
which is the point of non-normality and you
00:01:52.250 --> 00:02:03.860
can find these decreasing sequence of radii,
positive radii so that you know if you takeÖand
00:02:03.860 --> 00:02:09.729
you will be able to findÖsee the fact that
the family is not normal means what? It means
00:02:09.729 --> 00:02:14.190
that the family is not sequentially compact,
normally sequentially compact. I mean our
00:02:14.190 --> 00:02:20.050
definition of normal is normally sequentially
compact and that is the correct version of
00:02:20.050 --> 00:02:29.599
compactness for us okay when you are looking
at a family of analytic functions or Meromorphic
00:02:29.599 --> 00:02:34.420
functions the correct version of compactness
is normal sequentially compactness that is
00:02:34.420 --> 00:02:40.930
given every sequence you should find a normally
convergence subsequence and when you say a
00:02:40.930 --> 00:02:52.720
family is not normal what you are saying is
that you are saying that there is a sequence
00:02:52.720 --> 00:03:00.510
or which you cannot find any normally convergence
of sequence okay and you have toÖ and Zalcman's
00:03:00.510 --> 00:03:05.190
Lemma ratio that you can find such a sequence
and that is the sequence here f n you can
00:03:05.190 --> 00:03:06.360
find this sequence.
00:03:06.360 --> 00:03:14.730
It is a non-normal family in fact the sequence
itself forms are non-normal family, so that
00:03:14.730 --> 00:03:19.670
if you take the members of the sequence and
then you take the corresponding zoomed functions
00:03:19.670 --> 00:03:27.350
okay, so g n is the zooming of f n centred
at Z n and with the magnification factor 1
00:03:27.350 --> 00:03:37.680
by Epsilon n okay. Then this zoom family converges
normally on the whole complex plane to a non-constant
00:03:37.680 --> 00:03:44.200
Meromorphic function g okay and the point
is that the non-constant C of the Meromorphic
00:03:44.200 --> 00:03:48.620
function reflects the fact that it is spherical
derivatives is not 0 because the moment the
00:03:48.620 --> 00:03:52.810
spherical derivatives of a Meromorphic function
is 0 it means it has to be constant right,
00:03:52.810 --> 00:04:00.990
so this non-constant C of g as a Meromorphic
function is further you know fixed by this
00:04:00.990 --> 00:04:06.020
fact that the spherical derivative at the
origin is 1 and all the spherical derivatives
00:04:06.020 --> 00:04:09.230
are bounded by 1 okay.
00:04:09.230 --> 00:04:14.730
So this is Zalcman's Lemma so the point about
this Lemma is that the familyÖif a family
00:04:14.730 --> 00:04:21.099
of Meromorphic functions on a domain is not
normal it gives you a non-normal point Z not
00:04:21.099 --> 00:04:28.680
and it gives you a non-normal sequence in
the family which violates normality in a neighbourhood
00:04:28.680 --> 00:04:36.600
of Z not that is the whole point alright and
I have explained to you that what happens
00:04:36.600 --> 00:04:41.810
if the family were normal, if the family were
normal what would happen is that no matter
00:04:41.810 --> 00:04:49.000
what Z not you choose and the sequence Z n
you choose like this and you choose these
00:04:49.000 --> 00:04:52.930
any radii Epsilon in going to 0 okay.
00:04:52.930 --> 00:04:58.610
The zoom function will always converge normally
to a constant Meromorphic function I mean
00:04:58.610 --> 00:05:07.240
to a constant function okay covert so the
normality of the family will tell you that
00:05:07.240 --> 00:05:12.960
always the zoom functions will be constant
and the non-normality of the family is reflected
00:05:12.960 --> 00:05:20.620
by being able to find a sequence for which
the zoom functions not converge to normally
00:05:20.620 --> 00:05:26.190
to a constant function but actually they converge
normally were non-constant Meromorphic functions
00:05:26.190 --> 00:05:33.990
okay. So it is the limit function that matters,
if it is normal glow zoom function will always
00:05:33.990 --> 00:05:40.780
converge to a constant, if it is not normal
I can find a situation where the zoom functions
00:05:40.780 --> 00:05:46.660
converge to a non-constant Meromorphic functions
that is the whole point okay.
00:05:46.660 --> 00:05:58.190
So let us see a proof of this the proof is
tricky, so as I have mentioned in the reference
00:05:58.190 --> 00:06:11.400
material the text book that I am following
is that of okay and the proof as is mentioned
00:06:11.400 --> 00:06:16.740
that is tricky as you will see, so we will
make a couple of reductions the 1st thing
00:06:16.740 --> 00:06:25.430
is that you know what is given to me is that
the family is not normal okay and now you
00:06:25.430 --> 00:06:30.880
know we have already proof Martyís theorem
which is a characterisation of normality namely
00:06:30.880 --> 00:06:38.699
it says that a family is normal if and only
if you know if you take the family of spherical
00:06:38.699 --> 00:06:43.340
derivatives is normally uniformly bounded
okay.
