Academics

Differential Geometry Seminar | Recovering contact forms from boundary data

Time:2023-10-31, Tuesday 21:00 - 22:00

Venue:Zoom Meeting ID: 271 534 5558 Passcode: YMSC Zoom link: https://us06web.zoom.us/j/2715345558?pwd=eXRTTExpOVg4ODFYellsNXZVVlZvQT09

Organizer:Jialong Deng, Akito Futaki

Speaker:Gabriel Katz MIT

Abstract:

Let $X$ be a compact connected smooth manifold with boundary. The paper deals with contact $1$-forms $\beta$ on $X$, whose Reeb vector fields $v_\b$ admit Lyapunov functions $f$. We prove that any odd-dimensional $X$ admits such a contact form.

We tackle the question: how to recover $X$ and $\beta$ from the appropriate data along the boundary $\partial X$? We describe such boundary data and prove that they allow for a reconstruction of the pair $(X, \beta)$, up to a diffeomorphism of $X$. We use the term ``holography" for the reconstruction. We say that objects or structures inside $X$ are {\it holographic}, if they can be reconstructed from their $v_\b$-flow induced ``shadows" on the boundary $\partial X$.

For a given $\beta$, we study the contact vector fields $u$ on $X$ that preserve it invariant. Integrating $u$, we get a $1$-parameter family of contactomorphisms $\{\Phi^t(u)\}_{t \in \mathbb R}$ which maps Reeb trajectories to Reeb trajectories. This leads to estimates from below of the number of $\Phi^t(u)$-fixed Reeb trajectories.

We also introduce numerical invariants that measure how ``wrinkled" the boundary $\partial X$ is with respect to the $v_\beta$-flow and study their holographic properties under the contact forms preserving embeddings of equidimensional contact manifolds with boundary. We get some ``non-squeezing results" about such contact embedding, which are reminiscent of Gromov's non-squeezing theorem in symplectic geometry.

DATEOctober 31, 2023
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