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IUM, Spectral Geometry
Exam. 07.12.2013.
This is a take­home exam. The deadline for solutions is the lecture on December, 21.
The point­to­grade transfer for IUM students is the following: #50 points = 5, #40
points = 4, #30 points = 3.
The point­to­grade transfer for HSE students is the following: #50 points = 10, #47
points = 9, #44 points = 8, #40 points = 7, #37 points = 6, #33 points = 5, #30 points
= 4.
Problem 1. Let e 1 , . . . , e n be a local orthonormal basis in vector fields in a
neighbourhood of a point p on a manifold M, and c 1 (t), . . . , c n (t) be geodesics such
that c i (0) = p and c # i (0) = e i . Prove that #f(p) = -
n
#
i=1
d 2
dt 2 f(c i (t))| t=0 . (5 points).
Problem 2. We know that the Dirichlet spectrum of a domain has monotonicity
property, i.e.
if# 1
## # R n , then #
i(# 1 ) # #
i(# .
Prove that the Neumann spectrum of a domain does not have monotonicity
property, i.e. provide examples where the spectrum increases and decreases when
one goes from a domain to a subdomain (10 points).
Problem 3. Find upper and lower bounds for Dirichlet eigenvalues of the
domain ABCDEF, where A = (0, 0), B = (0, 2), C = (1, 2), D = (1, 1), E = (2, 1),
F = (2, 0). (up to 10 points depending on found bounds).
Problem 4. We proved Faber­Krahn theorem saying that the disc minimizes the
first Dirichlet eigenvalue #
1(# among all planar domains of the same area. Prove
that the disjoint union of two discs of the same radius minimizes #
2(# among all
planar domains of the same area. (10 points).
Problem 5. Prove that the disjoint union of n discs of the same radius cannot
minimize Dirichlet eigenvalues
#n(# for all n (5 points).
Problem 6. Prove that the spectrum of the square of side length 1 with the
Dirichlet condition on three sides and the Neumann condition on the remaining
side coincides with the spectrum of the rectangular triangle of catheti length # 2
with the Dirichlet conditions on the hypotenuse and one cathetus and the Neumann
condition on another cathetus (10 points).
Problem 7. Construct an isometric immersion of the cli#ord torus in spheres
using eigenfunctions of # corresponding to # 1 . Prove that the metric g Cl on the
Cli#ord torus is extremal for the functional # 1 (T 2 , g). Find the value # 1 (T 2 , g Cl )
(5 points).
Problem 8. Construct suitable isometric immersions of the cli#ord torus in a
sphere using eigenfunctions of # and prove that the metric g Cl on the Cli#ord torus
is extremal for infinite set of functionals # j (T 2 , g). Find at least three such values
of j (10 points).
Problem 9 # . Using the same approach as in problem 7 prove that the metric g eq
on the equilateral torus is extremal for the functional # 1 (T 2 , g). Find # 1 (T 2 , g eq ).
Prove that the metric g Cl on the Cli#ord torus is not maximal for the functional
# 1 (T 2 , g) (20 points).
Problem 10 # . Find the minimal isometric immersion of the real projective
plane RP 2 equipped with the canonical metric into a sphere using eigenfunctions
correponding to the first eigenvalue. Prove that this is one of Veronese maps.
(25 points).
Problem 11. Using your solution of problem 10 find an upper bound for
# 1 (RP 2 , g) for any metric g. (10 points).