00:06:43.340 --> 00:06:50.240
So recall if you take a family of analytic
functions okay the condition that that family
00:06:50.240 --> 00:06:56.910
is normal that it is normally sequentially
compact is by Montelís theorem equivalent
00:06:56.910 --> 00:07:04.389
to the family being normally uniformly bounded
and if you consider Meromorphic functions
00:07:04.389 --> 00:07:09.639
you get the analog as theorems which is Martyís
theorem with says that the condition for normality
00:07:09.639 --> 00:07:19.130
is that a family of spherical derivatives
is normally uniformly bounded. So spherical
00:07:19.130 --> 00:07:22.350
derivatives being normally uniformly bounded
is equivalent to normality of the family,
00:07:22.350 --> 00:07:28.360
so the family is not normal you have a violation
of the bounded normal uniform boundedness
00:07:28.360 --> 00:07:30.990
of spherical derivatives and what does that
mean?
00:07:30.990 --> 00:07:40.080
It means that there is a compact set on which
this spherical derivatives can become unbounded,
00:07:40.080 --> 00:07:46.289
so this means by Martyís theorem you can
actually find a sequence of points okay and
00:07:46.289 --> 00:07:51.400
functions such that the corresponding actions
at those points its spherical derivatives
00:07:51.400 --> 00:07:59.130
go to infinity plus infinity okay, so that
is the 1st step so let me write this by Martyís
00:07:59.130 --> 00:08:22.569
theorem there exist a sequence of w n tending
to w naught in D and functions f n in the
00:08:22.569 --> 00:08:44.420
family f such that f n hash of w n goes to
plus infinity and at w n are in a compact
00:08:44.420 --> 00:08:56.190
subset of D okay. So I can find is just because
of Martyís theorem because Martyís theorem
00:08:56.190 --> 00:09:03.459
says that you know normality is equivalent
to the spherical derivatives being uniformly
00:09:03.459 --> 00:09:05.920
bounded on compact subsets okay fine.
00:09:05.920 --> 00:09:12.420
Now we will make a couple of reduction what
we will do is for convenience we will assume
00:09:12.420 --> 00:09:19.370
that you know w naught is actually the origin
okay you assume w naught is the origin and
00:09:19.370 --> 00:09:25.600
how can we do that? You can do that by simply
translating the domain so that you make w
00:09:25.600 --> 00:09:32.119
naught the origin, so you translate the domain
by minus w naught you will get a new domain
00:09:32.119 --> 00:09:36.279
and you look at the functions there, the translated
functions. So without loss of generality what
00:09:36.279 --> 00:09:41.440
you can do is, you can assume that w naught
is the origin okay that is one thing and the
00:09:41.440 --> 00:09:47.060
2nd thing is that you can also assume that
the moment you assume w naught is the origin,
00:09:47.060 --> 00:09:51.260
so the origin is the point of D okay then
of course there is a small disk surrounding
00:09:51.260 --> 00:09:57.490
the origin which is also at D because after
all D is an open set and by using a scaling
00:09:57.490 --> 00:10:02.619
transformation you assume that the unit disk
along with the boundaries also at D okay.
00:10:02.619 --> 00:10:07.670
So these are you scale the domain I mean you
scaled the domain and you translate the domain
00:10:07.670 --> 00:10:13.259
so that you can assume without loss of generality
that the compact set you are looking at where
00:10:13.259 --> 00:10:19.319
you got this sequence w n is actually the
unit disk okay along with the boundary unit
00:10:19.319 --> 00:10:28.899
circle and the sequence actually converges
to the origin okay, so we will make these
00:10:28.899 --> 00:10:35.899
assumptions without any loss of generality,
so let me write that down. Without loss of
00:10:35.899 --> 00:10:58.170
generality we may assume w naught equal to
0 and mod Z less than or equal to 1 is in
00:10:58.170 --> 00:11:04.639
D okay, so for this all you have to do is
that you have 2 translate D by minus w naught
00:11:04.639 --> 00:11:11.820
and then you have 2 scaled be suitably so
that the unit disk which is a neighbourhood
00:11:11.820 --> 00:11:21.769
of w naught equal to 0 is inside D alright
fine, so you can do this. So you see my picture
00:11:21.769 --> 00:11:29.689
is now like this so here is my I think I will
have toÖ okay so let me go down.
00:11:29.689 --> 00:11:35.889
So here is my picture, so I have so this is
the complex plane and I have this, this is
00:11:35.889 --> 00:11:49.720
the origin so this is unit disk, this is unit
disk and well this is inside D, so you know
00:11:49.720 --> 00:12:07.019
your domain D is contains the unit disk, so
this is D and well this is the Z plane okay
00:12:07.019 --> 00:12:17.370
and of course there are these so there is
the sequence of points w n , w n plus 1 that
00:12:17.370 --> 00:12:27.160
is tending towards the origin okay and what
you are given is that okay fine. So I have
00:12:27.160 --> 00:12:35.839
this, now what you do is that youÖ So the
tricky thing is that so you know what I am
00:12:35.839 --> 00:12:43.689
looking for? I am looking for I have to extract
this sequence Z n okay which goes to Z not
00:12:43.689 --> 00:12:50.709
and I have to extract this sequence of functions
which you know which has this property that
00:12:50.709 --> 00:12:55.509
the zoomed functions okay they converge to
a non-constant a Meromorphic function.
