Seminar on Geometry

Schedule of talks

Abstracts/Notlar

Berger spheres are introduced by Marcel Berger. These are odd dimensional spheres, which are squashed in a direction. The shrinking direction can be provided by the circle action on the complex Euclidean space. In other words, it corresponds to shrinking or expanding the Hopf circles in an odd dimensonal sphere. Consequently, the resulting Riemannian metric is different than the round metric. In these seminar series, we understand the eigenvalues of the Laplacian on Berger spheres.

Ingredients of the individual seminars are as follows:

BS-1: Introduction, metrics on the 2-sphere.

BS-2: Berger sphere metrics.

BS-3: Spectrum of the complex projective space.

BS-4: Jacobi Fields on Spheres.

BS-5: Orthogonal, graded decomposition of the Berger Eigenspace.

BS-6: Nontriviality of Eigenspaces.

BS-7: Dimension counting for Eigenspaces.

BS-8: Dimension counting continued.

BS-9: First eigenvalue of the Laplacian on Berger Spheres.

BS-10:

1. Tanno, Shûkichi. The topology of contact Riemannian manifolds.
   Illinois J. Math. 12 (1968), 700–717.


2. Tanno, Shûkichi. The first eigenvalue of the Laplacian on spheres.
   Tohoku Math. J. (2) 31 (1979), no. 2, 179–185.

In this learning seminar series, we will give an introduction to minimal submanifolds of the higher dimensional spheres. In particular we give estimates on the index of the Jacobi operator. Talk about applications on the Plateau's problem and Bernstein conjecture.

Ingredients of the individual seminars are as follows:

B-1: Index of Spheres as totally geodesic minimal submanifolds.

B-2: Jacobi fields on totally geodesic minimal spheres in higher dimensional spheres.

B-3: Negative definite index form.

B-4: Killing Fields on the sphere.

B-5: Using Killing Fields on the sphere to find nullity.

B-6: An extrinsic rigidity theorem.

B-7: Rigidity theorem for higher codimension.

B-8: Rigidity theorem for higher codimension 2.

B-9: Sphere rigidity theorem.

B-10: Proof of the rigidity theorem.

B-11: Reformulation of the rigiditty theorem.

B-12: Laplacian of the second fundamental form on a sphere.

B-13: Laplacian of the second fundamental form on a hypersurface of a sphere.

B-14: Estimates on the shape operator.

B-14: Scalar curvature rigidity.

1. James Simons. - Minimal varieties in riemannian manifolds.
   Ann. of Math. (2), 88:62–105, 1968.

The Plateau problem can be stated as follows: Given an (n−1)-manifold(surface) as a boundary in an (n+k)-manifold, find an n-surface that is bounded by that boundary and has minimal area. The problem was first posed by Lagrange in 1760, and named after the Belgian Physicist Joseph Plateau, who studied soap films and observed several laws of their geometric properties.

Depending on the conditions we impose on the boundary and enclosing surface, the ambient manifold M, the codimension k and the interpretation of ”bounded by Γ”, we have variants of the Plateau problem. In this talk we are mainly focused on the oriented codimension one Plateau problem: Given a closed oriented immersed (n−1)-surface in the Euclidean (n+1)-space, find a oriented bounding surface which has minimal area among other candidates. To better understand the bounding condition, consider the following example. Take two parallel circles in R3 that are closed to each other. The oriented solution will be a catenoid if the two components are equipped with different orientations, and two disks if the two components are given the same orientation. Also, we know from the example that the oriented solution may not be minimizing among all surfaces that span.

To solve the Plateau problem, one wants to take a minimizing sequence of surfaces Σi, and hope that Σi converges to some minimal surface Σ. However, in general we do not have convergence as the area bound is not strong enough to control the surface. In the same spirit as the weak solution of a PDE, we want to find a space of ”weak manifolds” in which a notion of ”mass” is defined, and has nice functional analysis properties:

1. The space has good compactness property, so for a mass-minimizing sequence Σi, we can find a convergent subsequence.

2. The mass functional is lower semicontinuous, so the limit is minimizing.

3. The ”weak solution” generated above is actually regularity, thus a ”classical solution”.

In 1960, Federer and Fleming came up with a very powerful setting, called integral currents, which is suitable for the discussion of the oriented Plateau problem.

We will be using the following resources.

References:

1. Jenny Harrison and Harrison Pugh. - Plateau’s Problem: What’s Next.
   In: arXiv:1509.03797v2 2016.


2. Leon Simon. - Geometric Measure Theory.
   Stanford Univ. lecture notes.


3. James Simons. - Minimal varieties in riemannian manifolds.
   Ann. of Math. (2), 88:62–105, 1968.

Craig : An extremal Kähler metric is a canonical Kähler metric, introduced by E. Calabi, which is somewhat more general than a constant scalar curvature Kähler metric. The existence of such a metric is an ongoing research subject and expected to be equivalent to some form of geometric stability of the underlying polarized complex manifold (M,J,[ω]) –the Yau-Tian-Donaldson conjecture. Thus it is no surprise that there is a moment map, the scalar curvature (A. Fujiki, S. Donaldson), and the problem can be described as an infinite dimensional version of the familiar finite dimensional G.I.T.

I will describe how the moment map can be used to describe the local space of extremal metrics on a symplectic manifold. Essentially, the local picture can be reduced to finite dimensional G.I.T. In particular, we can construct a course moduli space of extremal Kähler metrics with a fixed polarization [ω]∈H2(M,ℝ) , which is an Hausdorff complex analytic space

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Activities are supported by University of Bonn and Boğaziçi University