Alright so so what we did in the last lecture was you know try to explain how the notion
of normal family makes sense for a domain which includes the point at infinity okay
and the technique has that we always follow is that you make a two-piece definition okay
you make one definition for a domain punctured at infinity which will be a domain in the
usual plane and then to take care of a neighbourhood of infinity what you do is that you invert
the variable Z to 1 by w and instead of considering Z equal to infinity you have to consider w
equal to 0, so you take you consider a neighbourhood of 0 okay which is again it is also a domain
in the usual complex plane okay, so therefore you are able to treat the case of normality
at infinity okay.
So with that we saw that both Montelís theorems and Martyís theorem hold good for domain
which probably include which may include the point at infinity okay that is domain in the
extended complex plane and of course I also told you that it is a matter of terminology
that a family being normally sequentially compact that is sequentially compact with
respect to normal convergence that is sometimes abbreviated to simply normal family okay,
so now what we are going to do is you know as I told you we have toÖ We have come very
close to the proof of the Picard theorem which was the main aim of this all these lectures
okay and but before that there is something called there is a theorem which is called
as Zalcman's Lemma which we have to use okay and see this is got to do with characterisation
of non-normality okay, so in order to you know motivate this Zalcman's Lemma okay what
we are going to do is we are going to try to analyse what happens to a normal family
at the point okay.
So so let me make this definition we define normality of a family at a single point to
be normality in a small neighbourhood of that point in an open disk containing that point
okay, so this is just like definition of analyticity, so you say a function is analytic if it is
differentiable not only at a point but in a small neighbourhood of that point, so in
the same way normality is also defined at a point by requiring the property in a small
open disk surrounding that point or an nonempty open set containing that point okay.
Well of course any open set contains that point is nonempty, so the reason for this
definition is that normality is a local property okay. If you have for example a cover of your
space which is countable and if your domain can be covered by countably many open sets
and on each of these open sets if your family is normal then on the whole domain also the
family will be normal because you can useÖBecause of accountability you can use a diagonalization
argument okay, so normality is a local property okay.
So the 1st thing I am going to start with is so I will put the heading as motivationÖlet
me use a different color, motivation for Zalcman's Lemma
which is actually a characterisation of non-normality, so what we will do is we
will make the following definition, let script F be a family of Meromorphic functions
defined in a neighbourhood of a point Z naught okay and of course this point Z naught could
also be the point at infinity but let us assume that Z naught is a point in the plane, the
case of you know how to treat the case at infinity okay, so suppose this is a family
of Meromorphic function defined in a neighbourhood of Z naught that means it is defined in an
open disk containing Z naught okay.
We say the family script F is normal at Z naught if it is normal in a neighbourhood
of Z naught in an open neighbourhood of Z naught okay, so well that is the definition
and so the question is suppose a familyÖ And of course a family is normal on a domain
if it is normal at every point okay. Further family is normal on a domain if and only if
it is normal at each point, so I should not say further in fact this is a remark, so this
is a remark that you can check. It just says that the notion of normality is a local property,
normality is a local property. Fine so here is the question, the question as I now have
a family defined in a neighbourhood of a point okay it is a family of Meromorphic functions
and it is given to me that this family is normal okay and I want to analyse how the
family behaves at that point okay, so the clue to Zalcman's Lemma are at least an understanding
or motivation for Zalcman's Lemma is trying to understand this okay.
So what we will do is let us so I will call this as local analysis of normal families,
so let script F be Meromorphic let script F be a family of Meromorphic functions on
mod Z minus Z naught less than or equal to rho okay, so you say mod Z minus Z naught
less than or equal to rho is actually a closed set, it is a compact set okay that is provided
Z naught is a point in the usual complex plane in Z naught is infinity then you know you
have to rewrite this asÖ You should not use the Euclidean metric what you have to use
the spherical metric okay but I am assuming for simplicity that Z naught is a point in
the plane okay and this is the closed end bounded set I am taking a compact set because
I can choose such rho certainly it is a family of Meromorphic functions at the point Z naught
so it is defined in the neighbourhood of Z naught, so it is defined on a small enough
disk which contains Z naught and if you take a small enough radius then the closed disk
centred at Z naught of that radius will also be in the domain where the Meromorphic functions
is defined where the family is defined.