00:12:55.509 --> 00:13:04.879
So you see the trick is the following, so
what you do is you put R n to be the maximum
00:13:04.879 --> 00:13:14.149
over mod Z less than or equal to 1 of f n
hash of Z and then you multiplied by 1 minus
00:13:14.149 --> 00:13:24.519
mod Z okay, so you do this. So this is the
tricky part, the trick is you see what is
00:13:24.519 --> 00:13:31.859
given to you is these functions f n hash they
are spherical derivatives of thoseÖf n hash
00:13:31.859 --> 00:13:36.471
is just the spherical derivatives of f n and
of course you know spherical derivatives is
00:13:36.471 --> 00:13:40.239
continuous mind you, the spherical derivatives
of Meromorphic function is continuous function.
00:13:40.239 --> 00:13:46.480
It is continues nonnegative real valued function
okay it is a positive function at worst it
00:13:46.480 --> 00:13:53.829
can be 0 alright which is what happens if
the function is a constant okay but the point
00:13:53.829 --> 00:13:59.959
is that mind you the spherical derivatives
has no problem at poles when the Meromorphic
00:13:59.959 --> 00:14:04.290
function as poles there is no problem with
the spherical derivatives unlike the usual
00:14:04.290 --> 00:14:07.850
derivatives, the usual derivative is not defined
at a pole because it is a singular point but
00:14:07.850 --> 00:14:11.769
a spherical derivative is defined at a pole
and we have already seen that if it is a pole
00:14:11.769 --> 00:14:16.389
of higher-order then the spherical derivative
is 0 if it is a pole of order 1 namely a simple
00:14:16.389 --> 00:14:20.029
pole then the spherical derivative is 2 divided
by modulus of the residue at that pole okay.
00:14:20.029 --> 00:14:27.939
So this spherical derivatives is a nice continuous
function okay non-negative real valued function
00:14:27.939 --> 00:14:31.170
and you are looking at this function on this
domain mod Z less than or equal to 1 which
00:14:31.170 --> 00:14:36.559
is a compact set, so if you are looking at
a continuous function on a compact set, continues
00:14:36.559 --> 00:14:40.639
real valued function on a compact set you
know the function is of course it will be
00:14:40.639 --> 00:14:44.759
uniformly continuous and it will attain its
maximum and minimum therefore this maximum
00:14:44.759 --> 00:14:50.359
is well-defined okay and the point is that
you see what is given to me is that these
00:14:50.359 --> 00:14:57.609
f n hash they become larger and larger okay
at points which are getting closer and closer
00:14:57.609 --> 00:14:58.619
to the origin.
00:14:58.619 --> 00:15:04.639
See as an tends to infinity w n converges
to 0 okay that means as n tends to infinity
00:15:04.639 --> 00:15:10.579
w n goes closer and closer to 0 and what is
f n hash of w n that is going to infinity
00:15:10.579 --> 00:15:17.299
that means f n hash attains larger values
closer and closer to the origin as n becomes
00:15:17.299 --> 00:15:26.549
large alright thereforeÖso you know what
one does is that it could happen that the
00:15:26.549 --> 00:15:34.840
maximum values of f n hash could also be taken
close to the boundary but if you go close
00:15:34.840 --> 00:15:39.470
to the boundary this quantity becomes very
small, if you go closer to the boundary of
00:15:39.470 --> 00:15:45.329
the unit disk, the quantity 1 minus mod Z
will become very small and that will offset
00:15:45.329 --> 00:15:55.369
this the value of f n hash at that point okay
so heuristically this is the reason for multiplying
00:15:55.369 --> 00:16:02.239
by 1 minus mod Z okay instead of just considering
the maximum of f n hash .
00:16:02.239 --> 00:16:11.079
So mind you 1 minus mod Z is also a continuous
real valued function, nonnegative real valued
00:16:11.079 --> 00:16:17.149
function inside the unit disk, so there is
no problem about it okay, so the product is
00:16:17.149 --> 00:16:22.850
of course continues real valued function so
it has a maximum okay. Now you have to make
00:16:22.850 --> 00:16:33.899
a series of observations, the 1st thing is
suppose Z n is such that
00:16:33.899 --> 00:16:44.660
mod Z n is to 1 and R n is attain at Z n,
so R n is f n hash of Z n times 1 minus mod
00:16:44.660 --> 00:16:55.800
Z n okay. So R n which is the maximum is attained
at some Z n okay, so look at that Z n and
00:16:55.800 --> 00:17:02.129
this is the Z n that I actually want or probably
a subsequence of that as you will see. See
00:17:02.129 --> 00:17:16.870
the 1st thing is note that you see R n is
greater than or equal to you know f n hash
00:17:16.870 --> 00:17:30.800
of w n into 1 minus mod w n this happens because
R n mind you is the maximum of f n hash of
00:17:30.800 --> 00:17:35.860
Z into 1 minus mod Z, so if you put Z equal
to w n, so the maximum value will always be
00:17:35.860 --> 00:17:37.390
greater than any of the other values.
