WEBVTT
Kind: captions
Language: en
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Alright so you see I am continuing with this
motivation for Zalcman's Lemma alright and…
00:00:57.690 --> 00:01:11.729
So let me recall what we were doing, so let
me use a different color. So basically you
00:01:11.729 --> 00:01:19.330
are trying to understand the behaviour of
a normal family at that point okay that is
00:01:19.330 --> 00:01:24.720
the so understanding that is you know the
key to understanding the statement of Zalcman's
00:01:24.720 --> 00:01:32.280
Lemma okay, how does a normal family of Meromorphic
functions how does it behave at a point, at
00:01:32.280 --> 00:01:39.970
a given point? So you know the important outcome
of this analysis is that you know you can
00:01:39.970 --> 00:01:46.690
characterise normality at a point and normality
at the point is defined as normality in the
00:01:46.690 --> 00:01:52.320
neighbourhood of that point in some neighbourhood
open disk surrounding that point okay and
00:01:52.320 --> 00:01:58.520
then so you see this means that normality
can be defined locally and it is in fact a
00:01:58.520 --> 00:02:00.890
local property like analyticity okay.
00:02:00.890 --> 00:02:09.050
So we start with a family script F these are
Meromorphic functions defined in a neighbourhood
00:02:09.050 --> 00:02:14.410
of the point Z not, I am assuming the point
Z naught is in the complex plane it could
00:02:14.410 --> 00:02:18.130
have been also a point in the extended complex
plane namely it could have been the point
00:02:18.130 --> 00:02:22.340
at infinity okay. All arguments will work
but you will have to make the right modifications
00:02:22.340 --> 00:02:29.720
so you know how to do that because we know
how to deal with the point at infinity and
00:02:29.720 --> 00:02:32.570
taking in finite limits and things like that
okay.
00:02:32.570 --> 00:02:47.700
So alright so what I told you last time is
that we start with this family script F and
00:02:47.700 --> 00:02:54.810
what you can do is given a function small
f and script F okay you want to study that
00:02:54.810 --> 00:03:02.510
function in neighbourhood of the point Z naught
okay so what you do is that you know you take
00:03:02.510 --> 00:03:09.099
this neighbourhood of this point which is
actually it is a disk centred at Z naught
00:03:09.099 --> 00:03:14.560
radius rho okay you take this neighbourhood
and I am also assuming that the boundary of
00:03:14.560 --> 00:03:23.181
that disk is in the domain where the function
are defined okay, so and the reason why I
00:03:23.181 --> 00:03:31.090
am including the boundary is because I want
a compact set okay.
00:03:31.090 --> 00:03:40.159
The disk along with the boundary forms a compact
set is a closed and bounded set okay and well
00:03:40.159 --> 00:03:47.239
so suppose I want to study a particular function
small f in the family script F its behaviour
00:03:47.239 --> 00:03:53.510
very close to the point Z naught then what
I do is I zoom into the point Z naught and
00:03:53.510 --> 00:03:57.280
try to study the behaviour of the function
and how do I do this zooming, I do this zooming
00:03:57.280 --> 00:04:07.450
in this way so I define this zoomed function
here okay g so given f I zoom f to get a new
00:04:07.450 --> 00:04:14.220
function g okay and what is this zooming?
You are zooming the function f centred at
00:04:14.220 --> 00:04:20.840
Z naught and the zoom factor is 1 by Epsilon
where Epsilon is supposed to be small positive
00:04:20.840 --> 00:04:23.480
quantity, so that 1 by Epsilon is large.
00:04:23.480 --> 00:04:31.539
In fact Epsilon is to be taken less than rho
okay and then you define this new function
00:04:31.539 --> 00:04:38.740
you define this new function which is for
this f you define this function g which is
00:04:38.740 --> 00:04:50.490
a zoomed function. g is just you take f look
at its behaviour in a small disk around Z
00:04:50.490 --> 00:04:58.849
naught and then you zoomed the behaviour by
a factor 1 by Epsilon okay. So what you do
00:04:58.849 --> 00:05:05.330
is that so this is how the zooming is defined
the zoom function g of Zeta is f of Z naught
00:05:05.330 --> 00:05:09.990
plus Epsilon Zeta which means that actually
what you have done is you are actually translating
00:05:09.990 --> 00:05:21.180
Z naught to the origin okay and then what
happens is that this whole disk centred at
00:05:21.180 --> 00:05:28.390
Z naught radius rho along with the boundary
translates to disk centred at the origin with
00:05:28.390 --> 00:05:30.170
radius rho by Epsilon okay.