So I am taking without loss of this closed set because the reason is I want to compact
set so that I can you know I can get uniform convergence, whenever I talk about convergence
you know uniform convergence it happens only on compact sets its normal convergence and
whenever I am talking about uniform boundedness it happens only on compact sets, so everything
I need a compact set that is the reason why I am doing this okay, so what it means is
that a family is defined on bigger open set which contains this closed set okay. Now you
see what we want to do is we want to really so you know let me draw a diagram, so here
is Z naught alright and here is this there is this disk centred at Z naught radius rho.
So this is the this centred at Z naught radius rho and the point is you give me function
small f in the family script F okay and what I want to really do is I want to really analyse
this function in that small disk okay I want to analyse this function, this function in
this small disk and how do I do it? So what I do is you know I have to look closer at
the small disk, so what I actually do is zoom in okay and how do I zoom in? So it is by
scaling okay, so what I do is that I put I define g for every function f in the family
I define this g and what is this G? This g is define with the new variable Zeta and this
is defined as f of Z naught plus Epsilon times Zeta I put this condition I make this
definition, so what is happening is that this is in the Z plane, so the variable here is
Z so this is also the complex plane and then I have another complex plane okay and this
is the Zeta plane.
So the variable here is Zeta alright and what I do is I take this unit disk okay I take
this unit disk, this is unit disk mod Zeta less than 1 okay I will take this disk and
now I am defining this function g as a function of Zeta okay and what is it is actually you
know you have just magnified the behaviour of Ö The value of g at Zeta is the value
of f at Z naught plus Epsilon times Zeta, so what it means is that you know if you start
withÖ And of course I am assuming that Epsilon is less than rho okay, so you know if I take
this disk of radius one centred at the origin with variable Zeta then if I multiply by Epsilon,
if I take Epsilon Zeta it will become disk of radius Epsilon okay, so it will be smaller
disk like this okay and this will have radius Epsilon okay and then if I had Z naught to
it I am just translating it to give me a small disk of radius Epsilon centred at Z naught
that is what this does okay.
So you are just multiplying Epsilon to Zeta then you are adding Z naught so that is a
translation by Z naught okay. So basically what you are doing is that when you do this
you know you are actually looking at this you are looking at this small disk centred
at Z naught radius Epsilon and so you know in principle for what values of Zeta is g
defined, so you will see that g is defined forÖshould be . See if mod Zeta is less than
rho by Epsilon okay then mod Epsilon Zeta will be less than rho and mod Z naught plus
Epsilon Zeta will be less than therefore also less than rho okay.
See mod Zeta is less than rho by Epsilon means that you know mod Epsilon Zeta is less than
rho okay and what does this tell you? This tells you that the distance of Epsilon Zeta
from the origin is at most rho and therefore if you add Z naught to Epsilon Zeta the distance
of Z naught plus Epsilon Zeta to Z naught is at most rho, so you see this g is defined
on this disk and you see this disk is actuallyÖsee this disk is larger see because you see rho
the original disk centred at Z naught has radius rho alright. Now the radius has become
rho by Epsilon and Epsilon being smaller rho by Epsilon should be larger if you think of
Epsilon to be you know Epsilon is less than rho, so rho by Epsilon is greater than 1 okay.
So what you have done is by this transformation this disk centred at Z naught radius rho okay
if you want to study the function value the values of the function f there the behaviour
of the function f there, what you have done is you have actually zoomed it by constructing
this function G, so g is a kind of zooming function okay it is like you have a microscope
or telescope and you have 10 X zoom 20 X zoom you see that also in camera these days okay
you have 5X zoom and so on, so this is so many X zoom and the zooming factor is 1 by
Epsilon okay. Original disk of radius rho has now become zoom to disk of radius rho
by Epsilon okay and well the point is thatÖ And of course rho by Epsilon will be slightly
bigger okay you will get so this is Epsilon, this thing will be rho by Epsilon and it is
in this bigger disk open disk that the function g is defined and studying this function g
in this bigger disk is the same as studying the functions f in the original disk okay.
