Geometry & Analysis Seminar Fall 2017

I will introduce asymptotically hyperbolic Einstein manifolds and the impact to the study of conformal geometry. Then I will talk on our recent work on asymptotically hyperbolic Einstein manifolds based on the so-called AdS/CFT correspondence. I will present a proof for a sharp volume comparison theorem for asymptotically hyperbolic Einstein manifolds, which will imply not only the rigidity theorem for hyperbolic space in general dimension but also curvature estimates for asymptotically hyperbolic Einstein manifolds. In particular, as a consequence of our curvature estimates, one now knows that the asymptotically hyperbolic Einstein metrics with conformal infinities of sufficiently large Yamabe constant have to be negatively curved.

Daniel Cristofaro-Gardiner, University of California, Santa Cruz

Two or Infinity

I will present recent joint work showing that for any nondegenerate contact form on a closed three-manifold, the associated Reeb vector field has either two or infinitely many distinct embedded closed orbits as long as the associated contact structure has torsion Chern class. A key ingredient in the proof is an identity relating the lengths of certain sets of Reeb orbits to the volume of the three-manifold.

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Thursday, November 9, 2017

Longzhi Lin, University of California, Santa Cruz

Energy Convexity of Intrinsic Bi-harmonic Map and Its Heat Flow

The theory of harmonic map and its higher dimensional analogues (e.g. bi-harmonic map) has been a classic and intensely researched field in PDE and geometric analysis. In this talk, we will discuss an energy convexity for intrinsic bi-harmonic map and its heat flow with small intrinsic bi-energy from the 4-disc to spheres. This in particular yields the uniqueness of intrinsic bi-harmonic maps on the 4-disc with small bi-energy and the uniform convergence of the intrinsic bi-harmonic map heat flow on the 4-disc with small initial bi-energy. The energy convexity for harmonic maps with small energy was proved earlier by Colding-Minicozzi (c.f. Lamm-Lin) and the uniform convergence of the harmonic map heat flow with small initial energy was proved earlier by myself. This is a recent joint work with Paul Laurain.

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Thursday, November 16, 2017

Hanming Zhou, University of California, Santa Barbara

In this talk we consider the motion of a classical colored spinless particle under the influence of an external Yang-Mills potential $A$ on a compact manifold with boundary of dimension $\geq 3$. We show that under suitable convexity assumptions, we can recover the potential $A$, up to gauge transformations, from the lens data of the system, namely, scattering data plus travel times between boundary points. The talk is based on joint work with Gabriel Paternain and Gunther Uhlmann.

In the process of gravitational collapse, singularities may form, which are either covered by trapped surfaces (black holes) or visible to faraway observers (naked singularities). In this talk, with three different approaches coming from hyperbolic PDE, quasilinear elliptic PDE and dynamical system, I will present results on four physical questions: i) Can “black holes” form dynamically in the vacuum? ii) To form a “black hole”, what is the least size of initial data? iii) Can we find the boundary of a “black hole” region? Can we show that a “black hole region” is emerging from a point? iv) For Einstein vacuum equations, could singularities other than black hole type form in gravitational collapse?

François Monard, University of California, Santa Cruz

Introduction to Scattering, Continued

Following the introductory talk I gave on 10/20/2017 at the Quantum Mechanics and Geometry Seminar earlier this quarter, on scattering and scattering poles, I will continue on introducing the main players of scattering problems, focusing on potential scattering on the real line. In the last talk, we proved the meromorphic continuation of the resolvent of 'Laplace + potential' across its continuous spectrum, and the existence of scattering poles, and we sketched their impact on long-term asymptotics of solutions to a wave equation.

Time permitting, this time I will cover:
- the absence of 'embedded eigenvalues' (=no eigenvalues inside the continuous spectrum)
- distorted plane waves and the generalization of the Fourier transform
- the scattering matrix and the wave operators
- inverse scattering