00:05:30.170 --> 00:05:35.330
So 1 by Epsilon is the scaling factor and
mind you you should think of it like this,
00:05:35.330 --> 00:05:40.920
Epsilon is small so 1 by Epsilon is large,
so rho by Epsilon is greater than rho so you
00:05:40.920 --> 00:05:47.270
have actually zoomed okay and so the behaviour
of the zoom function is being studied here
00:05:47.270 --> 00:05:58.520
okay and in this dish and I am calling this
zoomed function g, so this is a very very
00:05:58.520 --> 00:06:03.560
simple thing and the point is that the difference
between f and g is just you know it is
00:06:03.560 --> 00:06:12.100
a bilinear transformation, it is actually…
It is translation and it involves translation
00:06:12.100 --> 00:06:17.919
and it involves scaling okay. So what you
have done is you have taken the variable Zeta
00:06:17.919 --> 00:06:23.550
your scale did by Epsilon okay you multiplied
it by Epsilon mind you Epsilon is a positive
00:06:23.550 --> 00:06:28.419
real quantity and then you have translation
by Z naught okay so the g and f are of the
00:06:28.419 --> 00:06:32.390
same type of action okay.
00:06:32.390 --> 00:06:36.310
G is analytic if and only if f is analytic,
g is Meromorphic if and only if f is Meromorphic
00:06:36.310 --> 00:06:44.639
and if g has a pole at a certain point then
f will also have a pole of the same order
00:06:44.639 --> 00:06:50.750
at the corresponding point okay and conversely.
So g at f are literally the same function
00:06:50.750 --> 00:06:55.080
except that you have made a change of variable
alright but the point is that g is close up
00:06:55.080 --> 00:07:01.800
look of f you are looking at f very close
in a neighbourhood of Z naught that is the
00:07:01.800 --> 00:07:09.090
whole point okay. Now what you do is that
this is what you do if you have a single function
00:07:09.090 --> 00:07:13.190
but what you could have done is? You could
have done this to a sequence of functions.
00:07:13.190 --> 00:07:21.139
So the pointers if I knew that this family
script F is normal then suppose you assume
00:07:21.139 --> 00:07:27.800
that the family is normal in this closed disk
which means it is actually it is normally
00:07:27.800 --> 00:07:35.660
in an open set which contains this closed
disk including the boundary okay and suppose
00:07:35.660 --> 00:07:41.770
this family is normal you know normality is
the correct notion of compactness that we
00:07:41.770 --> 00:07:47.270
require. Normality actually means normally
sequentially compact, it means that given
00:07:47.270 --> 00:07:53.190
any sequence you can find a subsequence which
converges normally okay. So converges normally
00:07:53.190 --> 00:07:59.270
means converges uniformly on compact subsets,
so assuming that the family script F is normal
00:07:59.270 --> 00:08:06.220
and I want to study the behaviour of the family
at a given point Z naught okay.
00:08:06.220 --> 00:08:12.479
So if I take a sequence f n in this family
then because of normality you can find a subsequence
00:08:12.479 --> 00:08:20.280
f n k that will converge normally on this
disk in fact it will converge uniformly on
00:08:20.280 --> 00:08:25.620
that this because that is a closed disk okay.
Normal convergence means it is uniform on
00:08:25.620 --> 00:08:33.130
compact subsets and this disk centred at Z
naught radius rho is you know is a compact
00:08:33.130 --> 00:08:39.250
subset and again at the back of your mind
you should remember that you know this argument
00:08:39.250 --> 00:08:42.440
will work given if Z naught is a point at
infinity, the only thing is that if Z naught
00:08:42.440 --> 00:08:51.430
is a point at infinity you will have to think
of this you should write this disk you know
00:08:51.430 --> 00:08:59.870
if you want with the spherical metric or you
must invert the variable and look at the neighbourhood
00:08:59.870 --> 00:09:08.170
of 0 okay, so you can deal with the case when
Z naught it is the point at infinity also
00:09:08.170 --> 00:09:14.350
okay but in any case given the sequence f
n I have the subsequence which converges normally,
00:09:14.350 --> 00:09:20.549
so let me call the limit function as f and
then we have already seen this whenever you
00:09:20.549 --> 00:09:23.699
take a normal limit of Meromorphic functions
the limit is either identically infinity or
00:09:23.699 --> 00:09:25.400
it is Meromorphic okay.