00:17:37.390 --> 00:17:47.770
So I will get this but then you see as n tends
to infinity you see this goes to 1 okay because
00:17:47.770 --> 00:17:56.419
of w n tends to 0 and this fellow goes to
infinity okay because that is the original
00:17:56.419 --> 00:18:02.059
assumption. The f n hash the spherical derivatives
go to infinity okay that is how we pick the
00:18:02.059 --> 00:18:07.909
sequence w n because it was violating normality,
while letting the conditions of Martyís theorem
00:18:07.909 --> 00:18:13.510
okay. So you see what is happening is that
this will tell you that you know R n will
00:18:13.510 --> 00:18:19.510
tend to plus infinity, so this R n are becoming
bigger and bigger and bigger okay that is
00:18:19.510 --> 00:18:30.250
something that you have to understand first.
Now you look at this so you know if you
00:18:30.250 --> 00:18:37.429
look at this definition of Z n okay what it
will tell you is that he f n hash of Z n will
00:18:37.429 --> 00:18:46.150
also go to infinity because you see if you
take R n this is f n hash of z n times 1 minus
00:18:46.150 --> 00:18:52.180
mod z n and this is certainly you know greater
than this is less than or equal to f n hash
00:18:52.180 --> 00:19:00.620
Z n because you know after all 1 minus mod
Z n is less than or equal to 1 okay.
00:19:00.620 --> 00:19:12.360
So this is going to plus infinity as n tends
to infinity will imply that the f n hash of
00:19:12.360 --> 00:19:23.000
Z n will also go to plus infinity okay. So
this implies that this goes to infinity, plus
00:19:23.000 --> 00:19:30.990
infinity as n tends to infinity okay, so what
you have done is? You have got this from the
00:19:30.990 --> 00:19:38.789
sequence w n which goes to 0 to w naught you
have cooked of this other sequence z n okay
00:19:38.789 --> 00:19:45.309
and the point is that the spherical derivatives
at the Z n also go to infinity, plus infinity
00:19:45.309 --> 00:19:52.700
is like the spherical derivatives at the w
n go to plus infinity okay but the point is
00:19:52.700 --> 00:19:58.330
that of course the sequence Z n that you have
got that need not be convergent it is just
00:19:58.330 --> 00:20:03.590
the sequence okay but anyway it is a sequence
inside the unit disk and you know the unit
00:20:03.590 --> 00:20:07.400
disk is compact sequentially compact therefore
there is a convergence of sequence therefore
00:20:07.400 --> 00:20:11.090
without loss of generality can assume that
this sequence of Z n is actually convergent
00:20:11.090 --> 00:20:12.409
okay.
00:20:12.409 --> 00:20:29.350
So we will make their assumption without loss
of generality we assume Z n converges to Z
00:20:29.350 --> 00:20:37.590
not you know Z not also of course in the unit
disk because unit disk is closed okay. Of
00:20:37.590 --> 00:20:43.590
course when I say unit disk I am also including
the boundary is not the open unit disk okay.
00:20:43.590 --> 00:20:49.480
Fine so we have gotten hold of the sequence
actually alright and now the point is that
00:20:49.480 --> 00:20:55.950
we haveÖso you know what is our aim? Our
aim is you have to get this sequence of functions
00:20:55.950 --> 00:21:00.220
and you have to get this sequence of points
such that and then you have to get a certain
00:21:00.220 --> 00:21:06.580
sequence of radii okay such that the zoom
functions they converge to a non-constant
00:21:06.580 --> 00:21:07.580
Meromorphic functions.
00:21:07.580 --> 00:21:16.900
So where do you get those sequence of decreasing
radii okay and that comes very simply, so
00:21:16.900 --> 00:21:26.620
what you do is you do put Epsilon n to be
1 by f n hash of Z n okay and then this will
00:21:26.620 --> 00:21:31.049
of course go to 0 as it will go to 0 plus
as n tends to infinity that is because the
00:21:31.049 --> 00:21:37.009
f n hash of Z n is going to plus infinity
alright, so this will serve as the zooming
00:21:37.009 --> 00:21:46.870
radii, so now everything is in place we have
gotten what we want and so let me write this
00:21:46.870 --> 00:22:01.190
down since R n is f n hash of Z n times 1
minus mod z n what you will get is that? You
00:22:01.190 --> 00:22:07.049
will get Epsilon n R n is equal to 1 minus
mod Z n okay because Epsilon n is just defined
00:22:07.049 --> 00:22:14.760
to be 1 by f n hash of Z n and now what you
do is that you do the following thing.