So you know I will write it like this I will write g of Zeta is equal to zoom I will think
of it as zooming, zoom the function f centred at Z naught by a magnification factor 1 by
Epsilon and use the variable Zeta okay I will use this notation okay, so when I say zoom
f Z naught 1 by Epsilon Zeta you are zooming the function f centred at Z naught, so your
zoom is centred at Z naught okay and 1 by Epsilon is the magnification factor and Zeta
is the variable of the zoom function okay. Now wellÖ So this is how you zoom into the
behaviour of function at a point okay, now I am given that this function, I am given
that this family script F is normal suppose I am given it is normal at Z naught suppose
I am given normal in the closed disk centred at Z naught radius rho which means that it
is actually normal in an open set which contains that closed disk okay. Suppose that happens
and let us see what it means for us.
So let me continue like this suppose now that you know script F is normal in mod Z minus
Z naught less than or equal to rho okay. By that I mean it is normal in a slightly bigger
open disk okay. Then so what does normality means, it means actually normally sequentially
compact is the correct notion of compactness for as when we do complex analysis okay, so
what it means is that you give me any sequence of functions in script F I can always find
a subsequence which converges normally okay it is normal sequential compactness, it is
sequential compactness with respect to normal convergence. So given any sequence f n in
script F we can find a subsequence let me call that as f n k that converges normally
in mod Z minus Z naught less than rho okay I can put less than or equal to okay.
In fact mod Z minus Z naught less than or equal to rho is actually compact therefore
it will converge even uniformly there okay. So it converges normally so what happens is
that you will get this subsequence of f n k of Z it will go to some function f Z a and
f is a normal limit of Meromorphic functions so you know what is going to happen you have
body seen this a normal limit of Meromorphic function is either identically infinity or
it is also a Meromorphic function it could even be analytic okay, so then of course then
as we have seen either f is identically infinity or f is Meromorphic this is something that
we have seen but the point I want to emphasise is that is the following issue, you see you
have these functions f n k they are converging to f alright and what is happening is that
now you know mind you I have the zoomed function okay I have these zoomed functions by zooming
factor Epsilon is than rho.
So look at the zoom functions, so what happens is that I take g N k which is actually the
zooming ofÖso g n k of Zeta is the zooming of f n k centred at Z naught with the magnification
factor 1 by Epsilon and I am using the variable Zeta and similarly I get zoomed function for
the limit function also, so I get g also is equal to zoomed function of f centred at Z
naught magnification factor 1 by Epsilon my variable is Zeta okay, so this is g of Zeta
so I have that and of course you know if f n k converges to f then the zoomed function
of the f n k namely the g n k will converge to the zoomed function of f which is g and
this is just because the zooming is just the scaling followed by a translation okay.
So after all what is the zooming? The zooming is you multiply, you multiply Zeta by Epsilon
and then you add Z naught to it okay that is just a in fact it is actually a bilinear
transformation it is a linear fractional transformation, it is a Mobius transformation okay so it will
preserve all properties convergence everything okay. So you get this, now you see so let
me write this, this is normally so here also I will get so this will imply that this will
also converge normally okay and the point I want to make is that you see I have assumed
that the original family is normal okay and we have seen namely Martyís theorem that
normality is equivalent to spherical derivatives being normally bounded okay.
So if you recall you know Martyís theorem says that a family of Meromorphic functions
on a domain, the domain could even be a domain in the extended complex plane it could contain
the point at infinity. Such a family is normally if and only if the spherical derivatives of
those functions, the family of spherical derivatives is normally uniformly bounded so it should
be on every compact subset of the domain the family of spherical derivatives should have
uniform bound okay. So by Martyís theorem you know that the family script F itself has
if you take the spherical derivatives than that has uniform bound.