00:09:25.400 --> 00:09:33.640
And now look at what happens to the zoomed
functions, so if f n k converges to f normally
00:09:33.640 --> 00:09:41.150
then the zoomed functions for the sequence
will converge to the zoomed function of the
00:09:41.150 --> 00:09:51.819
limit okay, so this is obvious okay because
the differences is just a change of variable
00:09:51.819 --> 00:09:58.339
given by bilinear transformation consisting
of a scaling and the translation okay, so
00:09:58.339 --> 00:10:04.160
this g n k will converge normally to g okay
and all this is happening mind you because
00:10:04.160 --> 00:10:11.020
you have zooming factor is 1 by Epsilon it
is all happening here, it is happening in
00:10:11.020 --> 00:10:16.260
the open disk centred at the origin radius
rho by Epsilon which is greater than rho okay
00:10:16.260 --> 00:10:30.561
and well this is where it is happening. Now
you see the point is that whenever a family
00:10:30.561 --> 00:10:36.939
of Meromorphic functions converges then the
family of spherical derivatives will also
00:10:36.939 --> 00:10:38.330
converge okay.
00:10:38.330 --> 00:10:44.900
Now this is just saying that you know the
taking the spherical derivative will preserve
00:10:44.900 --> 00:10:51.010
the convergence in the normal normal convergence
okay. So f n k converges to f normally, so
00:10:51.010 --> 00:10:56.600
what will happen is that the you will get
this which is that the spherical derivatives
00:10:56.600 --> 00:11:05.380
of f n k is converge normally to the spherical
derivatives of f okay and you already know
00:11:05.380 --> 00:11:10.990
that because f n k converges normally to F,
g n k converges normally to g alright and
00:11:10.990 --> 00:11:18.720
therefore the spherical derivatives of g n
k hash they will converge normally to g hash
00:11:18.720 --> 00:11:27.819
okay but the point is that you see the origin
family the family script F you have assumed
00:11:27.819 --> 00:11:29.459
is normal.
00:11:29.459 --> 00:11:36.780
So this family script F is normal and Marty’s
theorem tells you that there are families
00:11:36.780 --> 00:11:42.430
normal if and only if the spherical derivatives
are normally uniformly bounded okay and since
00:11:42.430 --> 00:11:49.630
we are already looking at a compact set okay
there is a uniform bound for all the spherical
00:11:49.630 --> 00:11:53.490
derivatives of the functions in the family
mind you, you have to use the spherical derivatives
00:11:53.490 --> 00:11:57.040
because there are functions of family are
not analytic functions they are Meromorphic
00:11:57.040 --> 00:12:03.791
functions, so you cannot talk about usual
derivatives at a poles okay but then you can
00:12:03.791 --> 00:12:06.080
talk about spherical derivatives at a pole.
00:12:06.080 --> 00:12:14.620
So by Marty’s theorem mind you tells you
that normal sequential compactness is equivalent
00:12:14.620 --> 00:12:20.420
to normal uniform bounded of the spherical
derivatives okay and it is a generalisation
00:12:20.420 --> 00:12:28.800
of the Montel’s theorem or analytic functions
which says that you know the normal sequential
00:12:28.800 --> 00:12:35.020
compactness is equivalent to uniform boundedness,
normal uniform boundedness of the original
00:12:35.020 --> 00:12:42.410
family okay but the point is when you go to
Marty’s theorem you go to from the uniform
00:12:42.410 --> 00:12:48.300
boundedness of the normal uniform boundedness
of the given family you switch to the normal
00:12:48.300 --> 00:12:55.200
uniform boundedness of the family of spherical
derivatives okay and that is the quantum jump
00:12:55.200 --> 00:13:01.480
that you make from Montel’s theorem for
analytic functions to you when you go to Marty’s
00:13:01.480 --> 00:13:06.810
theorem which is for Meromorphic functions,
so so there is this uniform bound m which
00:13:06.810 --> 00:13:11.260
works for all functions in the family script
F.
