WEBVTT
Kind: captions
Language: en
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Alright so let me continue with this discussion
about Marty’s Theorem, okay which is basically
00:01:14.250 --> 00:01:23.080
an analog of Montel’s Theorem except that
your working with not with analytic functions
00:01:23.080 --> 00:01:25.229
but with meromorphic functions, okay.
00:01:25.229 --> 00:01:37.850
So let us look at let us again look at the
statement so you have this domain D in the
00:01:37.850 --> 00:01:46.810
complex plane and you have this family script
F of meromorphic functions on D so the script
00:01:46.810 --> 00:01:55.470
m D is the you know it is the set of meromorphic
functions on D and mind you we are considering
00:01:55.470 --> 00:02:03.700
these as continuous functions from D into
the extended complex plane, okay. So you are
00:02:03.700 --> 00:02:10.530
able to do that because meromorphic function
normally at a point which is a pole which
00:02:10.530 --> 00:02:15.700
is it goes to infinity, okay but then you
allow the value infinity and you declare the
00:02:15.700 --> 00:02:18.310
value at the pole to be infinity so it becomes
continuous.
00:02:18.310 --> 00:02:24.360
So the set of meromorphic functions is a subset
of this the set of continuous functions from
00:02:24.360 --> 00:02:34.310
D to the extended complex plane C union infinity
which as a metric space given the spherical
00:02:34.310 --> 00:02:38.840
metric we think of as the just the Riemann
sphere, okay with infinity corresponding to
00:02:38.840 --> 00:02:45.430
the north pole, right. And so you have this
family script F of meromorphic functions you
00:02:45.430 --> 00:02:52.299
can either use a word collection or family
or subset whichever you prefer but the point
00:02:52.299 --> 00:02:55.010
is when is this family compact?
00:02:55.010 --> 00:03:04.400
So in this case you know normally I should
since the word normal is used technically
00:03:04.400 --> 00:03:12.740
I should say usually compactness is equivalent
to sequential compactness and then that means
00:03:12.740 --> 00:03:16.740
that you know saying something is compact
is same as saying that every sequence has
00:03:16.740 --> 00:03:22.840
a convergent subsequence. So if you want to
say that the family script F is sequentially
00:03:22.840 --> 00:03:28.250
I mean it is compact you will like to say
it is sequentially compact if you want to
00:03:28.250 --> 00:03:29.630
say it is compact.
00:03:29.630 --> 00:03:35.130
And then you would like to which means that
you know given any sequence of functions in
00:03:35.130 --> 00:03:42.110
this family you are able to extract a subsequence
which converges. Now what kind of convergence?
00:03:42.110 --> 00:03:50.090
If we are usually the convergence that we
worry about is the uniform convergence but
00:03:50.090 --> 00:03:54.430
ofcourse in the case of analytic functions,
the meromorphic functions you will not get
00:03:54.430 --> 00:03:59.799
uniform convergence on the whole domain you
will get only uniform convergence on compact
00:03:59.799 --> 00:04:03.340
you will get only uniform convergence on compact
subsets and that is called normal convergence,
00:04:03.340 --> 00:04:04.340
okay.
00:04:04.340 --> 00:04:12.459
So in other words the compactness of the family
F is of as normal sequential compactness which
00:04:12.459 --> 00:04:18.840
means that every sequence in F admits a subsequence
that converges uniformly on compact subsets
00:04:18.840 --> 00:04:24.720
of the domain, okay so this is what compactness
for us means and Marty’s Theorem says that
00:04:24.720 --> 00:04:33.280
this is the same as the family of spherical
derivatives of F namely you take for each
00:04:33.280 --> 00:04:41.960
small F in script F you take its spherical
derivative F hash small f hash and you get
00:04:41.960 --> 00:04:48.630
this family script F hash this is the family
of spherical derivatives and that should be
00:04:48.630 --> 00:04:53.790
normally uniformly bounded which means that
it is uniformly bounded on compact subsets.
00:04:53.790 --> 00:05:06.169
So in some sense boundedness of derivatives
is equivalent to compactness I mean if you
00:05:06.169 --> 00:05:11.060
want to say it in a nutshell boundedness of
derivatives is equivalent to compactness,
00:05:11.060 --> 00:05:19.200
okay and so there are a couple of aspects
that I want to stress between this and the
00:05:19.200 --> 00:05:25.330
original Montel Theorem see the original Montel
Theorem was for analytic functions, okay so
00:05:25.330 --> 00:05:30.199
you took instead of taking a family of meromorphic
functions as we have done now, if you are
00:05:30.199 --> 00:05:37.960
taken analytic functions, okay then we would
have put the condition that the family is
00:05:37.960 --> 00:05:41.659
uniformly bounded the family itself is uniformly
bounded, okay.
00:05:41.659 --> 00:05:52.290
And the and there the uniform boundedness
of the family on compact subsets that would
00:05:52.290 --> 00:05:56.340
be equivalent to the family being normally
sequentially compact that is the usual Montel
00:05:56.340 --> 00:06:02.870
Theorem, okay and the way we work there is
you have the uniform if you restrict to a
00:06:02.870 --> 00:06:10.250
compact set you have uniform boundedness of
the family, okay then from the uniform boundedness
00:06:10.250 --> 00:06:15.039
of the family you derive equicontinuity because
from the uniform boundedness of the family
00:06:15.039 --> 00:06:20.650
you get uniform boundedness of the derivatives
and that is because of the fact that the derivatives
00:06:20.650 --> 00:06:24.220
are expressed in terms of the original functions
using the Cauchy integral formula and you
00:06:24.220 --> 00:06:26.590
can make an estimation that are the Cauchy
estimates.