00:22:14.760 --> 00:22:28.419
You put g you take the zoom functions, so
you take g n of Zeta to be well f n so it
00:22:28.419 --> 00:22:38.919
is just you zoom the function f n centred
at Z n with the magnification factor 1 by
00:22:38.919 --> 00:22:46.759
Epsilon n and use the variable Zeta, so this
is going to be f n of Z n plus Epsilon n times
00:22:46.759 --> 00:22:51.150
Zeta, so this is the zoom functions this is
the family of zoom functions okay mind you
00:22:51.150 --> 00:22:58.450
we have to find the family of zoom functions
which converge normally to non-constant Meromorphic
00:22:58.450 --> 00:23:06.029
function this will be that family okay and
where is this defined you see you know this
00:23:06.029 --> 00:23:22.820
is defined for mod Zeta less than R n you
see what is happening is that, so the diagram
00:23:22.820 --> 00:23:29.909
is like this you have this unit disk this
is the origin, this is one and then you have
00:23:29.909 --> 00:23:39.870
Z n somewhere here okay and then if you take
the small disk centred at Z n its radius will
00:23:39.870 --> 00:23:47.769
be 1 minus mod z n, this radius will be 1
minus mod Z n but this one minus mod Z n is
00:23:47.769 --> 00:23:52.230
as I have written above it is just Epsilon
n R n okay.
00:23:52.230 --> 00:24:02.000
So if I think of a variable Zeta here okay
then you know the maximum distance of Z n
00:24:02.000 --> 00:24:08.960
to Zeta can be Epsilon n R n okay and that
means that the maximum value of Zeta can be
00:24:08.960 --> 00:24:14.909
up to R n because I have reduced you know
I have actually used the scaling factor 1
00:24:14.909 --> 00:24:22.889
by Epsilon n okay. So but the point is look
at these functions g n, the zoom functions.
00:24:22.889 --> 00:24:27.220
The zoom functions are defined on mod Zeta
less than R n and mind you R n tends to plus
00:24:27.220 --> 00:24:33.299
infinity, so what it means is that as before
the zoom functions are eventually defined
00:24:33.299 --> 00:24:47.700
on any compact subset of the plane okay, so
g n is defined for n sufficiently large on
00:24:47.700 --> 00:24:58.290
any compact subset of the complex plane and
here of course the complex plane you are looking
00:24:58.290 --> 00:25:04.280
at is the Zeta plane mind you your brought
in this new variable, the zoomed variable
00:25:04.280 --> 00:25:05.870
Zeta okay.
00:25:05.870 --> 00:25:17.669
So now the fact is that g n does the job that
is all you have to verify and how does one
00:25:17.669 --> 00:25:27.659
do that? Mind you we want to show that you
know g n converges normally to a non-constant
00:25:27.659 --> 00:25:33.289
Meromorphic function okay that is what you
want to show that the whole point. Now again
00:25:33.289 --> 00:25:39.929
use Martyís theorem of course g n are also
Meromorphic because g n are just you know
00:25:39.929 --> 00:25:46.460
obtained from f n by translation and scaling
okay g n is just f n translated, see you take
00:25:46.460 --> 00:25:59.210
the variable of f n okay and you know
you translate that we will by minus Z n and
00:25:59.210 --> 00:26:05.950
then you divide by scale it by 1 by Epsilon
and you will get f n okay. So f n have been
00:26:05.950 --> 00:26:14.370
obtained from g n by a translation and scaling,
so g n are also Meromorphic okay and well
00:26:14.370 --> 00:26:15.370
and what am I trying to show?
00:26:15.370 --> 00:26:20.289
I am trying to show that the g n converge
normally but again I can apply Martyís theorem
00:26:20.289 --> 00:26:25.169
to show that the g n converge normally I will
have to only show that the g n are normally
00:26:25.169 --> 00:26:28.940
uniformly bounded I mean the spherical
derivatives of the g n are normally uniformly
00:26:28.940 --> 00:26:35.190
bounded okay. So that is what I check okay
and that is just an estimate, so how do I
00:26:35.190 --> 00:26:42.759
check that? See you will see that g n hash
so you know what will happen is g n hash if
00:26:42.759 --> 00:26:51.120
you calculate g n hash of Zeta this is spherical
derivative of g n of Zeta, mind you this isÖso
00:26:51.120 --> 00:26:58.519
I will have to take the spherical derivatives
of f n of Z n plus Epsilon n Zeta, so this
00:26:58.519 --> 00:27:03.009
is the spherical derivative I have to take
okay but then taking the spherical derivative
00:27:03.009 --> 00:27:08.360
you know will be the same as taking the spherical
derivative of f n and then I will get a multiplication
00:27:08.360 --> 00:27:09.360
factor Epsilon n.
