Mathematics Department - Complex Analysis and Geometry Seminar - Spring 2017

# Complex Analysis and Geometry Seminar - Spring 2017

### Organizer(s)

Xiaojun Huang, Jian Song, Sagun Chanillo, Hanlong Fang

## Friday, April 28th

### "On holomorphic isometries from the Poincar'e disk into bounded symmetric domains of rank at least two"

Time: 10:30 AM
Location: Hill 705
Abstract: In general, it is difficult to classify all holomorphic isometries from the unit disk into bounded symmetric domains​ with respect to their Bergman metrics up to normalizing constants, especially when the target is of rank at least three. Thus, in this talk we will first consider holomorphic isometries from the Poincar'e disk into the product of the unit disk and the complex unit n-ball for n at least two with respect to certain canonical K"ahler metrics and obtain a complete characterization of all such holomorphic isometries. On the other hand, I will also talk about how our study would provide new examples of holomorphic isometries from the Poincar'e disk into irreducible bounded symmetric domains of rank at least 2 except for type-IV domains.

This is a joint work with Yuan Yuan.

## Friday, April 21st

### "Rigidity theorems for invariant harmonic functions on bounded symmetric domains"

Time: 10:30 AM
Location: Hill 705
Abstract: Let $Delta_g$ be the Laplace-Beltrami operator in Bergman metric in the unit ball in $C^n$. Then the boundary value problem:

$$Delta_g u=0, hbox{ in } B_n; quad u=phi hbox{ on }d B_n$$

has a unique solution

$$u(z)=P[phi]=int_{d B_n} {(1-|z|^2)^n over |1-langle z, w angle|^{2n}} phi(w) dsigma(w)$$

It well known that even if $phiin C^infty(d B_n)$, $P[phi]$ may not be in $C^n(overline{B_n})$. A well known theorem of R. Graham says that if $u$ is invariant harmonic in $B_n$ and $C^n(overline{B_n})$, then $u$ must be pluriharmonic in $B_n$.

In this talk, I will present a joint work with R-Y. Chen, we try to generalize Graham's theorem to the bounded symmetric domains.

## Friday, March 31st

### "Hyperbolicity of some moduli spaces of polarized manifolds"

Time: 2:00 PM
Location: Hill 705
Abstract: In this talk, we would explain some joint work with Wing-Keung To on hyperbolicity of moduli spaces of polarized algebraic manifolds, including families or moduli spaces of Kaehler-Einstein manifolds of negative scalar curvature or trivial scalar curvature, and some log-general type manifolds for quasi-projective versions. Classically, results on moduli space of Riemann surfaces of genus at least two have been obtained by Ahlfors, Royden and Wolpert with Weil-Petersson metric. Study of moduli of higher dimensional manifolds in terms of Weil-Petersson metrics began with a work of Siu thirty years ago. Eventually, we construct a negatively curved Finsler metric on such moduli from which results in hyperbolicity follow naturally. We also apply the related techniques to study questions related to a conjecture of Viehweg.

## Friday, March 24th

Special Complex Analysis and Geometry Seminar

### "Sparse domination of singular integral operators"

Time: 10:30 AM
Location: Hill 705
Abstract: It is discovered recently by Lacey (and refined by Lerner) that Calder'on-Zygmund (CZ) operators, which are a priori non-local, can be dominated pointwisely by a class of local, positive, sparse averaging operators. This in particular implies sharp weighted norm inequalities for CZ operators. In a series of joint works with A. Culiuc and F. Di Plinio, we show that sparse domination can be obtained in contexts well beyond CZ theory, such as for rough homogeneous singular integrals, Bochner-Riesz multipliers, and even modulation invariant multilinear singular integrals including the bilinear Hilbert transforms. Many new sharp weighted estimates for these operators then follow immediately.

## Friday, February 17th

### "Metric spaces of non-positive curvature and applications in several complex variables"

Time: 10:30 AM
Location: Hill 705
Abstract: In this talk I will discuss how to use ideas from the theory of metric spaces of non-positive curvature to understand the behavior of holomorphic maps between bounded domains in complex Euclidean space. Every bounded domain has an metric, called the Kobayashi metric, which is distance non-increasing with respect to holomorphic maps. Moreover, this metric often satisfies well-known non-positive curvature type conditions (for instance, Gromov hyperbolicity or visibility) and one can then use these conditions to understand the behavior of holomorphic maps.

Some of what I will talk about is joint work with Gautam.

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