Exact values of the Bloch constants $B_k(M)$ and $B_{\mathrm{loc}}(M)$ for holomorphic mappings on the unit ball of $\mathbb{C}^n$

For holomorphic mappings on the unit ball $\mathbb{B}^n\subset\mathbb{C}^n$, Wang and Liu (2007) introduced Bloch-type constants $B_k(M)$ for the families $\beta_k(M)$ of mappings whose critical points have multiplicity at least $k$ and bounded Bloch seminorm, together with the constant $B_{\mathrm{loc}}(M)$ for the locally biholomorphic family obtained as $k\to\infty$, and proved sharp determinant distortion inequalities that yield explicit two-sided bounds for these constants. The exact values of $B_k(M)$ and $B_{\mathrm{loc}}(M)$ — in particular whether the integral lower bounds are optimal — remain undetermined. This problem asks for the exact determination of these constants, a several-variable generalization of the classical open problem of the exact value of Bloch's constant.

Background

Bloch's theorem in one complex variable states that every holomorphic function $f$ on the unit disk $\mathbb{D}$ normalized by $f'(0)=1$ maps some subdomain of $\mathbb{D}$ univalently onto a disk (a so-called schlicht disk) of radius at least $b$ for a universal constant $b>0$; the largest such $b$ is the Bloch constant $B$. The exact value of $B$ is a famous open problem: Bonk (1990) proved $B\ge\sqrt{3}/4$ by establishing a sharp distortion theorem for Bloch functions (holomorphic functions with finite Bloch seminorm $\sup_{z\in\mathbb{D}}(1-|z|^2)|f'(z)|$), refined distortion theorems were obtained by Liu and Minda (1992) and Bonk, Minda and Yanagihara (1997), and Liu and Ponnusamy (2022) record that the precise values of $B$ and the related Landau constant remain unknown despite decades of improved bounds.

In several complex variables the naive analogue fails: as Wang and Liu note, Bloch's theorem fails for unrestricted holomorphic mappings on the unit ball $\mathbb{B}^n=\{z\in\mathbb{C}^n:|z|<1\}$ unless restrictive hypotheses are added, so positive universal constants can only be sought for structured subclasses. The natural several-variable quantity measuring the growth of a mapping $f:\mathbb{B}^n\to\mathbb{C}^n$ is its Jacobian determinant $\det f'(z)$. Wang and Liu (2007) introduced the prenorm

$$\|f\|_0=\sup_{z\in\mathbb{B}^n}(1-|z|^2)^{\frac{n+1}{2n}}|\det f'(z)|^{\frac{1}{n}},$$

which controls determinant growth exactly as the one-variable Bloch seminorm controls $|f'|$, and for each integer $k\ge1$ the class $H_k(\mathbb{B}^n)$ of holomorphic mappings each of whose critical points (zeros of $\det f'$) has multiplicity at least $k$. This is a grading of all holomorphic mappings: $H_1(\mathbb{B}^n)$ is the class of all holomorphic mappings $\mathbb{B}^n\to\mathbb{C}^n$, while as $k\to\infty$ the classes decrease to the locally biholomorphic maps (those with $\det f'(z)\ne0$ everywhere, i.e. locally invertible at every point). Combining the multiplicity grading with the automorphism-invariant Bloch seminorm $\|f\|_{\mathcal{B}}=\sup_{\varphi\in\mathrm{Aut}(\mathbb{B}^n)}\|\partial(f\circ\varphi)/\partial z(0)\|$, where $\mathrm{Aut}(\mathbb{B}^n)$ is the automorphism group of the ball, they defined for $M\ge1$ the families $\beta_k(M)=\{f\in H_k(\mathbb{B}^n):\|f\|_{\mathcal{B}}\le M\}$ and the Bloch-type constants

$$B_k(M)=\inf\{r(f):f\in\beta_k(M),\ \det f'(0)=1\},$$

where $r(a,f)$ is the radius of the largest Euclidean ball centered at $f(a)$ that is the one-to-one image under $f$ of a subdomain of $\mathbb{B}^n$ containing $a$ (a schlicht ball) and $r(f)=\sup_{a\in\mathbb{B}^n}r(a,f)$; $B_{\mathrm{loc}}(M)$ is the corresponding constant for the locally biholomorphic family $\beta_{\mathrm{loc}}(M)$, the limit of $B_k(M)$ as $k\to\infty$.

