Geometric Analysis Learning Seminar

from Tuesday
January 01, 2019 to Friday
May 31, 2019
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Friday
February 01, 2019

4:00 PM - 6:00 PM
P-131 Math Tower
TBA, Stony Brook University
Organizational meeting + TBA


Friday
February 08, 2019

4:00 PM - 6:00 PM
P-131 Math Tower
Lisandra Vazquez, Stony Brook University
Isometric Embeddings and the Implicit Function Theorem

The main goal of this talk is to present the proof of the Nash Embedding Theorem which states that every Riemannian manifold is realizable as a submanifold of some Euclidean space by a smooth isometric embedding. The problem of isometric embeddings has been around since the introduction of the concept of an abstract manifold. Yet as conceptually fundamental as Nash’s result is, it is in the proof that Nash gives us one of the most important techniques in geometric analysis, known today as the Nash-Moser iteration or hard implicit function theorem. Thus we hope to give a brief history of the problem of isometric embeddings, from the local results of Cartan and Janet to the theorem of Nash and prepare the stage for the proof of Nash’s theorem by detailing the Nash-Moser iteration and its applications to solving nonlinear PDE’s.

References:

- Nash, J. “The Imbedding problem for Riemannian Manifolds”, Annals Math., Vol 63, No. 1, 1956
- Han, Q., Hong, J-X. “Isometric Embedding of Riemannian Manifolds in Euclidean Spaces”
- Krantz, S. , Parks, H. , “The Implicit Function Theorem”


Friday
February 22, 2019

4:00 PM - 6:00 PM
P-131 Math Tower
Demetre Kazaras, Stony Brook University
Bray's proof of the Penrose Conjecture


Friday
March 01, 2019

4:00 PM - 6:00 PM
P-131 Math Tower
Vardan Oganesyan, Stony Brook University
Second variation and stability of minimal Lagrangian submanifolds in Kahler manifolds

We say that an immersed minimal submanifold is stable if the second variation of the volume is nonnegative under all deformations. We will prove two Theorems.

1) Suppose that M is a Kahler manifold with non-positive first Chern class. Then any minimal Lagrangian submanifold is stable.

2) Assume that M is Kahler-Einstein with Einstein constant k. Then an immersed Lagrangian L is stable under Hamiltonian deformations if and only if the first eigenvalue of the Laplacian of L is greater than or equal to k.

These theorems were proved by Oh and Le in 1990.


Friday
March 08, 2019

4:00 PM - 6:00 PM
P-131 Math Tower
Saman Habibi Esfahani, Stony Brook University
On Existence, Regularity and Singularity of Harmonic Maps

We study an existence and regularity theory for Harmonic maps between Riemanian manifolds. We review an existence theorem by Eells and Sampson, for compact target manifolds with nonpositive curvature. Then we see from a theorem by Schoen and Uhlenbeck that a bounded energy minimizing map is regular except for a closed set S of Hausdorff dimension at most n-3.

References:

A regularity theory for harmonic maps by Schoen and Uhlenbeck

Theorems on Regularity and Singularity of Energy minimizing maps by Leon Simon


Friday
March 15, 2019

4:00 PM - 6:00 PM
P-131 Math Tower
Jingchen Hu, Shanghai Tech University
Existence of Einstein–Hermitian metrics in stable holomorphic vector bundles

In algebraic geometry, there is a notion of a (semi)stable bundle over a projective variety. On the other hand, in differential geometry, there is a notion of Einstein–Hermitian vector bundles over the complex manifold. It turns out these ideas coincide.

In this talk, show the existence of Einstein-Hermitian metrics on stable holomorphic vector bundle over a Kahler manifold.

Reference: K.Uhlenbeck, S.T.Yau On the existence of hermitian‐yang‐mills connections in stable vector bundles


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