# use Sard's theorem to prove Maximum Modulus Principle

Discussion in 'General Math' started by grape, Nov 12, 2004.

1. ### grapeGuest

I think one can use Sard's theorem to prove the maximum modulus Principle
in complex analysis, here is the brief skeleton:

1. let a U be region and f takes on "a" in int(U) as a maximum Modulus
point, thus it's locally vanishing to a zero-messurable point, which by
sard's theorem, the det f'(a) vanishes too, then if f is a
diffeomorphism(analytic functions are diffeomorphisms), then at point a,
it will vanish to a 1 or 0-manifold on C(det f'(a) =0, not an onto
diffeomorphism that transfers to an 2-manifold), but any conformal mapping
has non-vanishable det, (coz it's a 2-manifold on C, for every conformal
mapping coz it's angle preserving).

thus, conformal maps won't satisfy the maximum Modulus condition; now the
only analytic functions left are constant functions, thus only constant
functions satisfy the maximum Modulus condition.

it's my brief proof sketch. Hope you spot any errors.

THANK YOU

grape, Nov 12, 2004

2. ### The World Wide WadeGuest

You don't need Sard's Theorem for this.
No, analytic functions are not always diffeomorphisms. You think z -> z^2
is a diffeormorphism of the open unit disc to itself?
There are too many errors to catch them all.

The World Wide Wade, Nov 13, 2004

3. ### grapeGuest

a zero-measurable point, otherwise maximum can not take place.
consider it as F(z)= |f(z)| the detF'(a)=0 |f(a)| is zero-measurable by
Sard's theorem; it implies that det f'(a)=0 thus f'(a) is a singular
matrix. thus it's up to 0 or 1 diffeomorphism.

not bad if I use it. actually sard's theorem give assertion that detF'=0
when it takes on maximum, then it must be a 1 or 0 manifold.
ok, I shouldn't say it's a diffeomorphism, but rather, it's up to
diffeomorphism, and exist a diffeomorphism F that F(z)=z'^2, and F
compose of f1:z->unit disc, f2:unit disc ->unit disc by z'^2.
then F is a diffeomorphism.
the diffeomorphism, is irrelavent here, I just want to show it's a
vanished zero-measurable manifold.
hope you know understand what's the relation between manifolds and
diffeomorphism.

all my point is that the det vanishes at the maxmum point, but conformal
mappings have no such vanishing det. thus only constant functions satisfy
it.
say them ,or elsewise, keep quiet.

grape, Nov 13, 2004