**Title:** On the isometric immersion of pseudo-spherical surfaces via evolution and hyperbolic equations

**Speaker:** Nabil Kahouadji

**Speaker Info:** Northwestern University

**Brief Description:**

**Special Note**:

**Abstract:**

On one hand, the class of PDEs describing pseudo-spherical surfaces, which has been defined and studied in depth in a foundational paper by Chern and Tenenblat, contains a large subclass of equations enjoying remarkable integrability properties, such as the existence of infinite hierarchies of conservation laws, Bäcklund transformations and associated linear problems, and have been completely classified. On the other hand, a classical theorem in the theory of surfaces states that any pseudo-spherical surface can be locally isometrically immersed into a three-dimensional Euclidean space. This theorem is however largely an existence result, which does not give an explicit expression for the second fundamental form of the local isometric immersion.I will report on a recent joint work with Niky Kamran and Keti Tenenblat on the classification of the equations describing pseudo-spherical surfaces, for which the components of the second fundamental form of the local isometric immersion depend on a jet of finite order of u.

These results will show among other things that, when viewed through the prism of the local isometric immersions associated to solutions, one equation occupies a special position within the class of equations describing pseudo-spherical surfaces.

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