So let me write that down by Martyís theorem script F has uniform bound, script F hash,
so when I put script F upper hash means the family of spherical derivatives of the functions
in f okay. This has a uniform bound in mod Z minus Z naught less than or equal to rho
I have this okay because this mod Z minus Z naught less than or equal to rho is a compact
set alright and on a compact set I have a uniform bound for spherical derivatives and
that is equivalent to normality okay this is something that I have this is Martyís
theorem.
So what does it mean? It means that if you take any function f so let me not
use f let me use h because I have already use f for the limit, if you take a h in f
then h hash this is the spherical derivatives of h than m for all h and same m right, so
you have this you have uniform bound alright and this bound applies to all members of spherical
derivatives of all members of the family f script F so it applies also to the f n case
okay, so in particular what you will have is you know, you will have that f n k hash
the spherical derivatives you see these things are going to be bounded by m it is going to
happen because all these f n case are anyway in the family okay but you know we have also
seen this earlier if a family of Meromorphic functions converges normally to limit function
then the spherical derivatives will also converge okay.
Taking the spherical derivative will respect convergence okay, so you also have from here
if you want, so I need to go like this that f N k if I take the spherical derivatives
that will go to f hash okay you have this okay and of course you know if I by the same
token if I do it to the g n k the spherical derivatives of g n k will go to the spherical
derivatives of g okay mind you g n k is just f n k translated and scaled, so when you translate
and scale a function its nature does not change, so if you take a Meromorphic function and
translate and scale it you will again get a Meromorphic function okay.
After all translating and scaling is not going to change the nature of singularities okay
and if you translate and scale an analytic function you will again get an analytic function
okay and so on and so forth, so g n k are also Meromorphic functions and mind you you
have to remember that I am considering the f n k in mod Z minus Z naught less than or
equal to rho but I am actually considering the g n k in mod Zeta less than rho by Epsilon
okay that is the zoomed disk centred at the origin where the zoom functions are being
looked at, so let me write this here on mod Zeta less than rho by Epsilon okay this is
the Epsilon less than rho alright so this is a quantity greater than 1, fine.
So I have this and I shall have the same thing for these guys I will also have g n k hash
going to g hash alright and of course it is again normal
this also normal okay and the point is that but there is an inequality coming out, see
what is g n k? g n k of Zeta is by definition g n k of Z naught is Epsilon Zeta this is
the zoomed function, so what is g n k hash of Zeta, what is it? Say it is by definition
it is supposed to be 2 times the modulus of the derivative of g n k with respect to Zeta
divided by 1 plus modulus of g n k of Zeta the whole square, this is the definition of
spherical derivative okay and but then this is what is the g n k dash Zeta in the numerator?
It is the derivative of g n k with respect to Zeta it is derivative with respect to Zeta
okay and mind you this formula for spherical derivative also works when Zeta is a pole
okay we have seen that you know by continuity if the pole is of higher-order then the spherical
derivative at the pole is 0, if the pole is a simple pole then spherical derivatives at
the pole is actually 2 divided by modulus of the residue of the pole which is something
that we have seen already okay, so there is no problem if Zeta is a pole okay this formula
works but then you know if I differentiate g n k with respect to Zeta it is the same
as differentiating f n k and then you know by the chain rule differentiating the argument
of the f n k with respect to Zeta and that will bring me a multiple of Epsilon.
So what I will get is, I will get Epsilon times f n k hash of Z this is what I will
get okay not Z in fact I have to plug-in the right substitution scale to variable Z naught
plus Epsilon so this is what I will get okay but the f n k are all bounded by what? The
f n k are bounded by m alright so this is bounded by Epsilon m okay so the moral of
the story is that because you because of your scaling okay. See what you must understand
is that you zoomed the small disk centered at Z naught radius rho to a larger disk okay
you scaled and you got the zoom function but what has happened is that the bound for the
spherical derivative have become smaller because it has got multiplied by this the inverse
of the zooming factor. Zooming factors 1 by Epsilon where Epsilon is very small okay then
bound for the zoom function becomes much more smaller okay.
