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This is the Past Exam Paper of Mathematical Tripos which includes Solitons and Instantons, Smooth Function, Scalar Field Theory, Derrick Scaling Arguments, Bogomolny Equations, Topological Degree, Sigma Model Lumps etc. Key important points are: Riemann Surfaces and Discrete Groups, Proper Subdomain, Complex Plane, Principal Branch, Convergent Power Series, Holomorphic Function, Sequence of Points, Hyperbolic Metric, Mobius Transformations
Typology: Exams
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Thursday 7 June 2007 1.30 to 4.
Attempt FOUR questions.
There are SIX questions in total.
The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
1 Let D be a proper subdomain of the complex plane. For each point zo ∈ D set
δ(zo) = inf{|zo − w| : w ∈ C \ D}.
Show that there is a point wo ∈ C \ D with δ(zo) = |zo − wo|.
The function f (z) =
z − zo z − wo
is holomorphic. The principal branch of its logarithm is given by the convergent power series
log f (z) =
k=
k
zo − wo z − wo
)k
on the set {z : |z − wo| > δ(zo)}. Does it converge uniformly on this set? Does it converge locally uniformly?
For any o > 0, show that there is a natural number K for which the function
Eo(z) =
z − zo z − wo
exp
k=
k
zo − wo z − wo
)k]
is holomorphic on D, has a simple zero at zo and satisfies
| log Eo(z)| < o for |z − wo| > 2 δ(zo).
Deduce that, for any sequence (zn) of distinct points in D that satisfies δ(zn) → 0 as n → ∞, there is a holomorphic function f : D → C that has simple zeros at each zn and no other zeros.
Explain why this implies that every meromorphic function on D can be written as the quotient of two holomorphic functions on D.
2 Let u : D → R be a continuous function on the closed unit disc D which is harmonic on the open unit disc. Show that the value of u at any point z ∈ D is given by a Poisson integral of the boundary values u(w) for w ∈ ∂D.
A continuous function u : Ω → R on a domain Ω ⊂ C has the mean value property if, for each z ∈ Ω there exists r(z) > 0 with {w : |w − z| < r(z)} ⊂ Ω and
u(z) =
∫ (^2) π
0
u(z + reiθ^ )
dθ 2 π
for 0 < r < r(z). Prove that, if such a function has a local maximum at z ∈ Ω, then it is constant on a neighbourhood of z. Prove that u has the mean value property if, and only if, u is harmonic.
Paper 11
5 Define a Perron family of continuous subharmonic functions on a Riemann surface R. Prove that the supremum of such a Perron family is either +∞ on all of R or else a harmonic function on R. Give examples to show that both cases arise.
Let u : D → R be continuous and subharmonic on the unit disc D. Show that the least harmonic majorant of u is given by
r→lim 1 −
∫ (^2) π
0
u(reiθ^ )
r^2 − |z|^2 |z − reiθ^ |^2
dθ 2 π
= sup r< 1
∫ (^2) π
0
u(reiθ^ )
r^2 − |z|^2 |z − reiθ^ |^2
dθ 2 π
6 What does it mean to say that a simply-connected Riemann surface is hyperbolic. Give an example of such a surface and prove that it is indeed hyperbolic.
Write an essay describing the proof that a simply-connected, hyperbolic, Riemann surface is conformally equivalent to the unit disc.
Paper 11