00:06:26.590 --> 00:06:32.840
So uniform boundedness of the family on a
compact subset will give rise to uniform boundedness
00:06:32.840 --> 00:06:37.740
of the derivatives on the compact subsets
and uniform boundedness of the derivatives
00:06:37.740 --> 00:06:45.449
always gives rise to equicontinuity. So you
get along with uniform boundedness on a compact
00:06:45.449 --> 00:06:50.219
subset you get equicontinuity for free if
you are looking at analytic functions, okay.
00:06:50.219 --> 00:06:56.530
But you see Marty’s Theorem is slightly
different what is happening is whereas in
00:06:56.530 --> 00:07:01.659
Montel's Theorem uniform boundedness on compact
subsets of the family is equivalent to the
00:07:01.659 --> 00:07:08.289
family being normally sequentially compact.
In Marty’s Theorem it is not uniform boundedness
00:07:08.289 --> 00:07:14.759
on compact normal uniform boundedness of the
family but it is actually normal uniform boundedness
00:07:14.759 --> 00:07:17.919
of the spherical derivatives, okay.
00:07:17.919 --> 00:07:24.340
So you move from the family in some sense
you move from the boundedness of the family
00:07:24.340 --> 00:07:31.509
to the boundedness of the derivatives that
is the switch, okay and the point is that
00:07:31.509 --> 00:07:37.740
in a sense this is stronger than the original
Montel Theorem because in the original Montel
00:07:37.740 --> 00:07:46.880
Theorem if you know if you are looking at
a family of analytic functions and suppose
00:07:46.880 --> 00:07:53.289
you know that their derivatives are normally
uniformly bounded suppose I am not given that
00:07:53.289 --> 00:07:57.680
the family itself is uniformly bounded but
suppose I am given just the derivatives are
00:07:57.680 --> 00:08:00.439
normally uniformly bounded, okay.
00:08:00.439 --> 00:08:05.340
Then what happens is if the usual derivatives
are normally uniformly bounded then it also
00:08:05.340 --> 00:08:08.710
happens that the spherical derivatives are
normally uniformly bounded because of this
00:08:08.710 --> 00:08:16.890
reason, okay because you see if you the spherical
derivatives are bounded by 2 times they bound
00:08:16.890 --> 00:08:22.150
for the normal derivatives the usual derivatives.
So if the usual derivatives are bounded, okay
00:08:22.150 --> 00:08:24.020
then the spherical derivatives are bounded.
00:08:24.020 --> 00:08:30.009
So if you take a family of analytic functions
on a domain such that the usual derivatives
00:08:30.009 --> 00:08:34.419
are all uniformly bounded on compact subsets
that is normally uniformly bounded then also
00:08:34.419 --> 00:08:41.590
you will get you know a normal sequential
compactness, okay because of Marty’s Theorem
00:08:41.590 --> 00:08:48.090
but the only thing is that now you could have
you know because you are considering these
00:08:48.090 --> 00:08:52.600
as a meromorphic functions you could have
the extreme case that all these analytic functions
00:08:52.600 --> 00:09:01.230
go to infinity, okay and by that I mean they
go to the function which is infinity on all
00:09:01.230 --> 00:09:05.490
points of the domain which is also considered
as a continuous functions, okay and mind you
00:09:05.490 --> 00:09:12.040
for such for that function the spherical derivative
is 0 because it is a constant function, okay.
00:09:12.040 --> 00:09:20.220
So now what I want to say is so this is one
aspect that when you move from Montel's Theorem
00:09:20.220 --> 00:09:25.400
to Marty’s Theorem you are actually moving
from uniform boundedness on compact subsets
00:09:25.400 --> 00:09:32.870
of the family of functions to the uniform
boundedness of the derivatives, okay and because
00:09:32.870 --> 00:09:37.860
you are worrying about meromorphic functions
usual derivatives will not work. For example
00:09:37.860 --> 00:09:41.889
at poles so you will have to look at spherical
derivatives, okay now that is one aspect.
00:09:41.889 --> 00:09:49.300
Now here is another important aspect, see
if you know that these Montel's Theorem for
00:09:49.300 --> 00:09:57.000
example is actually deeper version or it is
an application of the Arzela-Ascoli Theorem,
00:09:57.000 --> 00:10:01.510
okay and what is the philosophy original what
is the philosophy of the original Arzela-Ascoli
00:10:01.510 --> 00:10:07.280
Theorem? The philosophy is that if you want
to say a family of functions is compact which
00:10:07.280 --> 00:10:11.339
is same as saying sequentially compact namely
you want to extract a convergent subsequence
00:10:11.339 --> 00:10:13.339
from any given sequence.
00:10:13.339 --> 00:10:17.579
See you will have to put the conditions of
the family being equicontinuous and uniformly
00:10:17.579 --> 00:10:22.450
bounded that is why the Arzela-Ascoli Theorem
is often referred at as uniform boundedness
00:10:22.450 --> 00:10:30.940
principle, okay. So you need uniform boundedness
plus you need equicontinuity together to give
00:10:30.940 --> 00:10:37.690
you sequential compactness, alright. If you
are working with analytic functions uniform
00:10:37.690 --> 00:10:46.829
boundedness is enough, okay because equicontinuity
will come out as a immediately it will come
00:10:46.829 --> 00:10:53.980
out for free because you have the Cauchy integral
formula, okay.