00:13:11.260 --> 00:13:16.830
So it will work also for these f n k, it is
bound for not the functions but it is bound
00:13:16.830 --> 00:13:21.240
for their derivatives okay spherical derivatives,
so all the spherical derivatives of the f
00:13:21.240 --> 00:13:29.850
n k are bounded by m and then well what happens
is therefore now if you take the limit, so
00:13:29.850 --> 00:13:35.829
there is a chain rule that works in here,
there is a chain rule, the spherical derivatives
00:13:35.829 --> 00:13:43.100
of the zoom function g n k hash is Epsilon
times the spherical derivatives of the f n
00:13:43.100 --> 00:13:51.530
k the spherical derivatives of g n k is
Epsilon times spherical derivatives of f n
00:13:51.530 --> 00:14:04.290
k because this… So
the idea is that you are zooming by a factor
00:14:04.290 --> 00:14:11.770
1 by Epsilon okay you are going closer to
the point by you are zooming into the point
00:14:11.770 --> 00:14:19.630
by a factor 1 by Epsilon okay but then this
spherical derivative becomes smaller okay.
00:14:19.630 --> 00:14:25.299
The spherical derivatives get multiplied by
the factor Epsilon mind you Epsilon is a small
00:14:25.299 --> 00:14:31.660
quantity, so multiplying by Epsilon makes
the quantity smaller okay whereas 1 by Epsilon
00:14:31.660 --> 00:14:37.939
is a zooming factor, so as you zoom closer
your spherical derivative is going to become
00:14:37.939 --> 00:14:43.650
smaller okay so that is what is happening.
So for the zoom function if the original function
00:14:43.650 --> 00:14:50.520
have this bound m then the zoomed function
have the bound to Epsilon m and then because
00:14:50.520 --> 00:14:57.120
the zoom function g the spherical derivatives
of the zoom functions converges normally to
00:14:57.120 --> 00:15:03.600
the spherical derivatives of the limit function
which is zoom limit function what it will
00:15:03.600 --> 00:15:09.110
tell you is that the limit function also if
you take its spherical derivatives is bounded
00:15:09.110 --> 00:15:12.309
by Epsilon times m okay.
00:15:12.309 --> 00:15:23.300
Now the point is that you know all this happens
in this domain which is the disk centred at
00:15:23.300 --> 00:15:29.949
the origin radius rho by Epsilon alright and
mind you, you must remember that as I make
00:15:29.949 --> 00:15:35.020
smaller and smaller and smaller I am zooming
closer and closer to the point Z naught but
00:15:35.020 --> 00:15:41.790
then the zoomed function are in larger and
larger disk centred at the origin because
00:15:41.790 --> 00:15:47.510
rho by Epsilon becomes larger and larger as
Epsilon becomes smaller. So now the point
00:15:47.510 --> 00:15:54.900
is that you repeat this game okay by changing
the zoom factor okay, so what you do is instead
00:15:54.900 --> 00:16:04.160
of taking single Epsilon you take a sequence
of Epsilons which go to 0 okay and of course
00:16:04.160 --> 00:16:10.320
from the right namely that you take sequence
of positive Epsilons and go to 0 okay.
00:16:10.320 --> 00:16:17.829
So if you repeat the game with that what you
do is that now you know the nth function g
00:16:17.829 --> 00:16:26.860
n okay and well I think I have not written
it correctly last time, the nth the function
00:16:26.860 --> 00:16:39.829
g n is the zooming of the nth function f n
by the factor 1 by Epsilon n centred at Z
00:16:39.829 --> 00:16:45.720
naught with the variable Zeta, so there should
have been an f n here alright. So GN is the
00:16:45.720 --> 00:16:52.060
zooming of f n alright but the only thing
is that now the zooming factor is also dependent
00:16:52.060 --> 00:16:59.660
on M, so f n is zoomed by a factor 1 by Epsilon
m and I am calling it as g m okay.
00:16:59.660 --> 00:17:06.660
Just what we did some time ago was keeping
all this Epsilon n the same Epsilon but now
00:17:06.660 --> 00:17:14.350
I am making the Epsilon n smaller as n becomes
larger okay. Well so if by doing this what
00:17:14.350 --> 00:17:22.689
happens is mind you if I make the Epsilon
n smaller than the domains on which the zoom
00:17:22.689 --> 00:17:37.620
function work these domains becomes bigger
okay so you know I get these domains okay
00:17:37.620 --> 00:17:45.440
mod Zeta lesser than I think there was a rho
there mod Zeta less than rho by Epsilon n
00:17:45.440 --> 00:17:53.310
and these are increasingly larger disk as
Epsilon n goes to 0 these disk cover the whole
00:17:53.310 --> 00:18:01.580
plane okay and in fact is that means the n
times to infinity you are getting your
00:18:01.580 --> 00:18:06.040
you will cover the whole plane in particular
you will cover any compact subset of the plane.