00:27:09.360 --> 00:27:12.750
You know the spherical derivatives becomes
smaller for the zoom functions in the spherical
00:27:12.750 --> 00:27:19.220
derivative it become smaller by the inverse
of the zooming factor. The zooming factors
00:27:19.220 --> 00:27:24.450
1 by Epsilon n, so the inverse of zooming
factor is Epsilon n okay and so you know this
00:27:24.450 --> 00:27:31.049
is just change rule of differentiation, so
this is Epsilon n times f n hash of Z n plus
00:27:31.049 --> 00:27:39.570
Epsilon n times Zeta this is what you get
alright and now you see what you must understand
00:27:39.570 --> 00:27:53.649
is that now I have this inequality because
you know f n hash z n times 1 minus mod Z
00:27:53.649 --> 00:27:58.330
n is R n and that is the maximum value okay.
00:27:58.330 --> 00:28:04.909
So recall that we have this definition of
R n here. R n is f n hash z n times 1 minus
00:28:04.909 --> 00:28:12.179
mod z n and mind you that is the maximum value
of this quantity RN is actually the maximum
00:28:12.179 --> 00:28:21.200
value of f n hash of Z multiplied by 1 minus
mod Z okay therefore what we can see is that
00:28:21.200 --> 00:28:30.980
R n is certainly going to be greater than
or equal to the value of f n hash times 1
00:28:30.980 --> 00:28:41.549
minus mod Z for any mod Z for any Z in
the unit disk okay, so for Z I will put this
00:28:41.549 --> 00:28:49.450
so I can put Z n plus Epsilon n Zeta and here
I will get Z n plus Epsilon n Zeta, so this
00:28:49.450 --> 00:28:56.240
is correct okay because in fact when you put
Zeta equal to 0 the value on the right is
00:28:56.240 --> 00:29:06.399
actually R n, R n is the maximum value okay.
So you have this but you see now I can use
00:29:06.399 --> 00:29:14.860
this to get aÖso this is the quantity here
and this is the quantity that is appearing
00:29:14.860 --> 00:29:20.230
here okay which multiplied by Epsilon is g
n hash of Zeta.
00:29:20.230 --> 00:29:24.369
So I can use this to get a bound for g n hash
of Zeta, so what will I get? I will get g
00:29:24.369 --> 00:29:31.889
n hash of Zeta is equal to Epsilon n times
this but this thing and this rectangle but
00:29:31.889 --> 00:29:36.820
this thing in this rectangle is less than
or equal to R n by 1 minus mod Z n plus Epsilon
00:29:36.820 --> 00:29:42.860
n Zeta, so I will get this is less than or
equal to Epsilon n R n by 1 minus mod Z n
00:29:42.860 --> 00:29:51.389
plus Epsilon n Zeta alright but then you see
inequality mod Z n plus Epsilon n Zeta is
00:29:51.389 --> 00:29:57.980
less than or equal to mod Z n plus Epsilon
n mod Zeta okay and therefore what I will
00:29:57.980 --> 00:30:06.470
get is that this is also less than or equal
to Epsilon n R n by 1 minus mod Z n minus
00:30:06.470 --> 00:30:13.669
Epsilon n mod Zeta I will get this right and
now mind you go backÖso here is why this
00:30:13.669 --> 00:30:21.820
proof is tricky, this one minus mod Z n mind
you is Epsilon n R n okay that has to be trickily
00:30:21.820 --> 00:30:27.490
used, so this one minus mod Z n I can put
Epsilon n R n then you can see this Epsilon
00:30:27.490 --> 00:30:32.140
n is coming out both the numerator and denominator
and gets cancelled so you see I get Epsilon
00:30:32.140 --> 00:30:41.490
n R n by Epsilon n R n minus Epsilon n mod
Zeta and this becomes that is less than or
00:30:41.490 --> 00:30:52.600
equal to RN by R n minus mod Zeta okay and
you seeÖso this is what? This is the estimate
00:30:52.600 --> 00:30:56.090
for the spherical derivative of g n right.
00:30:56.090 --> 00:31:06.549
And see the point is that if mod Zeta is less
than say some R okay there exist a n large
00:31:06.549 --> 00:31:15.690
enough such that R n is going to be greater
than R okay because after all the R n tend
00:31:15.690 --> 00:31:25.529
to plus infinity okay, so beyond a certain
stage all the R n are greater than R, so that
00:31:25.529 --> 00:31:35.990
means that you know so I can divide by R n
and let me put equal to here I will get 1
00:31:35.990 --> 00:31:46.559
by 1 minus mod zeta by R n and if mod Zeta
is less than R and R n is greater than R okay
00:31:46.559 --> 00:31:59.289
then g n is defined on mod Zeta less than
R okay then g n hash zeta is defined on mod
00:31:59.289 --> 00:32:05.899
zeta less than R okay because mod Zeta less
than R is contained in mod Zeta less than
00:32:05.899 --> 00:32:11.899
R n and mod Zeta less than R n is the domain
of g n okay.