00:10:53.980 --> 00:11:00.850
Now in the case of Marty’s Theorem there
is a slight advantage the advantage is that
00:11:00.850 --> 00:11:12.500
if you see I have if I look at it in one direction
that is why is it that the uniform boundedness
00:11:12.500 --> 00:11:17.620
normal uniform boundedness of derivatives
should give me a normal sequential compactness,
00:11:17.620 --> 00:11:22.190
okay what you can guess immediately is that
always boundedness of the derivatives gives
00:11:22.190 --> 00:11:27.279
rise to equicontinuity it always give rise
to equicontinuity.
00:11:27.279 --> 00:11:34.050
So even on a compact set if you want to extract
a convergent subsequence from a given sequence,
00:11:34.050 --> 00:11:38.940
okay you would like to apply Arzela-Ascoli
Theorem. So what is missing? What is missing
00:11:38.940 --> 00:11:45.110
is uniform boundedness because if you want
to apply Arzela-Ascoli Theorem you need uniform
00:11:45.110 --> 00:11:50.660
boundedness together with equicontinuity so
that you can extract from any given sequence
00:11:50.660 --> 00:11:52.320
as convergent subsequence.
00:11:52.320 --> 00:11:57.470
So if I restrict to a compact set what I if
I assume that the derivatives spherical derivatives
00:11:57.470 --> 00:12:04.820
are bounded, okay I can expect only equicontinuity,
okay I will not get the I will get equicontinuity
00:12:04.820 --> 00:12:10.279
of the given family of functions but I cannot
get I do not seem to be getting uniform boundedness
00:12:10.279 --> 00:12:15.899
of the family, but here is where the beautiful
thing is you do not need any uniform boundedness,
00:12:15.899 --> 00:12:16.899
okay.
00:12:16.899 --> 00:12:22.930
The reason is because the values are being
taken in a compact metric space, okay see
00:12:22.930 --> 00:12:26.889
the values are being taken as far as meromorphic
functions are concerned where are values being
00:12:26.889 --> 00:12:30.759
taken the values are being taken in the extended
complex plane extended complex plane mind
00:12:30.759 --> 00:12:35.540
you is identified as a Riemann sphere and
is a compact metric space, okay and you know
00:12:35.540 --> 00:12:46.230
a compact metric space is ofcourse bounded
it is totally bounded, it is bounded, okay
00:12:46.230 --> 00:12:47.230
it is complete, okay.
00:12:47.230 --> 00:12:51.360
So there is no unboundedness phenomena that
is going to occur in a compact metric space,
00:12:51.360 --> 00:12:57.300
okay. So this uniform boundedness condition
is not necessary that is the whole point,
00:12:57.300 --> 00:13:05.540
okay. So what I want to say is that your Arzela-Ascoli
Theorem in the Arzela-Ascoli Theorem okay
00:13:05.540 --> 00:13:12.850
we were looking at functions continuous functions
either real or complex valued on a compact
00:13:12.850 --> 00:13:14.440
metric space, okay.
00:13:14.440 --> 00:13:21.209
Now I am saying and there for sequential compactness
of a family of functions you needed both uniform
00:13:21.209 --> 00:13:29.410
boundedness and equicontinuity but if I instead
of looking at real or complex valued functions
00:13:29.410 --> 00:13:36.570
suppose I was looking at functions with values
in a compact metric space, okay that is the
00:13:36.570 --> 00:13:43.269
change I am making you try to look at functions
defined on a compact metric space and taking
00:13:43.269 --> 00:13:48.970
values in another compact metric space the
target is no real numbers or complex numbers
00:13:48.970 --> 00:13:52.470
but the target is another compact metric space.
00:13:52.470 --> 00:13:59.350
Then because the target is already compact
this uniform boundedness is not needed just
00:13:59.350 --> 00:14:06.000
equicontinuity is enough and it is equivalent
to sequential compactness that is the whole
00:14:06.000 --> 00:14:09.589
point that is the whole point, okay so what
I want to tell you is that when you go to
00:14:09.589 --> 00:14:14.589
Marty’s Theorem, okay you switch to the
uniform boundedness of the derivatives and
00:14:14.589 --> 00:14:21.240
you do not care about boundedness of the original
family of functions locally that is because
00:14:21.240 --> 00:14:26.319
how the functions are already taking values
in a compact metric space and you do not have
00:14:26.319 --> 00:14:29.470
to worry about it, okay.
00:14:29.470 --> 00:14:41.920
So let me explain the proof so whatever I
have circle here is to tell you that what
00:14:41.920 --> 00:15:02.750
this tells you is that if the family of derivatives
of a collection of a family of analytic functions
00:15:02.750 --> 00:15:24.079
functions is uniformly bounded then so is
the family of spherical derivatives
00:15:24.079 --> 00:15:28.129
so the boundedness of the ordinary derivatives
implies boundedness of spherical derivatives,
00:15:28.129 --> 00:15:35.670
okay so that is something that I am writing
here I think I have cramped it a little bit
00:15:35.670 --> 00:15:55.070
so let me get rid of this lemma and rewrite
it later, okay fine.