00:18:06.040 --> 00:18:12.730
So what we can say is that these zoom functions
g n you know you can talk about normal convergence
00:18:12.730 --> 00:18:22.450
on the whole plane okay, so actually what
happens is that the...anyway as before the
00:18:22.450 --> 00:18:31.530
f n k will converge to f okay and therefore
the f n k hash will converge to f hash where
00:18:31.530 --> 00:18:36.160
hash then the spherical derivatives and
then the corresponding zoom functions g n
00:18:36.160 --> 00:18:41.850
k hash will converge to g hash all these convergences
are all normal okay but the only thing is
00:18:41.850 --> 00:18:46.910
that this when you come to the zoom functions
this normal convergence is on all of C, it
00:18:46.910 --> 00:18:51.870
is on the whole complex plane because now
you have covered the whole complex plane you
00:18:51.870 --> 00:18:55.600
have covered every compact subset of the complex
plane by a sufficiently large disk centred
00:18:55.600 --> 00:19:05.020
at the origin where that means all the GN
k beyond a certain k they are all defined
00:19:05.020 --> 00:19:08.380
on any compact subset beyond a certain stage
okay.
00:19:08.380 --> 00:19:21.240
So and so you have this…and the point is
that you know the GN k if you take the spherical
00:19:21.240 --> 00:19:27.540
derivatives they are bounded by Epsilon n
k times the bound for the original function
00:19:27.540 --> 00:19:35.450
which is m okay and therefore this limiting
argument will tell you that the limit function
00:19:35.450 --> 00:19:41.280
g if you take its spherical derivative it
has to be 0 and therefore you get a constant
00:19:41.280 --> 00:19:47.910
function on all of C okay so you see the point
is therefore you know as you go closer and
00:19:47.910 --> 00:19:54.250
closer and closer to point and look at the
zoom functions okay then the zoom functions
00:19:54.250 --> 00:20:03.360
they tend to become constant that is what
it means, the zoom functions becomes constant.
00:20:03.360 --> 00:20:10.910
So to sum of all this all I am saying is that
you take a normal family you take a point
00:20:10.910 --> 00:20:21.740
where the family is normal and then what happens
is that as you take any sequence of in that
00:20:21.740 --> 00:20:28.110
family you can always find a convergence subsequence
if you study it at that point okay then the
00:20:28.110 --> 00:20:36.000
zoomed function will you know they will tend
to a constant function that is the whole point
00:20:36.000 --> 00:20:46.430
okay and Zalcman's Lemma is all about you
know being able to find zoom functions which
00:20:46.430 --> 00:20:55.720
converges to a non-constant Meromorphic functions
okay. So something opposite to this happens,
00:20:55.720 --> 00:21:00.440
so what I want to do 1st is that I need to
1st of all tell you that I can repeat this
00:21:00.440 --> 00:21:09.490
argument with not just one point Z naught
but I can even take the sequence of points
00:21:09.490 --> 00:21:11.010
tending to Z naught okay.
00:21:11.010 --> 00:21:25.510
So now what we do is now take sequence Z n
going to Z naught say the same disk mod Z
00:21:25.510 --> 00:21:30.540
minus Z naught less than or equal to rho if
you want okay. Well for that matter if you
00:21:30.540 --> 00:21:35.180
take any sequence Z n going to Z naught the
sequence beyond a certain stage will lie in
00:21:35.180 --> 00:21:41.190
that disk okay but I am assuming this whole
sequence of lies in that those disk centred
00:21:41.190 --> 00:21:49.940
at Z naught radius rho okay and what you do
is now you zoom at each Z n okay. See all
00:21:49.940 --> 00:21:55.340
this time we were zooming only at Z naught
but now what you do is you zoom at each Z
00:21:55.340 --> 00:22:06.770
m, so you define g n of Zeta to be the zooming
of you zoom the function f n but now centred
00:22:06.770 --> 00:22:12.780
at Z n and you know the zooming factor is
1 by Epsilon n and the new variable is Zeta,
00:22:12.780 --> 00:22:27.360
so this means that g n of Zeta is well it
is f n of Z n plus Epsilon n times Zeta.
