Semesterarbeit, Bachelorarbeit, and Diplom Topics / Reading Courses
Tom Ilmanen
I have listed some possible topics. The list is only a
selection -- in fact, I welcome your suggestions. Please feel
free to contact me at the address listed at the bottom.
General study areas (clickable if there is detail):
An Einstein manifold is a Riemannian manifold (M,gij)
such that the Ricci curvature is a constant multiple of the metric, namely
Rij=gij.
Two canonical sources:
[1] Besse, Arthur L, Einstein manifolds, Ergebnisse
der Mathematik und ihrer Grenzgebiete 10,
Springer, 1987.
[2] Joyce, Dominic, Compact manifolds with special holonomy,
Oxford Math. Monographs, Oxford University Press, 2000.
[1] P. Griffiths and J. Harris, Principles of algebraic geometry,
Wiley Classics Library, John Wiley & Sons, 1994.
Also see the appropriate sections in:
[2] A. L. Besse, Einstein manifolds, Ergebnisse
der Mathematik und ihrer Grenzgebiete 10, Springer, 1987.
[3] D. Joyce, Compact manifolds with special holonomy,
Oxford Mathematical Monographs, Oxford University Press, 2000.
For the all-important Yau estimates, see the corresponding chapter in
[4] G. Tian, Canonical metrics in Kähler geometry,
ETH Lectures in Math., Birkhäuser, 2000.
This would make a great semesterarbeit. Here are four beautiful
sources:
[1] J. R. Weeks, The shape of space, Marcel Dekker, 2002.
[2] P. Scott, "The geometries of 3-manifolds", Bull. London Math. Soc.
15 (1983) 401--487.
[3] W. P. Thurston, Three-dimensional geometry and topoolgy, vol 1,
Princeton Univ. Press, 1997.
[4] J. Milnor, "Curvatures of left invariant metrics on Lie groups",
Advances in Math. 21 (1976) 293--329.
A Lagrangian submanifold L is an n-dimensional
real submanifold of Cn such that for each x
in L,
TxL ⊥ J(TxL),
where J represents multiplication by i and TxL is the
tangent plane.
Form a complex n x n matrix from the components of a real basis of
TxL and take the determinant to yield a function
eiθ(x) on L called the Lagrangian angle.
Particularly interesting is the case when θ is constant -- then
L is called special Lagrangian. Under this condition, L is also
a minimal submanifold -- that is, L has vanishing mean
curvature vector H at point in L. (H(x) is perpendicular to L at
each point and can be described as the Laplacian of L written as a
graph over its tangent space at x.) The converse is also true if L is
connected.
More generally, special Lagrangian submanifolds may be defined in
Calabi-Yau manifolds (Ricci-flat Kähler manifolds), where they
have received a lot of attention recently because of their connection to
mirror symmetry.
If L is Lagrangian, we can try
to make it special Lagrangian by moving it in the direction given
by the vector field H.
A family Lt, t≥0, of submanifolds
is moving by mean curvature if
L_t satisfies the evolution equation
∂x/∂t=H,
for all points x∈ Lt and times t≥0.
Remarkably, the mean curvature flow preserves the property of being
Lagrangian. So if Lt ever converges to something static,
we have found a special Lagrangian submanifold. A major difficulty is that
Lt will typically develop singularities along the way...
There are many beautiful formulas for Lagrangian mean curvature flow
that yield rich geometric information about the formation of
singularities. And there are lovely analogies between Lagrangian mean
curvature flow and the Kähler-Ricci flow.
Several directions present themselves for study in a semesterarbeit
or diplomarbeit:
1) Singularities of Hamiltonian and Lagrangian stationary submanifolds.
See [1,2].
3) Self-similar solutions and singular Lagrangian mean curvature flow. See
[3,4] and references therein.
4) Analogies between Lagrangian mean curvature flow and
Kähler-Ricci flow). See references in [4] for LMCF and references
in [5] for KRF.
For some general background in Lagrangian minimal surfaces, see [6].
[1] R. Schoen and J. Wolfson, "Minimizing area among Lagrangian surfaces:
the mapping problem," J. Diff. Geom. 58 (2001) 1--86.
[2] J. Wolfson, "Lagrangian homology classes without regular
minimizers," J. Diff. Geom. 71 (2005) 307--313.
[3] A. Neves, "Singularities of Lagrangian mean curvature flow: monotone case",
http://arxiv.org/pdf/math.DG/0608401.
[4] A. Neves, "Singularities of Lagrangian mean curvature flow: zero-Maslov class case, http://arxiv.org/pdf/math.DG/0608399.
[5] M. Feldman, T. Ilmanen and D. Knopf, "Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons,"
http://www.math.ethz.ch/~ilmanen/papers/kaehler.ps.
[6] J.-M. Morvan, "Minimal Lagrangian submanifolds: a survey,"
in Geometry and topology of submanifolds III (Leeds, 1990),
World Sci. Publ., River Edge, NJ, 1991, 206--226.
A Ricci-flat manifold is a Riemannian manifold (M,g) whose Ricci
tensor vanishes:
Rij=0.