00:15:55.070 --> 00:16:06.639
So what I will do is I will try to give you
the proof of this so let us go in one direction
00:16:06.639 --> 00:16:17.720
so let me again rewrite the Arzela-Ascoli
Theorem is valid, okay in the sense that sequential
00:16:17.720 --> 00:16:22.250
compactness is same as equicontinuity you
do not worry about uniform boundedness, if
00:16:22.250 --> 00:16:27.090
you are looking at functions which are taking
values in continuous functions values in a
00:16:27.090 --> 00:16:31.140
compact metric space, okay if that is if you
replace real and complex numbers by a compact
00:16:31.140 --> 00:16:36.320
integrals that is the whole point so just
equicontinuity is enough, right and I will
00:16:36.320 --> 00:16:41.569
try to instead of trying to prove a theorem
in that generality I will even explain to
00:16:41.569 --> 00:16:44.550
you how you can get sequential compactness
so what you do is.
00:16:44.550 --> 00:16:54.509
So let us start this way suppose so maybe
I will use so suppose F is suppose F hash
00:16:54.509 --> 00:17:06.150
is normally uniformly bounded, okay suppose
it is normally uniformly bounded, what do
00:17:06.150 --> 00:17:10.940
I have to show? I have to show that it is
normally sequentially compact that means you
00:17:10.940 --> 00:17:18.380
will have to pick up given any sequence in
the family script F you have to show that
00:17:18.380 --> 00:17:24.600
there is a convergent subsequence, right convergence
in the sense of normal convergence that is
00:17:24.600 --> 00:17:28.480
convergence on compact subsets so that is
what I have to do, we need to so let me write
00:17:28.480 --> 00:17:29.480
that down.
00:17:29.480 --> 00:17:46.680
We need to show show that any sequence f 1,
f 2 and so on admits this sequence in ofcourse
00:17:46.680 --> 00:17:56.880
in I should not say well when I put subset
this is I am not writing this sequence as
00:17:56.880 --> 00:18:03.210
set, okay because there could be repetitions
in the sequence, okay so this is by this notation
00:18:03.210 --> 00:18:10.280
let me put let me put belongs to okay so this
I mean that f 1, f 2, etc is a sequence in
00:18:10.280 --> 00:18:22.090
F you have to show that any sequence admits
a convergent subsequence, subsequence and
00:18:22.090 --> 00:18:29.220
ofcourse it should be a normally convergent
subsequence that is something that converges
00:18:29.220 --> 00:18:35.150
on compact subsets, okay uniformly on compact
and ofcourse on compact subsets the convergence
00:18:35.150 --> 00:18:43.020
is uniform, alright so uniform convergence.
00:18:43.020 --> 00:18:49.970
So now so how do I go about this? So as usual
the moment usually if you have boundedness
00:18:49.970 --> 00:18:54.920
of the derivatives the first thing that you
do is you get equicontinuity of the family,
00:18:54.920 --> 00:19:00.320
okay that is always always you should remember
as a philosophy boundedness of the derivatives
00:19:00.320 --> 00:19:07.150
is a strong condition that will imply equicontinuity
of the original family. So what you do is
00:19:07.150 --> 00:19:11.330
that so that is what I am going to demonstrate
we will demonstrate that this family script
00:19:11.330 --> 00:19:12.810
F is equicontinuous, okay.
00:19:12.810 --> 00:19:24.800
So we will show script F is equicontinuous,
how do I do that you check equicontinuity
00:19:24.800 --> 00:19:40.340
at every point so what you do is fix z not
at D and disk a closed disk centred at z not
00:19:40.340 --> 00:20:01.370
at z not in D of sufficiently small radius,
okay so now you know so the situation is like
00:20:01.370 --> 00:20:08.050
this you have the you have the complex plane
and you have this you have some you have this
00:20:08.050 --> 00:20:16.220
domain D okay and so this is D, I am always
trying to draw a bounded domain but it will
00:20:16.220 --> 00:20:22.750
not be a bounded domain, okay because it is
an unbounded domain which I cannot show on
00:20:22.750 --> 00:20:24.220
a picture.
00:20:24.220 --> 00:20:32.770
So here is the domain D it is the boundary
is this dotted line and what I am having is
00:20:32.770 --> 00:20:43.020
a point z not in D and I am choosing a sufficiently
small disk such say of radius rho, okay rho
00:20:43.020 --> 00:20:48.390
sufficiently small so that the whole closed
disk is inside D okay the open disk with z
00:20:48.390 --> 00:20:54.340
not centre z not, radius rho along with the
boundary circle that is also in D, okay and
00:20:54.340 --> 00:21:02.940
what do I do I just so I remember that you
know my if you take a function f small f in
00:21:02.940 --> 00:21:08.740
script F mind you the function is being now
as going into the Riemann sphere, okay it
00:21:08.740 --> 00:21:14.920
is going into C union infinity and the C union
infinity is identified so I put a triple line,
00:21:14.920 --> 00:21:24.430
okay this is identified with Riemann sphere
so what is it? It is just so this is just
00:21:24.430 --> 00:21:33.940
S 2 the real two sphere in three space real
three space radius 1 centred at the origin.
00:21:33.940 --> 00:21:43.570
So it is this you know it is this thing so
this is the Riemann sphere and this points
00:21:43.570 --> 00:21:49.110
corresponds to the north pole which corresponds
to so this infinity corresponds to the north
00:21:49.110 --> 00:21:53.980
pole, okay. So here function is taking values
on the Riemann sphere that is how you think
00:21:53.980 --> 00:22:01.460
about it, right and now what is it that I
am given? I am given that I am given that
00:22:01.460 --> 00:22:07.990
the family I am given that the family of spherical
derivatives is normally uniformly bounded
00:22:07.990 --> 00:22:16.240
so that means it is uniformly bounded on compact
subsets of D and this closed disk centred
00:22:16.240 --> 00:22:23.090
at z not, radius rho is a compact subset of
D so it is uniformly bounded on that, okay.