Wang and Liu proved sharp distortion inequalities for these classes; for example, for $f\in H_k(\mathbb{B}^n)$ with $\|f\|_0=1$ and $\det f'(0)=1$,

$$|\det f'(z)|\ge\frac{\bigl(1-\sqrt{\tfrac{n+k+1}{k}}\,|z|\bigr)^k}{\bigl(1-\sqrt{\tfrac{k}{n+k+1}}\,|z|\bigr)^{n+k+1}} \quad\text{for}\quad |z|\le\sqrt{\tfrac{k}{n+k+1}},$$

together with the corresponding real-part inequality, and, passing to the limit $k\to\infty$, for locally biholomorphic $f$ the global bound $|\det f'(z)|\ge(1-|z|)^{-(n+1)}\exp\{-(n+1)|z|/(1-|z|)\}$; in dimension $n=1$ these reduce to the sharp one-variable distortion theorems of Liu and Minda. Integrating these estimates along curves yields two-sided bounds on the Bloch constants: $M^{1-n}\ge B_k(M)\ge$ an explicit positive integral depending on $n$ and $k$, and

$$M^{1-n}\ge B_{\mathrm{loc}}(M)\ge M^{1-n}\int_0^1\frac{(1+t)^{n-1}}{(1-t)^2}\exp\Bigl\{\frac{-(n+1)t}{1-t}\Bigr\}\,dt\ge\frac{M^{1-n}}{n+1}.$$

The distortion inequalities themselves are sharp, but the paper does not determine whether the induced Bloch-constant bounds are optimal, nor the exact constants, nor the extremal mappings. Subsequent work extended the distortion and Bloch-radius estimates to the unit polydisc, the Lie ball, general bounded symmetric domains (domains such as the ball and polydisc admitting a transitive automorphism group), homogeneous unit balls, and infinite-dimensional Banach balls, in every case producing bounds rather than exact constants, and the exact value of the classical one-variable Bloch constant remains open as well.

Problem Statement

For the unit ball $\mathbb{B}^n\subset\mathbb{C}^n$, determine the exact values of the Bloch constants

$$B_k(M)=\inf\{r(f):f\in\beta_k(M),\ \det f'(0)=1\},\qquad B_{\mathrm{loc}}(M)=\inf\{r(f):f\in\beta_{\mathrm{loc}}(M),\ \det f'(0)=1\},$$

for arbitrary integers $n\ge1$ and $k\ge1$ (with $B_{\mathrm{loc}}$ the locally biholomorphic limit $k\to\infty$) and real $M\ge1$. Here, following Wang and Liu (2007): $H_k(\mathbb{B}^n)$ is the class of holomorphic mappings $f:\mathbb{B}^n\to\mathbb{C}^n$ all of whose critical points (zeros of $\det f'$) have multiplicity at least $k$; $\|f\|_{\mathcal{B}}=\sup_{\varphi\in\mathrm{Aut}(\mathbb{B}^n)}\|\partial(f\circ\varphi)/\partial z(0)\|$ is the automorphism-invariant Bloch seminorm, where $\mathrm{Aut}(\mathbb{B}^n)$ is the automorphism group of the ball and $\|\cdot\|$ is the operator norm of the Jacobian matrix; $\beta_k(M)=\{f\in H_k(\mathbb{B}^n):\|f\|_{\mathcal{B}}\le M\}$; $\beta_{\mathrm{loc}}(M)=\{f:\mathbb{B}^n\to\mathbb{C}^n:\det f'(z)\ne0$ for all $z\in\mathbb{B}^n$, $\|f\|_{\mathcal{B}}\le M\}$ is the locally biholomorphic family; and $r(a,f)$ is the radius of the largest Euclidean ball centered at $f(a)$ that is the one-to-one image under $f$ of some subdomain of $\mathbb{B}^n$ containing $a$ (a schlicht ball), with $r(f)=\sup_{a\in\mathbb{B}^n}r(a,f)$. The single scientific objective is the exact determination of these universal schlicht-ball constants; establishing the exact value of each constant in particular decides whether the known two-sided bounds are optimal — e.g. whether $B_{\mathrm{loc}}(M)$ equals $M^{1-n}\int_0^1\frac{(1+t)^{n-1}}{(1-t)^2}\exp\{-(n+1)t/(1-t)\}\,dt$ — or can be improved. (Characterization of the extremal mappings, or of attainability of the infima, is a natural accompanying question but is not part of the completion criterion.)

References

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