This is the Euler-Lagrange equation of the Einstein-Hilbert
functional
E(g):=∫ Rg(x) dμg(x),
n≥2,
where Rg is the scalar curvature of g.
Ricci-flat manifolds are Riemannian analogs of vacuum space-times;
certain ones play a central role in string theory; and
they are static points of
Hamilton's Ricci flow for metrics on a manifold.
Compact Ricci-flat manifolds are fairly hard to come by, and their
properties are of great interest. Several kinds of Ricci flat
manifolds are known to exist: Kähler manifolds with vanishing first
Chern class (called Calabi-Yau manifolds), with holonomy group SU(n);
hyperkähler manifolds, with holonomy Sp(n); and manifolds with
exceptional holonomy groups G2 or Spin(7). See [1], [2] for
extensive information.
All known examples have special holonomy and carry a parallel spinor.
Recently Dai, Wang, and Wei [3] have proven that
any compact Ricci-flat manifold
that carries a parallel spinor is stable in the sense that
the Lichnerowicz Laplacian on symmetric 2-tensors, defined by
ΔLhik:=Δhik
+2Rijklhjl-Rijhjk
-Rkjhji
is nonpositive, that is, all its eigenvalues are nonpositive.
This implies that the metric is a stable fixed point
of the Ricci flow, at least at the linear level. See [5] for an
equivalent notion of stability involving the Perelman functional.
One task is to read paper [3] and explain it.
This is a fairly demanding semesterarbeit/bachelorarbeit because
you will need to understand spin bundles and spin connections,
more or less in advance; see [4]. It is also suitable as
a diplomarbeit, with wider foray into background and methods,
and contact with open questions.
[1] A. L. Besse, Einstein manifolds, Ergebnisse
der Mathematik und ihrer Grenzgebiete 10,
Springer-Verlag, 1987.
[2] D. Joyce, Compact manifolds with special holonomy,
Oxford Mathematical Monographs, Oxford University Press, 2000.
[3] X. Dai, X. Wang, and G. Wei,
"On the stability of Riemannian manifold with parallel spinors,"
Invent. Math. 161 (2005) 151--176.
[4] H. B. Lawson and M.-L. Michelsohn, Spin geometry,
Princeton Math. Series 38, Princeton Univ. Press, 1989.
[5] H. D. Cao, R. Hamilton, and T. Ilmanen,
"Gaussian densities and stability for some Ricci solitons,"
http://arxiv.org/pdf/math.DG/0404165
Let γt, t ≥ 0, be a family of embedded curves in
R2. We say that γt is evolving by
curvature if each point on the curve moves orthogonal to the curve
with speed equal to the curvature k of the curve, namely
∂x/∂t= k ν,
for all points x∈ γt and times t≥0, where
ν is a normal vector to the curve. For example, a circle shrinks
homothetically to a point.
Curvature flow is a kind of "heat equation" for curves,
and tends to smooth the curve out, but also permits singularity
formation, as can been seen from the shrinking circle. Matt Grayson
proved that any simple closed curve, no matter how wild,
possesses an evolution that remains smooth and remains a simple closed curve
until it shrinks to a "round" point.
(See [].)
Now suppose that instead of a curve we have a so-called network
-- a union of curved segments that meet only at their endpoints. They
generally meet in groups of three at an angle of 120
degrees: a so-called triple junction. Such a network might
model the interface between different crystal grains in an annealing
metal, for instance.
Unusual things can happen during the evolution: for example,
two triple junctions can plow into each other, changing the topology
of the curve. Some classical examples may be found in the Appendix of [1].
The question: prove the mathematical existence of the curvature
evolution of a given initial network, and study its properties --
in particular the nature of singularities. Are there finitely many
singularities in spacetime?
Recent progress on the problem appears in [2].
Suitable for a diplom problem.
[1] K. Brakke, The motion of a surface by its mean
curvature, Princeton Univ. Press, 1978.
[2] C. Mantegazza, M. Novaga and V. M. Tortorelli, "Motion
by curvature of planar networks, Ann. Sc. Norm. Super. Pisa Cl. Sci.
(3), at http://arxiv.org/archive/math under math.AP/0302164.
This makes an excellent reading course (or a semesterarbeit)
whenever it is not on the regular program of courses. Canonical sources:
[1] F. Morgan, Geometric measure theory: A beginner's guide,
Academic Press, 2000.
[2] L. Simon, Lectures in geometric measure theory, 1984.
The following are good background reading:
[3] L. C. Evans and R. F. Gariepy, Measure theory and fine properties of
functions, Studies in Adv. Mathematics, CRC Press, 1992.
[4] K. Falconer, Fractal geometry: Mathematical foundations and
applications, John Wiley & Sons, 2003. (Either of the predecessor
volumes is also good.)
[1] Hsiang, W.-Y., Lectures on Lie groups,
Series on University Mathematics #2, World Scientific Publishing Co.,
2000.
[2] Fulton, William and Harris, Joe, Representation theory:
A first course, Graduate Texts in Mathematics #129,
Readings in Mathematics, Springer-Verlag, 1991.
[1] Morita, Shigeyuki, Geometry of differential forms,
Translations of Math. Monographs, #201,
Iwanami Series in Modern Math., American Math. Soc., 2001.