00:22:23.090 --> 00:22:31.160
By hypothesis of normal uniform boundedness
of the family script F there exist an M such
00:22:31.160 --> 00:22:40.030
that the spherical derivatives of all the
spherical derivatives in the family are bounded
00:22:40.030 --> 00:22:55.810
by M so let me just put in mod z minus z not
less than or equal to so I have this, okay
00:22:55.810 --> 00:23:01.560
this is just the uniform boundedness of the
spherical derivatives restricted to this compact
00:23:01.560 --> 00:23:03.310
subset given by this right.
00:23:03.310 --> 00:23:11.180
Now what you do mind you that in this situation
since the functions are taking values in the
00:23:11.180 --> 00:23:15.360
extended complex plane, okay on the target
the target metric space is extended complex
00:23:15.360 --> 00:23:20.200
plane and the target metric is the spherical
metric that is what you have to remember,
00:23:20.200 --> 00:23:24.030
okay the target metric space is the extended
complex plane and on the extended complex
00:23:24.030 --> 00:23:28.570
plane the metric is the spherical metric it
is actually the spherical distance on the
00:23:28.570 --> 00:23:36.740
Riemann sphere transported by the of the Riemann
sphere with the extended plane, okay.
00:23:36.740 --> 00:23:39.950
So you should remember this is the big point
to remember you have to remember that whenever
00:23:39.950 --> 00:23:46.610
you are working with values in the extended
complex plane, okay the in the target space
00:23:46.610 --> 00:23:52.530
the extended complex plane the metric involved
is spherical metric. So if you keep that in
00:23:52.530 --> 00:23:59.220
mind this is what is going to happen. See
if I take two points suppose I take so let
00:23:59.220 --> 00:24:00.790
me use the different colour.
00:24:00.790 --> 00:24:10.120
Suppose I take two points say z 1 and z 2
okay inside this closed disk and I take the
00:24:10.120 --> 00:24:19.420
straight line segment from z 1 to z 2, okay
then and suppose I call this segment as L,
00:24:19.420 --> 00:24:26.750
okay then under if I take the image of the
segment straight line segment under this map
00:24:26.750 --> 00:24:33.860
f, where f is any function any meromorphic
function and the collection script F, okay
00:24:33.860 --> 00:24:37.280
what I am going to get is I am going to get
something on the on the Riemann sphere I am
00:24:37.280 --> 00:24:38.910
going to get something, okay.
00:24:38.910 --> 00:24:47.640
So it is going to be again it is going to
be a contour with starting point f of z 1
00:24:47.640 --> 00:24:54.420
and ending point f of z 2 mind you now f of
z 1 and f of z 2 are being thought of as points
00:24:54.420 --> 00:25:03.910
in the extended plane, okay and the image
contour is going to be just f of L, okay and
00:25:03.910 --> 00:25:13.220
what is the if you now you know you can you
know that from f z 1 to f z 2 on the Riemann
00:25:13.220 --> 00:25:17.900
sphere that is in the extended complex plane
the spherical distance is actually the shortest
00:25:17.900 --> 00:25:23.090
distance on the sphere, it is just the is
it the minor arch of the greater circle passing
00:25:23.090 --> 00:25:26.590
through f of z 1 and f of z 2 on the sphere,
okay.
00:25:26.590 --> 00:25:32.230
And so what you can write is the distance
the spherical distance between f z 1 and f
00:25:32.230 --> 00:25:39.310
z 2 this is certainly is the shortest distance
because it is the geodesic okay curves of
00:25:39.310 --> 00:25:44.110
shortest length on a surface occur geodesic
okay in general if you have a space with a
00:25:44.110 --> 00:25:50.280
metric then the if you give me two points
in the space it is not necessary that the
00:25:50.280 --> 00:25:54.330
straight line distance if it makes sense is
the shortest, okay there could be some other
00:25:54.330 --> 00:26:00.320
curves depending on the metric especially
you could have you could find the distance
00:26:00.320 --> 00:26:04.770
along the curve to be smaller than the straight
line distance in some cases.
00:26:04.770 --> 00:26:09.660
For example for spaces for negative curvature,
okay but in any case if you take a space where
00:26:09.660 --> 00:26:17.770
metric is defined on if you take two points
in the space then the shortest distance the
00:26:17.770 --> 00:26:21.650
curve of shortest distance from this point
to that point on the space is called geodesic
00:26:21.650 --> 00:26:27.210
and that is geodesic distance on the sphere
the geodesics are all given by the minor archs
00:26:27.210 --> 00:26:31.960
of the major surface, okay so that is the
spherical distance and this is certainly this
00:26:31.960 --> 00:26:41.010
is the smallest and so this is certainly less
than the length the spherical length so let
00:26:41.010 --> 00:26:55.400
me now abbreviate it spherical length of f
of L, okay said to be and well what is the
00:26:55.400 --> 00:27:00.510
spherical length of f of L you know that how
to get the formula for the spherical lengths
00:27:00.510 --> 00:27:06.440
the formula for the if you give me a curve
on the plane that is a contour on the plane
00:27:06.440 --> 00:27:13.150
then the length of the contour is just given
by integrating mod d z okay where z is very
00:27:13.150 --> 00:27:17.140
low you integrate mod d z from the initial
point of the contour to the final point of
00:27:17.140 --> 00:27:20.610
the contour you get the length of the arch
or contour on the plane.
