# Global Analysis Seminar 2019

Temple-Rutgers Global Analysis Seminar

Current contacts: Gerardo Mendoza (Temple) and Siqi Fu and Howard Jacobowitz (Rutgers)

The seminar takes place Friday 3:00 - 3:50 pm in Wachman 527 for talks at Temple, or 319 Cooper, Rm 110, for talks at Rutgers-Camden. Click on title for abstract.

• Friday May 3, 2019 at 10:30, Rutgers-Camden, Business and Sciences Building (BSB) 108

Wanke Yin, Wuhan University and Rutgers University

Let $M$ be a smooth real hypersurface in $\mathbb{C}^n$ with $n\geq 2$. For any $p\in M$ and any integer $s\in [1,n-1]$, Bloom in 1981 defined the following three kinds of integral invariants: invariant $a^{(s)}(M,p)$ defined in terms of contact order by complex submanifolds, invariant $t^{(s)}(M,p)$ defined by the iterated Lie bracket of vector fields and invariant $c^{(s)}(M,p)$ defined through the degeneracy of the trace of the Levi form. When $M$ is pseudoconvex, Bloom conjectured that these three invariants are equal. Bloom and Graham gave a complete solution of the conjecture for $s=n-1$. Bloom showed that the conjecture is true for $a^{(1)}(M,p)=c^{(1)}(M,p)$ when $n=3$. In this talk, I will present a recent joint work with Xiaojun Huang, in which we gave a solution of the conjecture for $s=n-2$. In particular, this gave a complete solution of the Bloom conjecture for $n=3$.

• Friday May 3, 2019 at 11:30, Rutgers-Camden, Business and Sciences Building (BSB) 108

Yuan Yuan, Syracuse University

I will talk about joint work with Gang Liu on the following rigidity problem. If the bisectional curvature of a Kähler manifold is bounded below by 1, and the diameter equals the diameter of the standard complex projective space, is the manifold isometric to the complex projective space?

• Friday May 3, 2019 at 14:00, Rutgers-Camden, Business and Sciences Building (BSB) 108

Howard Jacobowitz, Rutgers University

In 1998 the physicist Andrzej Trautman conjectured:

A three-dimensional CR manifold is locally realizable $\iff$ its canonical bundle admits a closed nowhere zero section.

A real three dimensional hypersurface in the two-dimensional complex space $\mathbb{C}^2$ is said to be a realizable CR manifold. The definition of a CR structure on an abstract three-dimensional manifold is essentially due to E. Cartan. A natural question is then: When can an abstract CR manifold be realized by a real hypersurface in $\mathbb{C}^2$.

In this talk we review the relevant mathematical definitions. Then we outline the construction due to Robinson and Trautman of the abstract CR manifold associated to an optical geometry, that is, to a shear-free congruence of null geodesics in a Lorentzian 4-space. A special case of the conjecture, already known in a different context, will be proved.

• Friday May 3, 2019 at 15:00, Rutgers-Camden, Business and Sciences Building (BSB) 108

Weixia Zhu, Rutgers University

We study how the spectrum of the $\bar\partial$-Neumann Laplacian behaves as the underlying domains are perturbed and establish several upper semi-continuity properties for the variational eigenvalues of the $\bar\partial$-Neumann Laplacian. In particular, we establish spectral stability of the $\bar\partial$-Neumann Laplacian on smoothly bounded pseudoconvex domains of finite type in $\mathbb{C}^n$. This talk is based on joint work with Siqi Fu.