Now that is what it says, now this is true for every m K but you know g n k hash converges
normally to g hash and each of the g n k is bounded by Epsilon m therefore g hash will
also be bounded by Epsilon M, so these 2 put together will tell you that g hash will also
be bounded by Epsilon m okay so this will happen alright, so that is because of normal
convergence okay. So the moral of the story is that so this is what is happening, so what
have we shown so far? You take a point Z naught a point where a family is normal okay and
then you take any sequence in the family you will get a convergence of sequence, normally
convergence of sequence you look at those functions in the subsequence okay and zoomed
them.
Then the zoomed functions their spherical derivatives becomes very small and in fact
the zoom functions if you take the limit of the zoom functions, the limit function will
be a Meromorphic function whose spherical derivatives is very small that is what it
says okay and now you know, so this is the first step of the argument. Now what I am
going to do is I am going to introduce a level of complication by doing the following thing,
what you do is instead of considering a single Epsilon you consider sequence of Epsilon going
to 0. Imagine you are considering I am using the same Epsilon here for all the functions
okay but suppose for g n I use an Epsilon n okay and such that the Epsilon n are going
to go to 0 okay that means I am doing ultra-zooming, I am zooming into smaller and smaller and
smaller and smaller neighbourhood of Z naught okay.
If I do that then what will happen is that you can guess what is going to happen? This
g hash will be 0 because see g hash will the thing on the left side g n k hash that
is going to be bounded by Epsilon n k m okay and if I take limit as N k tends to 0 Epsilon
NK is going to go to 0, so Epsilon NK m is going to go to 0 because m is anyway finite
quantity therefore g hash is going to go to 0 and what does it mean if g hash goes to
0? It means that g is a constant. A spherical derivative of function cannot be 0 unless
it is a constant.
It can be even be constant function that is uniformly infinity mind you for the constant
function which is uniformly infinity also the spherical derivatives is 0 okay in line
with the fact that the philosophy that whenever you have a constant function the derivative
of any type should be okay. So the moral of the story is that if you zoom in a family
of convergent, family of functions at a point okay then the limit function is going to be
constant that is the full point okay, so let me add that level of complication but now
there is something very nice.
What is happening is that so let me put this in a different color here whatever is happening
happens in mod Zeta less than rho by Epsilon okay this was the disk where everything is
happening okay but now you know if I make these Epsilon goes to 0 by choosing Epsilon
N which go to 0 something nice is going to happen as Epsilon N go to 0 rho by Epsilon
N go to infinity, so your disks are becoming bigger and bigger and bigger and the beautiful
thing is that any compact subset of the complex plane, the Zeta plane will be contained in
a sufficiently large disk and therefore you get a family of functions for which you can
talk about normal convergence on the whole plane okay, so this g will be defined
on the whole plane and you will get a constant function okay that is the whole point, so
let me write this down.
Now let us consider a sequence of Epsilon say Epsilon n going to 0, so Epsilon going
to 0 plus so they are all positive and you know let me say even I mean say even decreasing
okay anyway sequence going to 0 eventually it is going to be a decreasing sequence alright
strictly decreasing sequence you can think of it as smaller and smaller neighbourhoods
then what happens is that you we get for you know so here is the funny thing you define
g m all Zeta to be you zoom f so you take f you write Z naught plus Epsilon n Zeta,
this is actually the zooming of the function f centred at Z naught scaling factor is now
1 by Epsilon n and the variable I am using a Zeta okay and mind you this 1 by Epsilon
n is going to go to infinity because Epsilon n is going to 0 plus 1 by Epsilon n is going
to infinity that means I am actually I am doing ultra-zooming you know at the point
Z naught and I am looking at the zoom functions alright.