00:32:11.899 --> 00:32:17.429
So g n hash is defined and in fact it is not
only for n it is also for higher values of
00:32:17.429 --> 00:32:28.470
n okay, so you know let me write g n hash
of n plus m, m is equal to 1, 0, 1, 2 and
00:32:28.470 --> 00:32:38.389
so on, so all these g n are defined okay and
the point is in any case this quantity you
00:32:38.389 --> 00:32:46.480
get this estimate g n hash of zeta is bounded
by 1 by 1 minus R by R n which is bounded
00:32:46.480 --> 00:32:58.380
by 1 by 1 minus R okay. So and finally I have
gotten this 1 by 1 minus R without any condition
00:32:58.380 --> 00:33:06.080
on the subscript n small n and that is the
uniform bound for g n hash beyond a certain
00:33:06.080 --> 00:33:10.870
stage alright and that is it, now Martyís
theorem will tell you that g n hash to
00:33:10.870 --> 00:33:17.210
therefore it has to converge normally
okay.
00:33:17.210 --> 00:33:38.570
So by Martyís theorem g n admits a subsequence
that converges normally on the whole complex
00:33:38.570 --> 00:34:02.039
plane okay and without loss of generality
you may assume that the subsequence is g n
00:34:02.039 --> 00:34:12.810
itself okay, so after all I am interested
in a convergence of sequence and what I have
00:34:12.810 --> 00:34:18.700
got is a sequence which I know admits a convergence
of sequence, so without loss of generality
00:34:18.700 --> 00:34:22.750
I replace the sequence by the convergence
of sequence okay if I do not do this then
00:34:22.750 --> 00:34:29.240
I will have to use a double subscript okay
but it really does not matter but now this
00:34:29.240 --> 00:34:40.770
g n does the job because you see what happens
is that 1st of all this tells you this bound
00:34:40.770 --> 00:34:51.970
on g n hashes, so what you do is now you let
R n to tent to infinity okay then R by R n
00:34:51.970 --> 00:34:54.319
will go to 0 okay.
00:34:54.319 --> 00:34:59.070
So you let n tend to infinity then R n goes
to infinity R by R n goes to 0 and this quantity
00:34:59.070 --> 00:35:05.869
goes to 1 okay and that will tell you that
all the g n hash they are all bounded by 1
00:35:05.869 --> 00:35:15.290
okay so clearly you get all the g n hash are
all bounded by 1 that is one condition and
00:35:15.290 --> 00:35:21.560
what is the other thing. What about g n hash
of 0? If you calculate the g n hash of 0,
00:35:21.560 --> 00:35:30.119
g n hash of 0 is going to be what? So let
us go back to what we have here, so go to
00:35:30.119 --> 00:35:37.990
this formula here you put 0 is equal to 0,
g n hash of 0 as Epsilon n f n hash of Z n
00:35:37.990 --> 00:35:43.300
but you see Epsilon n f n hash of Z n is 1
because Epsilon n is actually 1 by f n hash
00:35:43.300 --> 00:35:51.930
of Z n, so g n hash of 0 is actually one okay.
See these are all little tricks that I mean
00:35:51.930 --> 00:35:57.040
they are all there okay you have to look at
them okay that is the reason why this proof
00:35:57.040 --> 00:35:58.040
is tricky.
00:35:58.040 --> 00:36:13.930
So this is actually 1 okay so if the g n converge
to g normally on C then the g n hash converges
00:36:13.930 --> 00:36:24.570
to g hash normally on C, so what this means
is that since the g n hash are all bounded
00:36:24.570 --> 00:36:34.670
by 1 the limit g hash also bounded by 1 and
since all the g n hash at 0 are equal to 1,
00:36:34.670 --> 00:36:41.460
g hash also at 0 will be equal to 1 just by
properties of limits and you have done with
00:36:41.460 --> 00:36:48.109
the proof of Zalcman's Lemma okay and we have
used Martyís theorem that is the whole point
00:36:48.109 --> 00:36:49.109
right.
00:36:49.109 --> 00:36:58.701
Now what I want you to understand is that
ash in this Zalcman's Lemma basically you
00:36:58.701 --> 00:37:05.960
have condition for non-normality of a family
okay and the fact is that the converse of
00:37:05.960 --> 00:37:12.510
Zalcman's Lemma is also true namely if you
have a family script F such that you are able
00:37:12.510 --> 00:37:19.619
to find a sequence Z n tending to Z not and
a sequence of radii Epsilon n and also sequence
00:37:19.619 --> 00:37:25.700
of functions such that the zoom family converges
normally to a non-constant Meromorphic function
00:37:25.700 --> 00:37:34.799
then the original family has to be not normal.