00:22:27.360 --> 00:22:38.110
So this is the thing but then I have to worry
about one has to worry about where this will…
00:22:38.110 --> 00:22:44.620
What about the domains of the GN okay you
will have to worry about that but point is
00:22:44.620 --> 00:22:49.110
that essentially as n tends to infinity you
are coming closer and closer to Z not, so
00:22:49.110 --> 00:22:56.350
the domains of g n is going to again cover
the whole complex plane okay, so one can write
00:22:56.350 --> 00:23:04.100
out the details for that. So you know the
picture is like this, the pictures is that
00:23:04.100 --> 00:23:12.370
you know so I have you know I have this Z
naught and you know there is so there is a
00:23:12.370 --> 00:23:24.400
Z n and then I have this…so inside Z naught
there is this big disk with radius rho where
00:23:24.400 --> 00:23:34.340
I am concentrating my attention and of course
even the boundary is included though I am
00:23:34.340 --> 00:23:36.790
putting a dotted line.
00:23:36.790 --> 00:23:55.870
Well and what I am doing is at Z n I am taking
a smaller disk with radius Epsilon n okay
00:23:55.870 --> 00:24:03.900
and then you know as Z n are tending to Z
not, so well Z n plus 1 is closer to Z naught
00:24:03.900 --> 00:24:10.450
if you want and there is and then I am
taking a much smaller disk and the radius
00:24:10.450 --> 00:24:16.250
of that disk is Epsilon m plus 1 okay so you
know I am coming I am taking smaller and smaller
00:24:16.250 --> 00:24:24.000
disk above points which are going closer and
closer and closer to Z naught and the point
00:24:24.000 --> 00:24:30.690
is that on this I have the zoom function okay
and then if I go to Z n plus 1 I have another
00:24:30.690 --> 00:24:42.120
zoom function it is g n plus 1 of Zeta and
this is f n plus 1 centred at Z n plus 1.
00:24:42.120 --> 00:24:47.860
So it is Epsilon n plus 1 Zeta so this is
just zooming of f n plus 1 centred at Z n
00:24:47.860 --> 00:24:52.470
plus 1 with a factor 1 by Epsilon and plus
1 and the variable Zeta okay.
00:24:52.470 --> 00:24:59.000
So I have this zoom function okay and well
of course you know I will have to assume that
00:24:59.000 --> 00:25:11.240
the Epsilon n are chosen in such a way that
the disk centred at Z n radius Epsilon n is
00:25:11.240 --> 00:25:17.090
within my big disk centred at Z naught with
radius rho okay but that will happen eventually
00:25:17.090 --> 00:25:21.320
even if you did not assume it beyond certain
stage it has to happen okay but we assume
00:25:21.320 --> 00:25:27.110
therefore with lot of generality that this
happens a diagram is like we have already
00:25:27.110 --> 00:25:36.210
shown it, so you know for example so I am
assuming that if you take Z 1 then I am taking
00:25:36.210 --> 00:25:44.060
the Epsilon 1 that I am taking is such that
the disk centred at Z 1 radius Epsilon 1 lies
00:25:44.060 --> 00:25:49.130
inside this big disk centred at Z naught radius
rho okay and the point is that if this does
00:25:49.130 --> 00:25:55.270
not work I can replace the Epsilon by some
Epsilon I primes which I can calculate okay.
00:25:55.270 --> 00:26:09.930
So the condition I want is that you know the
distance of Z naught where any point of the
00:26:09.930 --> 00:26:15.240
disk should not be greater than rho okay should
be less than rho that is the condition I want.
00:26:15.240 --> 00:26:22.680
So well let us assumed that the picture is
like this alright we do the same calculations
00:26:22.680 --> 00:26:34.640
as before you see f n you know you have this
f n k they converges normally to f on
00:26:34.640 --> 00:26:47.260
this is normally and well what happens is
that you will therefore get the spherical
00:26:47.260 --> 00:26:59.390
derivatives will converge to f normally
and you will also get that the zoom functions
00:26:59.390 --> 00:27:18.460
will also converge normally to g and so will
their spherical derivatives okay.
