Resum - holomorphic/harmonic forms
e
on compact Riemann surfaces
Notation.
M = a compact Riemann surface of genus g
with canonical homology basis cfw_a1 , , ag , b1 , , bg
if 1 j g
if g + 1 j 2g
j =
aj
bj g
H=
L2 (M ) harmonic
H=
L2 (M ) holomorphic
Th
Elliptic Curves
Elliptic curves have equations of the form w 2 = z 3 + az + b. For concreteness, we
look at
E=
(z, w ) C2
w2 = z 3 z
Let
:
EC
:
(z, w ) z
EC
(z, w ) w
be the projections from E onto the z and w axes, respectively. We shall often rewrite z
Resum - closed/exact/holomorphic/harmonic forms
e
Denition.
a) A 1form is (co)closed if is C 1 and d() = 0.
b) A 1form is (co)exact if = ()dF for some C 2 function F on M .
Proposition. Let be a C 1 1form.
a) If is (co)exact, then is (co)closed.
b) is (co
An Elliptic Function The Weierstrass Function
Denition W.1 An elliptic function f (z ) is a non constant meromorphic function on C
that is doubly periodic. That is, there are two nonzero complex numbers 1 , 2 whose ratio
is not real, such that f (z + 1 )
Elliptic Regularity
Let be an open subset of IRd . A measurable, locally square integrable function
is said to be a weak solution of Laplaces equation in if
(r) (r) dd r = 0
C0
for all
functions that are supported in . The theorem that any weak solution
Integration on Manifolds
This is intended as a lightning fast introduction to integration on manifolds. For a more
thorough, but still elementary discussion see
B. ONeill, Elementary Dierential Geometry, Chapter 4.
W. Rudin, Principles of Mathematical Ana
The RiemannRoch Theorem
Well, a Riemann surface is a certain kind of Hausdorf space. You know what a Hausdorf space is,
dont you? Its also compact, ok. I guess it is also a manifold. Surely you know what a manifold
is. Now let me tell you one non-trivial
Operators on Dierential Forms
In the following
cfw_U, is a coordinate patch for a Riemann surface M ,
f (x, y ), g (x, y ), u(x, u), v (x, y ) are functions on (U ) IR2
F is a 0-form on M ,
is a 1-form on M with
cfw_U, = f (x, y ) dx + g (x, y ) dy