It has to be in fact normality will be actually
00:37:34.799 --> 00:37:39.900
you know normality will be violated at the
point Z not. Z not is the point where a normality
00:37:39.900 --> 00:37:50.030
of the family is violated okay and in what
senseÖthe point is that at Z not the spherical
00:37:50.030 --> 00:37:57.549
derivatives become as you approach Z not through
by a Z n spherical derivatives becomes unbounded
00:37:57.549 --> 00:38:03.000
and the unboundedness of the spherical derivatives
is the same as non-normality because that
00:38:03.000 --> 00:38:06.260
is Martyís theorem okay.
00:38:06.260 --> 00:38:20.240
So let me probably say give this as a note
that the
00:38:20.240 --> 00:38:40.690
converse of Zalcman's Lemma
is true and why is that so, for if f is a
00:38:40.690 --> 00:38:55.099
family such that there exist sequence Z n
going to Z not with and there is a sequence
00:38:55.099 --> 00:39:04.359
of radii going to 0 with the zoom functions
and you have family of functions and there
00:39:04.359 --> 00:39:14.809
exist a sequence f n family of functions at
f such that the zoom functions g n which is
00:39:14.809 --> 00:39:25.290
zooming of f n centred at Z n, the magnification
factor 1 by Epsilon and in the new variable
00:39:25.290 --> 00:39:38.819
Zeta suppose this goes normally converges
to g Zeta on the complex plane with g hash
00:39:38.819 --> 00:39:47.130
of 0 equal to 1 and g hash is always less
than or equal to 1 suppose this happens okay
00:39:47.130 --> 00:39:51.609
then the family cannot be normal and why is
that true that is very very simple because
00:39:51.609 --> 00:39:59.490
you see then the script F is not normal. Why
is that true?
00:39:59.490 --> 00:40:07.470
It is very simple because you know g n hash
of Zeta as we just calculate we have seen
00:40:07.470 --> 00:40:16.069
is Epsilon n times f n hash of you know Z
n plus Epsilon n Zeta you know this and put
00:40:16.069 --> 00:40:27.440
Zeta equals to 0 what you will get is g n
hash of 0 is equal to Epsilon n into f n hash
00:40:27.440 --> 00:40:39.010
of Z n okay but you see g n hash of 0 g hash
of 0 is 1 and g n hash goes to g okay, so
00:40:39.010 --> 00:40:47.900
g n hash goes to g hash of 0 which is one
okay. So if you take any mind you all this
00:40:47.900 --> 00:40:57.220
Epsilon n are going to 0 okay, so this happens
as an tends to infinity okay, so what will
00:40:57.220 --> 00:41:04.300
it mean? See you have a product of 2 quantities
one of them is going to 0 but the product
00:41:04.300 --> 00:41:09.609
is bounded that means the other has to go
to infinity, so what you will get is that
00:41:09.609 --> 00:41:16.500
so this implies that this to go to plus
infinity okay and what does that mean?
00:41:16.500 --> 00:41:22.960
It means that you have violated the sequence
f n has violated the conditions of Martyís
00:41:22.960 --> 00:41:30.390
theorem you have found functions whose spherical
derivatives are going to plus infinity. Spherical
00:41:30.390 --> 00:41:34.079
derivatives are not bounded and where is this
happening? See is f n hash spherical derivatives
00:41:34.079 --> 00:41:39.270
of f n at Z n is becoming larger and larger
and larger going to plus infinity and the
00:41:39.270 --> 00:41:43.720
Z n are approaching Z not okay and mind you
Z n are all approaching Z naught, so what
00:41:43.720 --> 00:41:48.460
is happening is that if you look at a compact
neighbourhood of Z not an open disk closed
00:41:48.460 --> 00:41:57.030
disk containing Z not you see that on that
compact neighbourhood okay these f n hash
00:41:57.030 --> 00:42:02.210
are not going to be bounded uniformly because
they are going to plus infinity and now Martyís
00:42:02.210 --> 00:42:08.549
theorem will tell you therefore that this
even the f n that sequence itself as a family
00:42:08.549 --> 00:42:12.490
is not normal okay, so that it implies non-normality.
00:42:12.490 --> 00:42:16.240
So the converse of Zalcman's Lemma is also
true, so Zalcman's Lemma is actually an if
00:42:16.240 --> 00:42:23.780
and only if condition okay but the beautiful
thing about the Lemma is that you if a family
00:42:23.780 --> 00:42:30.500
is not normal the Lemma is able to guarantee
the existence of non-normal point and non-normal
00:42:30.500 --> 00:42:39.920
sequence at that point okay you get both the
point and sequence that violates normality
00:42:39.920 --> 00:42:49.760
okay. So what is now left is that I will have
to use this to prove Picard theorem and we
00:42:49.760 --> 00:43:01.770
will do that in the coming lectures, so let
me write this here, so this implies by Martyís
00:43:01.770 --> 00:43:19.369
theorem that f n, hence is not normal at Z
not okay, so I will stop here.