Workshop: Several complex variables and CR geometry (November 3 - 13, 2015)

After the foundation of several complex variables by K. Oka and H. Cartan, the $L^2$ method was developed to obtain effective results in complex analysis and geometry. An extension theorem with $L^2$ growth condition was obtained in this context. It turned out to be useful in complex analysis and geometry. Recently, based on an argument of Q. Guan and X.-Y. Zhou, J. Cao proved in a general context that an $L^2$ extension theorem implies the semipositivity of the direct image of relative canonical bundles. Vice versa, B. Berndtsson and L. Lempert showed that such a curvature property implies an $L^2$ extension theorem in an optimal form. Since this progress is closely related to the solution of a long-standing conjecture of N. Suita by Z. Błocki, Guan and Zhou, it might be a good occasion to give an overview of this development starting from the background of the questions. Several new questions will be raised, too.

We introduce a class of domains in $C^2$, that interpolate between the classical Hartogs triangle and the product domain $D \times D^*$, and discuss their Bergman theory. The main result is that the Bergman projection, $B$, of these domains is only bounded on $L^p$ for a restricted range of $p$ that shrinks to $2$ as the domains fill out $D \times D^*$. This is somewhat surprising as $B$ on $D \times D^*$ is known to map $L^p$ to itself boundedly for all $1 < p <\infty$. The results are joint work with Luke Edholm.

I will give a survey of recent joint work with E. M. Stein [LS-1]-[LS-5] concerning the $L^p$-regularity of the Szegő and Bergman projections, and other singular integral operators, in the non-classical context of domains with non-smooth boundary.

We study nonlinear automorphisms of Levi degenerate hypersurfaces of finite multitype. By recent results, the Lie algebra of infinitesimal CR automorphisms may contain a graded component consisting of nonlinear vector fields of arbitrarily high degree, which has no analog in the classical Levi nondegenerate case, or in the case of finite type hypersurfaces in $\mathbb C^2$. We analyze this phenomenon for hypersurfaces of finite Catlin multitype in complex dimension three. The results provide a complete classification of such manifolds. As a consequence, we show on which hypersurfaces 2-jets are not sufficient to determine an automorphism. (Joint work with Martin Kolar.)

By the classical Chern-Moser theory, there are no nonlinear automorphisms on any strongly pseudoconvex nonspherical hypersurface. Several recent results indicate that the case of Levi degenerate hypersurfaces is much more interesting. In this talk we will consider hypersurfaces of finite Catlin multitype in $\mathbb C^3$ and give a classification of such hypersurfaces which admit nonlinear infinitesimal automorphisms. We will also describe a common source of such vector fields in terms of existence of suitable mappings into hyperquadrics in $\mathbb C^K$, $K \geq 3$. This is joint work with Francine Meylan.

Stationary discs are holomorphic discs that are attached to a real submanifold $M$ of $\mathbb C^n$ and satisfy additional lifting properties, ensuring that their family is both finite dimensional and biholomorphically invariant. Because of that, such discs are suitable for studying mapping problems between smooth CR submanifolds. The usual notion of stationarity, however, is only well adapted for Levi non-degenerate submanifolds. Via the treatment of certain linear Riemann-Hilbert problems with singularities, we show how to construct stationary discs (in a generalized sense) for a class of smooth hypersurfaces of finite type, and then obtain finite jet determination results for CR maps between hypersurfaces of this class. This is joint work with Florian Bertrand.

In this talk, we discuss the preservation of certain analytic properties of the $\overline\partial$-Neumann operator, Bergman projection and Hankel operators on the intersection of pseudoconvex domains. (Joint work with Yunus Zeytuncu.)

Let $\varphi:\mathbb{C}^n\rightarrow[0,+\infty)$ be a smooth plurisubharmonic weight function, and let us denote by $L^2(\mathbb{C}^n,\varphi)$ the $L^2$ space with respect to $e^{-2\varphi}$-times Lebesgue measure. The associated weighted $\overline{\partial}$-problem consists in studying existence and properties of the solutions in $L^2(\mathbb{C}^n,\varphi)$ of the equation: $\overline{\partial} f=\omega$, where $\omega$ is a $\overline{\partial}$-closed $(0,1)$-form with coefficients in $L^2(\mathbb{C}^n,\varphi)$. A related question is that of studying the Bergman projection operator that maps orthogonally $L^2(\mathbb{C}^n,\varphi)$ onto its subspace consisting of holomorphic functions. These questions have been studied in depth in one complex variable (see in particular [1]), and it appears that their several variables counterparts present serious difficulties and require new ideas.

