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.

We study the Riemann tensor of Einstein spaces. Also the Jacobi operator for a minimal and totally geodesic submanifold and the find its relationship to the Ricci curvature. We will be using the following resources.

References:

1. Kalafat, Kelekçi and Taşdemir. Minimal submanifolds and stability in Einstein manifolds by .
   E-print available at arXiv:2406.03347 math.DG, 2024.

We study the mean curvature and second fundamental form of a subspace of a Riemannian manifold, and understand its behaviour under conformally rescaling of the Riemannian metric.

We discuss irreducible decompositions of the curvature tensor in manifolds with special holonomy. The metrics considered are Kähler, Einstein, Calabi-Yau, Hyperkähler, Quaternionic Kähler and other special holonomy metrics. We will be using the following resources. We will be using the following resources.

References:

1. Salamon, Simon. Riemannian geometry and holonomy groups.
   Pitman Res. Notes Math. Ser., 201 Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.

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.

In this learning seminar series, we will give an introduction to minimal submanifolds.

Ingredients of the individual seminars are as follows:

M-1: Minimal Surface equation and Calibrations

M-2: We derive the minimal surface equation for a graph and give an example of a Calibration.

M-3: We will talk on the relationship between minimal surfaces and harmonic functions.

M-4: We will see how meromorphic functions can be used to produce minimal surfaces. Weierstrass Representation Theorem.

M-5: Concrete examples will be given as application.

1. Colding, Minicozzi. - A course in minimal surfaces.
   GSM 121. American Mathematical Society. AMS. Providence, RI. 313 p. 2011.

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 curvature and the Jacobi operator of Berger spheres embedded in complex projective spaces. We will be using the following resources.

References:

1. Kalafat, Kelekçi and Taşdemir. Minimal submanifolds and stability in Einstein manifolds by .
   E-print available at arXiv:2406.03347 math.DG, 2024.

Alexei : Critical metrics for eigenvalues in Riemannian Geometry and minimal / harmonic maps:

Eigenvalues of a differential operator defined using a metric (Laplace operator, Dirac operator etc) are functionals on the space of Riemannian metrics. Investigation of critical metrics for these functionals is a natural approach in study of maximal / minimal metrics. It turns out that a relation between critical metrics and minimal / harmonic maps is a very powerful tool.

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