00:27:20.610 --> 00:27:25.350
But if you want to get the length of the image
of the arch what you will have to do is you
00:27:25.350 --> 00:27:29.370
have to multiply by the factor which is given
by the spherical derivative, okay if you multiply
00:27:29.370 --> 00:27:34.340
the ordinary derivative and if it is a analytic
function you will get the length of the image
00:27:34.340 --> 00:27:39.110
arch in the complex plane itself, okay that
is if you use modulus of the derivative of
00:27:39.110 --> 00:27:42.420
the analytic function as a scaling factor
but if you use the spherical derivative or
00:27:42.420 --> 00:27:46.550
scaling factor and you will take the spherical
derivative corresponding to meromorphic function
00:27:46.550 --> 00:27:51.170
then you will get the spherical length of
the image of this arch on the Riemann sphere,
00:27:51.170 --> 00:27:52.170
okay.
00:27:52.170 --> 00:28:00.030
So what is this? This is going to be just
integral from z 1 to z 2 of f hash of z mod
00:28:00.030 --> 00:28:05.880
d z this is the spherical derivative, alright
and what will happen is that you see now since
00:28:05.880 --> 00:28:10.950
you know now the point is that this integration
is being carried out from z 1 to z 2 and ofcourse
00:28:10.950 --> 00:28:15.560
this integration is over let me put L here
because this integration is along the straight
00:28:15.560 --> 00:28:21.330
line path from z 1 to z 2, okay and that path
lies inside this closed disk, okay and on
00:28:21.330 --> 00:28:24.790
this closed disk all the spherical derivatives
are all bounded by M.
00:28:24.790 --> 00:28:31.180
So you know mind you spherical length is always
a non-negative quantity, okay it is a non-negative
00:28:31.180 --> 00:28:35.430
real number, okay so what I will get is that
this is this is certainly less than or equal
00:28:35.430 --> 00:28:41.850
to M times mod z 1 minus z 2 this is what
I will get because I can replace this f hash
00:28:41.850 --> 00:28:50.200
of z by M because M is upper bound and the
integral from z 1 to z 2 mod d z is just the
00:28:50.200 --> 00:28:55.610
is just along the straight line segment is
just the length of that segment mod z 1 minus
00:28:55.610 --> 00:28:57.760
z 2, okay so I get this.
00:28:57.760 --> 00:29:03.140
But now what is the advantage what is the
advantage of this now it tells me I have got
00:29:03.140 --> 00:29:14.440
equicontinuity. See so for epsilon greater
than 0, okay if we choose for epsilon greater
00:29:14.440 --> 00:29:28.070
than 0, if we choose delta to be you know
epsilon by M okay then mod z 1 minus z 2 less
00:29:28.070 --> 00:29:35.480
than delta will imply that the spherical distance
between f z 1 and f z 2 is going to be less
00:29:35.480 --> 00:29:42.540
than epsilon I will get this inequality, given
epsilon greater than 0 whenever the distance
00:29:42.540 --> 00:29:46.880
between z 1 and z 2 is less than delta I can
find a delta such that whenever distance between
00:29:46.880 --> 00:29:51.210
z 1, z 2 is less than delta this is the spherical
distance between f z 1 and f z 2 is less than
00:29:51.210 --> 00:29:59.650
epsilon and this works for all f in the family
script F so long as z 1 and z 2 lie in that
00:29:59.650 --> 00:30:00.650
closed disk.
00:30:00.650 --> 00:30:05.450
So what have I got? I have got equicontinuity,
I have got a kind of uniform equicontinuity
00:30:05.450 --> 00:30:11.110
you can think of this as either equicontinuity
at z 1 or thinking at z 2 as a variable or
00:30:11.110 --> 00:30:15.980
you can think of equicontinuity at z 2 thinking
of z 1 as a variable in any case it is a uniform
00:30:15.980 --> 00:30:28.820
equicontinuity, okay. So what I have got is
that f from D f from this disk mod z minus
00:30:28.820 --> 00:30:42.890
z not less than or equal to rho to the extended
complex plane is equicontinuous and this but
00:30:42.890 --> 00:30:48.580
then ofcourse I can cover the source domain
D I can cover every point by such a closed
00:30:48.580 --> 00:30:53.440
disk lying in the domain therefore I have
got equicontinuity at every point so this
00:30:53.440 --> 00:30:59.290
implies that so and this is equicontinuity
f in script F. So basically what I am saying
00:30:59.290 --> 00:31:08.760
is that F is this family script F is equicontinuous
on D so I get equicontinuity, okay.
00:31:08.760 --> 00:31:13.280
So basically what I have done is I have just
shown that boundedness of the spherical derivatives
00:31:13.280 --> 00:31:17.390
gives me equicontinuity and that is a very
general philosophy whenever you have boundedness
00:31:17.390 --> 00:31:23.160
of the derivative you integrate and you get
equicontinuity that is a very general thing,
00:31:23.160 --> 00:31:24.160
alright.
00:31:24.160 --> 00:31:28.660
Now what I have to show what do I have to
show? I have this I started with this I have
00:31:28.660 --> 00:31:34.600
this sequence here in script F okay and I
will have to extract a subsequence which converges
00:31:34.600 --> 00:31:38.040
uniformly on compact subsets that is what
I have to do, what I want to indicate is that
00:31:38.040 --> 00:31:45.400
you can now do it exactly in the way you proved
equicontinuity and uniform boundedness implies
00:31:45.400 --> 00:31:50.851
sequential compactness in one way of the proof
of the Arzela-Ascoli Theorem, okay so what
00:31:50.851 --> 00:31:55.530
you do is you do so these are the steps, okay.