In the talk we present some results we obtained (see [2], [3], [4]) about the weighted $\overline{\partial}$-problem in $\mathbb{C}^n$, in particular:

1. new pointwise estimates for the integral kernel of the Bergman projection operator that extend those obtained by Christ,
2. sufficient and/or necessary conditions for solvability and compactness of the problem (i.e., compactness of the canonical solution operator) for certain classes of model weights previously studied by Nagel and Pramanik.

A key tool in our analysis is a new ``holomorphic uncertainty principle" inspired by the close relationship between the weighted Kohn Laplacian, an elliptic operator naturally arising in this context, and certain generalized Schrödinger operators.

We will survey a few semi-recent (~ last ten years) results concerning compactness or Sobolev estimates for $\overline{\partial}_{M}$. Here $M$ is a compact pseudoconvex orientable CR submanifold of $\mathbb{C}^{n}$ without boundary, of hypersurface type.

I use classical convexity tools, in particular John ellipsoids, to obtain a size estimate for the Szego kernel on the boundary of a class of unbounded convex domains in $\mathbb{C}^n$. Given a polynomial $b:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying a certain growth condition, I consider domains of the type $\Omega_b = \{ z\in\mathbb{C}^{n+1}\,:\, {\rm Im}[z_{n+1}] > b({\rm Re}[z_1],\ldots,{\rm Re}[z_n]) \}$.

Let $\Omega_t=\{\rho_t<0\}$, $t\in [0,1]$, be a family of bounded hyperconvex domains such that $\rho_t$ is Hölder continuous of order $\alpha$ in $t$. Let $K_t$ denote the Bergman kernel of $\Omega_t$. We show that $K_t(z,w)$ is Hölder continuous of order $\beta$ in $t$ for any $\beta<\alpha$.

Integral formulae yield good estimates for solutions of the $\overline\partial$ equation. The structure of their kernels gives, moreover, some insight into the nature of the problem. This has been worked out, over the years, in the case of strictly pseudoconvex domains with smooth boundary. More recently, there has been some progress in the case of non-smooth boundaries. I will report on some aspects of this theory.

We compute the essential spectrum of the complex Laplacian on the product of two Hermitian manifolds in terms of the spectral data of the factors. Applications to (non-)compactness of the minimal solution operators to the $\overline\partial$-equation are given, and this generalizes some previously known results.

In this talk I shall present recent results obtained in collaboration with J. Hounie on the improvement of the celebrated Hörmander's theorem on semi-global solvability for LPDO satisfying condition (P). In particular we show that in dimension 2 solutions can be found with the sharp loss of one derivative.

In 2013 Z. Blocki proved that the Suita conjecture, i.e. the inequality $c_{\Omega}^2(z)\leq \pi K_{\Omega}(z,z)$, $z\in\Omega$ holds for any planar domain $\Omega$. Here $c_{\Omega}$ denotes the logarithmic capacity of the complement of $\Omega$ whereas $K_{\Omega}$ denotes the Bergman kernel. The method used in the proof of the above inequality is of higher dimensional character and relies upon the estimates between the volume of the sublevel sets of the pluricomplex Green function and the Bergman kernel which holds in higher dimensional domains. This allows us to formulate and prove the higher dimensional version of the above inequality which is the following $F_{\Omega}(z):=\lambda(I_{\Omega}^A(z))K_{\Omega}(z,z)\geq 1$, where $\Omega$ is a pseudoconvex domain in $\mathbb C^n$ and $I_{\Omega}^A(z)$ is the Azukawa indicatrix at $z$. It is interesting that in the case of convex domain $\Omega$ an upper estimate for $F_{\Omega}$ also holds. Analysis of (non-trivial) examples of convex domains $\Omega$ with $F_{\Omega}\not\equiv 1$ will also be presented. The results presented in the lecture come from the joint work with Z. Blocki.

In this talk, I will discuss ongoing joint work with Gustavo Hoepfner and Ziad Adwan. I will introduce a class of functions that arose in the work of Boggess and myself when estimating the $\Box_b$-heat kernel on polynomial models. This class of functions captures the quantitative estimates on the Fourier transform needed to characterize exponential decay. I will discuss properties of these functions, give examples, and explore other applications.