The central purpose of the present paper is to study boundary behavior of squeezing functions on bounded domains. We prove that the squeezing function of a strongly pseudoconvex domain tends to 1 near the boundary. In fact, such an estimate is proved for the squeezing function on any domain near its globally strongly convex boundary points. We also study the stability of squeezing functions on a sequence of bounded domains, and give comparisons of intrinsic measures and metrics on bounded domains in terms of squeezing functions. As applications, we give new and simple proofs of several well known results about geometry of strongly pseudoconvex do- mains, and prove that all Cartan-Hartogs domains are homogenous regular. Finally, some related problems that ask for further study are proposed.

Properties of squeezing functions and global transformations of bounded domains

The Cartan–Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan–Hartogs domain $\Omega ^{B^{d_0}}(\mu )$ endowed with the canonical metric $g(\mu ),$ we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space $\mathcal {H}_{\alpha }$ of square integrable holomorphic functions on $\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) $ with the weight $\exp \{-\alpha \varphi \}$ (where $\varphi $ is a globally defined Kahler potential for $g(\mu )$) for $\alpha >0$, and, furthermore, we give an explicit expression of the Rawnsley’s $\varepsilon $-function expansion for $\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) .$ Secondly, using the explicit expression of the Rawnsley’s $\varepsilon $-function expansion, we show that the coefficient $a_2$ of the Rawnsley’s $\varepsilon $-function expansion for the Cartan–Hartogs domain $\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) $ is constant on $\Omega ^{B^{d_0}}(\mu )$ if and only if $\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) $ is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.

On canonical metrics on Cartan–Hartogs domains

Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan–von Neumann–Wigner classification of finite euclidean Jordan algebras is outlined with special attention to the 27 dimensional exceptional Jordan algebra $${\mathcal {J}}$$
 . The automorphism group $$F_4$$
 of $${\mathcal {J}}$$
 and its maximal Borel-de Siebenthal subgroups $$\frac{SU(3)\times SU(3)}{{{\mathbb {Z}}_3}}$$
 and Spin(9) are studied in detail and applied to the classification of fundamental fermions and gauge bosons. Their intersection in $$F_4$$
 is demonstrated to coincide with the gauge group $$S(U(2)\times U(3))$$
 of the Standard Model of particle physics.

Octonions, Exceptional Jordan Algebra and The Role of The Group $$F_4$$ F 4 in Particle Physics

This paper consists of two results dealing with balanced metrics (in S. Donaldson terminology) on nonconpact complex manifolds. In the first one we describe all balanced metrics on Cartan domains. In the second one we show that the only Cartan-Hartogs domain which admits a balanced metric is the complex hyperbolic space. By combining these results with those obtained in [13] (Kaehler-Einstein submanifolds of the infinite dimensional projective space, to appear in Mathematische Annalen) we also provide the first example of complete, Kaehler-Einstein and projectively induced metric g such that $\alpha g$ is not balanced for all $\alpha >0$.

Balanced metrics on Cartan and Cartan-Hartogs domains

The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains The purpose of this paper is twofold Firstly, for a Cartan-Hartogs domain $\Omega^{B^{d_0}}(\mu)$ endowed with the canonical metric $g(\mu)$, we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space $\mathcal{H}_{\alpha}$ of square integrable holomorphic functions on $(\Omega^{B^{d_0}}(\mu), g(\mu))$ with the weight $\exp\{-\alpha \varphi\}$ (where $\varphi$ is a globally defined Kahler potential for $g(\mu)$) for $\alpha>0$, and, furthermore, we give an explicit expression of the Rawnsley's $\varepsilon$-function expansion for $(\Omega^{B^{d_0}}(\mu), g(\mu))$ Secondly, using the explicit expression of the Rawnsley's $\varepsilon$-function expansion, we show that the coefficient $a_2$ of the Rawnsley's $\varepsilon$-function expansion for the Cartan-Hartogs domain $(\Omega^{B^{d_0}}(\mu), g(\mu))$ is constant on $\Omega^{B^{d_0}}(\mu)$ if and only if $(\Omega^{B^{d_0}}(\mu), g(\mu))$ is biholomorphically isometric to the complex hyperbolic space So we give an affirmative answer to a conjecture raised by M Zedda

On canonical metrics on Cartan-Hartogs domains

We study the complete K\"{a}hler-Einstein metric of a Hartogs domain $\widetilde {\Omega}$, which is obtained by inflation of an irreducible bounded symmetric domain $\Omega $, using a power $N^{\mu}$ of the generic norm of $\Omega$. The generating function of the K\"{a}hler-Einstein metric satisfies a complex Monge-Amp\`{e}re equation with boundary condition. The domain $\widetilde {\Omega}$ is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by $X\in\lbrack0,1[$. This allows to reduce the Monge-Amp\`{e}re equation to an ordinary differential equation with limit condition. This equation can be explicitly solved for a special value $\mu_{0}$ of $\mu$, called the critical exponent. We work out the details for the two exceptional symmetric domains. The critical exponent seems also to be relevant for the properties of other invariant metrics like the Bergman metric; a conjecture is stated, which is proved for the exceptional domains.

The Kaehler-Einstein metric for some Hartogs domains over bounded symmetric domains

We introduce the notion of "virtual Bergman kernel" and apply it to the computation of the Bergman kernel of "domains inflated by Hermitian balls", in particular when the base domain is a bounded symmetric domain.

Weighted Bergman kernels and virtual Bergman kernels

We introduce two classes of "egg type" domains, built on general bounded symmetric domains, for which we compute the Bergmann kernel in explicit form. We use the characterization of bounded symmetric domains through Jordan triple systems. The egg type domains are defined using the generic norm. A generalization of the Hua integral is computed; the result shows the existence of a special polynomial with integer or half-integer coefficients, attached to each irreducible bounded symmetric domain.

New classes of domains with explicit Bergman kernel

We give the presentation of exceptional bounded symmetric domains using the Albert algebra and exceptional Jordan triple systems. 
The first chapter is devoted to Cayley-Graves algebras, the second to exceptional Jordan triple systems. In the third chapter, we give a geometric description of the two exceptional bounded symmetric domains, their boundaries and their compactification.

/pdf/exceptional-symmetric-domains-fj8yhlprgp.pdf

Exceptional symmetric domains

A theorem of Harald Bohr (1914) states that if f is a holomorphic map from the unit disc into itself, then the sum of absolute values of its Taylor expansion is less than 1 for |z|<1/3. The bound 1/3 is optimal. This result has been extended in a suitable sense by Liu Taishun and Wang Jianfei (2007) to the bounded complex symmetric domains of the four classical series, and to polydiscs. The result of Liu and Wang may be generalized to all bounded symmetric domains, with a proof which does not depend on classification.

H. Bohr's theorem for bounded symmetric domains

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