00:31:55.530 --> 00:32:21.240
So what you do is we retrace we retrace the
steps in one way of the proof of the Arzela-Ascoli
00:32:21.240 --> 00:32:39.170
Theorem
to extract a normally convergent so I am using
00:32:39.170 --> 00:32:54.960
cgt for convergent as a an abbreviation subsequence
from the given sequence, okay. So what you
00:32:54.960 --> 00:33:15.380
do so I will put it as a star list so the
first thing is find a countable dense subset
00:33:15.380 --> 00:33:34.890
x 1, x 2, etc so you given given start with
a compact subset K of D given a compact subset
00:33:34.890 --> 00:33:42.260
K of D first find a countable dense subset,
okay here it is just the general statement
00:33:42.260 --> 00:33:45.000
that a compact metric space is separable,
okay.
00:33:45.000 --> 00:33:54.150
Then what you do is now you have now go back
and think about the the proof of the Arzela-Ascoli
00:33:54.150 --> 00:33:59.620
Theorem what you do is that you take the original
sequence you will apply it to x 1, okay and
00:33:59.620 --> 00:34:05.931
you apply it to x 1 you get all these real
or complex numbers, okay and now you will
00:34:05.931 --> 00:34:12.080
use the fact that the original sequence is
uniformly bounded to say that you have a sequence
00:34:12.080 --> 00:34:17.720
of bounded sequence and you will extract a
subsequence, okay any bounded sequence of
00:34:17.720 --> 00:34:24.010
real numbers or complex numbers admits a subsequence
convergent subsequence that is how you use
00:34:24.010 --> 00:34:25.010
it.
00:34:25.010 --> 00:34:31.869
But now you see look at the present situation
if I apply f 1, f 2 if I apply this sequence
00:34:31.869 --> 00:34:40.020
to x 1 mind you let me change just change
the notation to from x 1, x 2 if you want
00:34:40.020 --> 00:34:47.010
to z 1, z 2 because all my points are actually
my compact subset K is actually a point is
00:34:47.010 --> 00:34:51.119
a subset of D and all my points are complex
numbers so let me change it to z 1, z 2 and
00:34:51.119 --> 00:34:58.210
so on, okay. Now what I will do is I will
apply to z 1 I will apply the sequence, okay
00:34:58.210 --> 00:35:04.119
and I will get a convergent subsequence I
will get a convergent subsequence, why is
00:35:04.119 --> 00:35:11.640
that? That is because if I apply these functions
I am going to get a sequence of points on
00:35:11.640 --> 00:35:16.890
the Riemann sphere which is compact therefore
it is sequentially compact therefore every
00:35:16.890 --> 00:35:22.990
sequence gives me a convergent subsequence
you see so it works that is the whole point.
00:35:22.990 --> 00:35:38.839
So apply apply the sequence f 1, f 2 to z
1 what will you get? You get f 1 of z 1, f
00:35:38.839 --> 00:35:53.220
2 of z 1, f 3 of z 1 on the Riemann sphere,
okay but but this is compact it is a compact
00:35:53.220 --> 00:36:08.230
metric space so it is sequentially compact
and because of that what I will get I get
00:36:08.230 --> 00:36:29.609
a subsequence
f 11, f 12, f 13 such that f 1j of z 1 converges,
00:36:29.609 --> 00:36:36.829
okay so this is the key step okay this is
the key point of difference. When we were
00:36:36.829 --> 00:36:41.559
looking at real numbers or complex when we
are looking at real or complex valued functions,
00:36:41.559 --> 00:36:47.890
okay when you apply the sequence to a point
you got a sequence real sequence or a complex
00:36:47.890 --> 00:36:52.829
sequence but then you extracted a subsequence
because you know it is bounded and where from
00:36:52.829 --> 00:36:57.329
did the boundedness come, it came from the
uniform boundedness of the original family,
00:36:57.329 --> 00:36:58.329
okay.
00:36:58.329 --> 00:37:02.220
But now you do not need any uniform boundedness
in this case to extract a subsequence because
00:37:02.220 --> 00:37:07.289
the values are already being taken in the
extended complex plane which is compact and
00:37:07.289 --> 00:37:11.099
is already sequentially compact I do not need
anything more to extract a convergent subsequence
00:37:11.099 --> 00:37:16.009
that is the big difference, okay. Now what
you do is now you iterate, what you do is
00:37:16.009 --> 00:37:28.769
apply to z 2, okay and you get a subsequence
further subsequence
00:37:28.769 --> 00:37:42.109
which is f 21, f 22, f 23 and so on such that
if you take f 2j z 2 this converges, okay
00:37:42.109 --> 00:37:47.980
and you do this ad infinitum what you will
end up with is that you will end up with this
00:37:47.980 --> 00:38:02.309
matrix as usual so you know you will get this
you will get this f matrix of functions 13
00:38:02.309 --> 00:38:14.119
and so on f 21, f 22, f 23 and so on f 31,
f 32, f 33 and so on so it goes on like this
00:38:14.119 --> 00:38:18.619
and you know it is the diagonalization trick
that we used what we do is that we extract
00:38:18.619 --> 00:38:24.700
this diagonal subsequence, okay.
00:38:24.700 --> 00:38:26.990
Then what is the advantage of this diagonal
subsequence? This diagonal subsequence will
00:38:26.990 --> 00:38:32.240
give you a sequence which will converge at
all points of this dense subset this countable
00:38:32.240 --> 00:38:42.150
dense subset of K, okay so f 11 so g 1 is
equal to f 11, g 2 equal to f 22, g 3 equal
00:38:42.150 --> 00:39:05.369
to f 33 and so on is a subsequence that converges
on the countable dense subset z 1, z 2 of
00:39:05.369 --> 00:39:10.009
K, okay alright.