00:27:18.460 --> 00:27:26.500
So you will get all this as before mind you
the only thing now is that you have also changed
00:27:26.500 --> 00:27:33.110
the center of zooming okay. The centres of
the zooming are all different now okay but
00:27:33.110 --> 00:27:41.740
that does not change the calculation for the
spherical derivative okay because spherical
00:27:41.740 --> 00:27:47.230
derivative or the zoom functions with respect
to Zeta and as far as Zeta is concerned the
00:27:47.230 --> 00:27:50.670
Z n are all constant okay.
00:27:50.670 --> 00:27:58.320
So well again what you will get is that you
will get that again you will get that you
00:27:58.320 --> 00:28:06.690
know the original functions f n k their spherical
derivatives of bounded by m and this will
00:28:06.690 --> 00:28:16.550
tell you that the zoom functions their spherical
derivatives are going to be bounded by Epsilon
00:28:16.550 --> 00:28:20.971
the corresponding scaling factor times m I
mean not a scaling factor, the corresponding
00:28:20.971 --> 00:28:30.320
Epsilon times m and if you take a limit as
n tends to infinity you will get that the
00:28:30.320 --> 00:28:38.270
limit zoom function which is zoom function
of the limit function, that will have spherical
00:28:38.270 --> 00:28:47.430
derivative is 0 okay that is because the Epsilon
n k tends to 0 and and as a fixed constant,
00:28:47.430 --> 00:28:54.309
positive constant and as before the moment
you know that the spherical derivatives of
00:28:54.309 --> 00:28:57.860
Meromorphic function is 0 it has to be constant.
00:28:57.860 --> 00:29:03.700
So this will tell you that g is constant on
the complex plane mind you this constant could
00:29:03.700 --> 00:29:08.490
have very well-being the constant value infinity
that is all out okay because we are in the
00:29:08.490 --> 00:29:14.980
context of Meromorphic functions but the whole
point is and now you have got this important
00:29:14.980 --> 00:29:23.110
characterisation, you take a point Z naught
where a family is normal okay then what happens
00:29:23.110 --> 00:29:28.540
is that you take any sequence of point Z n
going to Z naught and you take a sequence
00:29:28.540 --> 00:29:34.929
of radii which is going to 0 the Epsilon n
okay then given any sequence in the family
00:29:34.929 --> 00:29:38.990
you can find a subsequence such that if you
take the subsequence and zoom it with respect
00:29:38.990 --> 00:29:45.110
to the sequence, the zoom functions converge
uniformly on compact subsets of C that is
00:29:45.110 --> 00:29:47.310
normally on C to a constant function.
00:29:47.310 --> 00:29:54.210
This is a behave of a normal family at a point
okay a normal family in the neighbourhood
00:29:54.210 --> 00:29:59.950
of a point okay and the amazing thing is that
you whole point is that this characterises
00:29:59.950 --> 00:30:05.900
normal families okay this is by this you
can actually characterise normal families
00:30:05.900 --> 00:30:15.470
okay. So let me write this down and the fact
that you can characterise normal families
00:30:15.470 --> 00:30:20.390
like this is actually the philosophy behind
Zalcman's Lemma and the way it works is it
00:30:20.390 --> 00:30:26.660
tells you when this is contradicted, so Zalcman's
Lemma will tell you that such a thing will
00:30:26.660 --> 00:30:32.070
not happen if the family is not normal okay.
00:30:32.070 --> 00:30:50.190
So let me write this, so proposition is well
let script F be normal at Z naught which means
00:30:50.190 --> 00:30:59.210
that it is a normal family in an open disk
containing Z naught and of course all my families
00:30:59.210 --> 00:31:13.200
of functions I Meromorphic okay. So let me
write here normal family of Meromorphic functions
00:31:13.200 --> 00:31:31.580
at Z naught okay given any sequence Z n tending
to Z naught and any sequence of radii Epsilon
00:31:31.580 --> 00:31:46.460
n tending to 0 plus okay for any sequence
f n in the family script F okay we can find
00:31:46.460 --> 00:32:06.549
a subsequence f n k such that the zoom sequence
of f n k with respect to Z n and Epsilon n
00:32:06.549 --> 00:32:13.040
that zoom sequence converges to a constant
function normally on all of the complex plane
00:32:13.040 --> 00:32:31.340
okay g n k of Zeta zooming of f n k with respect
to center Z N, z k scaling factor 1 by Epsilon
00:32:31.340 --> 00:32:41.960
k new variable Zeta which is by definition
you just take f n k of Z k is Epsilon k times
00:32:41.960 --> 00:32:45.590
Zeta this is the zooming okay.