Now see this is defined as I told you this is defined in mod Zeta less than rho by Epsilon
n and the point is that this goes to infinity this goes to plus infinity as N tends to infinity
because Epsilon N goes to 0 rho is a fixed quantity alright and note that what will happen
is that g n k hash will go to g hash as before okay this will be normal but the beautiful
thing is this will be normally on C on the whole of C that is the beautiful part. Earlier
this g n k hash going to g hash was on only on this bounded domain it was on mod Zeta
less than rho by Epsilon, it was normal on the convergence g n k hash going to g hash
wasÖit was normal convergence on this bounded domain but now since this Epsilons are becoming
smaller the g n are you being defined on bigger and bigger domains okay and that is an increasing
sequence of open disk centred at the origin that will eventually cover the whole plane.
So it will eventually cover any compact subsets of the plane and therefore on any compact
subset of the plane I can say that this sequence is going to converge normally because I will
have to consider the sequence only beyond a certain stage okay. See whenever you are
looking at convergence the 1st finitely no matter how large, the 1st finitely many terms
do not matter okay, so the moral of the story is that the advantage of taking smaller and
smaller Epsilons is that you are zooming into the point, you are zooming into the behaviour
of the functions at that point but then you get a limit function which is defined on the
whole plane, so you get a Meromorphic function on the plane, so this g it will be a Meromorphic
function on the whole plane okay g will be a Meromorphic function on the whole plane
and the nice thing is that as I was telling you the bound for the spherical derivatives
of g n will be what?
It will be Epsilon n k times m, so you will get g n k hash is bounded by Epsilon n k times
m okay because you know earlier we got the bound as Epsilon m for g n k okay but now
the Epsilon is Epsilon n k in this case so I will get this okay and this goes to g hash
and as k tends to infinity and well this was to 0 as k tends to infinity. So the moral
of the story is g hash 0 and this implies that g is a constant
and when I say g is a constant mind you g could be the constant function which is identically
infinity that is also allowed okay. So you know how do you argue that g is a constant
mind you g becomes GÖ anyway the limit function g is Meromorphic on the whole plane okay.
So it has only poles on the plane okay and outside the poles it is analytic function
okay but outside the poles the spherical derivatives is 0 means ordinary derivatives is 0 okay
and if ordinary derivative is 0 the function is constant, so except for these isolated
set of points which are poles everywhere else the function is constant and those points
cannot be there you cannot have on a single pole because if you have a pole as the points
approach the pole the function goes, the modulus of the function value goes to infinity. How
can it be constant in a deleted neighbourhood of a pole? So that forces the function has
to be identically constant that is how you get g is identically constant g is only a
Meromorphic function okay. So this is the important thing, the important thing is that
if you take a normal family, so what have we prove?
You take a point where a family is normal okay and you take any sequence in the normal
family you can find a normally convergence subsequence. If you take those sequence of
functions and do zoom in at the point you are able to constructÖsee the zoom functions
will converge to a constant function that is what it says okay. If you look at normally
convergent family of Meromorphic function zoomed at a point you will get only a constant
function limit that is what you are saying okay this is the behaviour of a normal family
at a point okay and what I want to do next is introduce another level of complication
okay and this next level of complication is to vary Z naught also okay.
So instead of considering a point Z naught all this time I had fixed Z naught but now
what you do is you take a sequence point Z N which goes to Z naught okay and at each
Z n you zoom to Epsilon N or the function f n and do this process okay and you will
still get the same result okay and the beautiful thing is that whatever you get there okay
that behaviour is good enough to characterise normal families okay and the point about Zalcman's
Lemma is the negativity of that statement okay, so roughly Zalcman's Lemma is a Lemma
that is actually a theorem which will explain when family is not normal it will give you
a sufficient conditions for a family to be not normal and guess what the condition will
be, the condition will be at you can do all this but the limit function that you get g
it will be non-constant that will be the only difference.
You will get a limit function which will be a non-constant Meromorphic function, so its
spherical derivatives will not be 0 okay and that is the characterisation of non-normality
okay. See this is the motivation for Zalcman's Lemma okay, so in my next lecture I will you
how to instead of considering Z naught consider a sequence ZN which goes to Z naught and have
the same argument okay and continue with Zalcman's Lemma alright, so I will stop here.