00:39:10.009 --> 00:39:19.510
And now what you do is I will not repeat those
steps now you use we have just now proved
00:39:19.510 --> 00:39:25.569
that all the functions in this family are
equicontinuous, okay we have just now proved
00:39:25.569 --> 00:39:35.380
that. So just use equicontinuity and on this
sequence of functions to hook up to show that
00:39:35.380 --> 00:39:43.079
this sequence is actually Cauchy on the whole
space, okay and therefore it is convergent,
00:39:43.079 --> 00:39:52.630
okay. So the moral of the story is that at
this point you use the equicontinuity of the
00:39:52.630 --> 00:39:59.190
family and mind you that equicontinuity came
from the boundedness of the spherical derivatives
00:39:59.190 --> 00:40:02.480
that is what you have to remember, okay.
00:40:02.480 --> 00:40:23.529
So use the equicontinuity of use the equicontinuity
of this family script F to get to show that
00:40:23.529 --> 00:40:39.430
to show that g 1, g 2, etc is convergent on
all of K, okay. So this is exactly as we did
00:40:39.430 --> 00:40:46.059
in the Arzela-Ascoli Theorem I am not going
to repeat it, okay. Now so what have we succeeded
00:40:46.059 --> 00:40:52.039
using what we have succeeded is given any
compact subset of D given a sequence, given
00:40:52.039 --> 00:41:00.250
any compact subset I am able to extract a
uniformly convergent subsequence, okay.
00:41:00.250 --> 00:41:07.059
But then what do I need? I need one if I change
the compact subset, okay my subsequence could
00:41:07.059 --> 00:41:11.530
change but I want one global subsequence which
works on every compact subset and how do you
00:41:11.530 --> 00:41:17.920
get that you again get by another diagonalization
argument, what you do is you fill up D by
00:41:17.920 --> 00:41:24.509
a sequence of increasing compact sets, okay
with the property that any compact subset
00:41:24.509 --> 00:41:29.420
is contained in one of one set of this sequence,
okay and then use again a diagonalization
00:41:29.420 --> 00:41:34.809
argument as we used in the proof of Montel's
Theorem to extract from this we have global
00:41:34.809 --> 00:41:39.299
subsequence which is going to be convergent
uniformly on every compact subset and that
00:41:39.299 --> 00:41:46.210
finishes the proof one way of proof of Marty’s
Theorem that boundedness of spherical derivatives
00:41:46.210 --> 00:41:53.030
implies if family is normally sequentially
compact, okay boundedness of the derivatives
00:41:53.030 --> 00:41:59.119
on compact subsets, okay so normal boundedness
of derivatives implies normal sequentially
00:41:59.119 --> 00:42:00.119
compactness.
00:42:00.119 --> 00:42:18.890
So let me write that as in the proof proof
of Montel's Theorem fill out D by an increasing
00:42:18.890 --> 00:42:39.049
sequence sequence of compact subsets and use
a diagonalization argument
00:42:39.049 --> 00:43:00.180
to extract a global subsequence that converges
uniformly on every compact subset of D so
00:43:00.180 --> 00:43:04.940
this proves one way, what is the other way
you have to show that if you have a normal
00:43:04.940 --> 00:43:11.180
family you will have to show that it is the
spherical derivatives are bounded and the
00:43:11.180 --> 00:43:16.640
other way is proved by contradiction, if the
spherical derivatives are not bounded, okay
00:43:16.640 --> 00:43:23.300
then I can extract a sequence I can find a
compact set and a sequence of functions and
00:43:23.300 --> 00:43:26.849
a sequence of points at which the spherical
derivatives are becoming bigger and bigger
00:43:26.849 --> 00:43:28.380
and bigger, okay.
00:43:28.380 --> 00:43:33.559
Now from this sequence of functions because
I assume normal sequential compactness I can
00:43:33.559 --> 00:43:41.599
also get a subsequence which converges, okay
if the functions you know if a family of functions
00:43:41.599 --> 00:43:46.240
converge meromorphic function converges to
a limit function then the family of spherical
00:43:46.240 --> 00:43:50.670
derivatives will also converge, okay but mind
you the spherical derivative of any function
00:43:50.670 --> 00:43:56.089
is always a finite quantity spherical derivative
of any function meromorphic function is only
00:43:56.089 --> 00:44:00.980
a finite quantity even if you take the function
which is uniformly infinity derivative is
00:44:00.980 --> 00:44:03.950
0, okay you will only get a finite quantity.
00:44:03.950 --> 00:44:09.809
So if this sequence of functions converges
to a function then the sequence of spherical
00:44:09.809 --> 00:44:14.240
derivatives converges to the spherical derivative
of the limit function and that is a finite
00:44:14.240 --> 00:44:20.599
quantity but on the other hand the original
sequence had points where the values were
00:44:20.599 --> 00:44:24.930
becoming larger and larger so that is a contradiction
so that contradiction will proof that you
00:44:24.930 --> 00:44:30.130
know if you assume that the family is normally
sequentially compact spherical derivatives
00:44:30.130 --> 00:44:34.349
have to be normally uniformly bounded that
is the other way for the proof of Marty’s
00:44:34.349 --> 00:44:38.279
Theorem, okay and with that we we are through
with the proof of Marty’s Theorem, alright.