00:32:45.590 --> 00:33:06.049
Now says that if this is the zoomed family
then g n k converges to a constant normally
00:33:06.049 --> 00:33:13.510
on the whole complex plane, so this is the
characterisation of… this is how normal
00:33:13.510 --> 00:33:22.390
families behave and the big deal about…so
this is normality at a point okay this is
00:33:22.390 --> 00:33:29.190
normality at a point Z naught which means
which by definition is normality in a small
00:33:29.190 --> 00:33:34.320
open disk containing Z naught that is what
it means okay and if you want to cover this
00:33:34.320 --> 00:33:40.970
to normality on a whole domain then you know
you have to say family is normal on a domain
00:33:40.970 --> 00:33:55.309
if it is normal at each point of the domain
okay. You can define it point wise, so now
00:33:55.309 --> 00:34:03.350
let me take Zalcman's Lemma which will tell
you and and you will appreciate about but
00:34:03.350 --> 00:34:08.169
you know with all this background that I have
giving you, you will see that Zalcman's Lemma
00:34:08.169 --> 00:34:10.230
is something natural to expect okay.
00:34:10.230 --> 00:34:20.070
So here is Zalcman's Lemma
and well it is call Zalcman's Lemma but it
00:34:20.070 --> 00:34:35.919
is actually a theorem. So what is a Lemma?
The Lemma is about non-normality okay, see
00:34:35.919 --> 00:34:46.329
I will tell you in a very short words, what
we just saw is that if you have a normal family
00:34:46.329 --> 00:34:53.140
then the zoom functions go to a constant okay.
Zalcman's Lemma tells you that if you have
00:34:53.140 --> 00:34:59.980
an family which is not normal you can get
hold of zoomed family of functions which will
00:34:59.980 --> 00:35:06.750
go to a non-constant Meromorphic function
that is all that is the whole point okay.
00:35:06.750 --> 00:35:10.960
So let me write that of course you know you
should always keep saying this in as little
00:35:10.960 --> 00:35:12.670
words as you can.
00:35:12.670 --> 00:35:17.091
So that you grab the main idea and then of
course when you write statements you will
00:35:17.091 --> 00:35:23.710
have to be you have to bring in lot of notation,
you have to worry about a lot of notation
00:35:23.710 --> 00:35:27.760
and then of course proves are even most slightly
complicated but the point is that you should
00:35:27.760 --> 00:35:34.090
always be able to zoom out you know and say
things in just few words because that is how
00:35:34.090 --> 00:35:39.930
you will map it in your memory and remember
it. So you do not lose track of the idea,
00:35:39.930 --> 00:35:57.500
so here is the Lemma suppose script F is a
family of Meromorphic functions on a domain
00:35:57.500 --> 00:36:09.160
D that is not normal.
00:36:09.160 --> 00:36:22.140
So here is a non-normal family then what happens
is that there exist sequence Z n converging
00:36:22.140 --> 00:36:42.260
to Z naught in D and Epsilon n going to 0
plus with and further a sequence f n in a
00:36:42.260 --> 00:36:56.340
family script F such that the zoom functions
g n of Zeta is equal to the zooming of f n
00:36:56.340 --> 00:37:06.040
centred at Z n scaling factor 1 by Epsilon
n new variable Zeta namely f n of Z n plus
00:37:06.040 --> 00:37:22.369
Epsilon n Zeta converges normally on C to
a non-constant that is the whole point to
00:37:22.369 --> 00:37:41.010
a non-constant Meromorphic functions g on
the whole complex plane such that a fact that
00:37:41.010 --> 00:37:45.790
you know it is a non-constant Meromorphic
function so you do not expect the spherical
00:37:45.790 --> 00:37:55.680
derivative to be 0, so the spherical derivatives
at the origin will be 1 and all the spherical
00:37:55.680 --> 00:38:10.740
derivatives is bounded by 1 okay, so this
is Zalcman's Lemma. It tells you that the
00:38:10.740 --> 00:38:18.119
behaviour is exactly opposite to what you
saw of a normal family okay, so we will try
00:38:18.119 --> 00:38:19.630
to prove this in the next lecture.