Squeezing Function and Invariant Metrics

• Squeezing Function is a biholomorphic invariant that measures the largest Euclidean ball embedded via an injective holomorphic map, serving as a key tool in complex analysis.
• It establishes quantitative comparisons among intrinsic metrics such as the Carathéodory, Kobayashi, and Bergman metrics, which aids in characterizing domain geometry.
• The invariant provides precise boundary behavior estimates and detects uniform squeezing properties and rigidity phenomena in both one and several complex variables.

The squeezing function of a bounded domain $D\subset \mathbb C^n$ is the biholomorphic invariant that records, at each point $z\in D$, the largest Euclidean ball that can be inscribed in the image of $D$ under an injective holomorphic map sending $z$ to the origin. In the literature the notation varies among $s_D$, $S_\Omega$, and $\sigma_D$. The invariant takes values in $(0,1]$, equals $1$ identically on the unit ball, and organizes a large body of results connecting holomorphic embeddings, invariant metrics, boundary regularity, finite-type geometry, and explicit uniformization problems in one and several complex variables (Deng et al., 2011).

1. Definition and foundational properties

For a bounded domain $D\subset\mathbb C^n$ and $z\in D$0, one considers injective holomorphic maps

$z\in D$1

and defines

$z\in D$2

Equivalent formulations normalize the target image by requiring $z\in D$3, or by writing the extremal ratio $z\in D$4 for inclusions $z\in D$5 (Ng et al., 2020).

Several basic properties are standard. The squeezing function is biholomorphically invariant, satisfies $z\in D$6, and $z\in D$7 at one point if and only if $z\in D$8 is biholomorphic to the unit ball. The supremum is attained: for each $z\in D$9 there exists an extremal embedding $D$0 with $D$1 and $D$2. The function $D$3 is continuous, indeed Lipschitz with respect to a natural Kobayashi-type distance (Deng et al., 2011).

The global lower envelope

$D$4

is the squeezing constant. The condition $D$5 is called the uniform squeezing property, also known as holomorphic homogeneous regularity (HHR). This means that there exists a single radius $D$6 such that every point of $D$7 can be sent to $D$8 by an injective holomorphic map into $D$9 whose image contains $z$0 (Kim et al., 2013).

In one complex variable the definition specializes to injective holomorphic maps into $z$1, and $z$2 can be expressed as the largest radius of a Euclidean disk centered at $z$3 contained in an extremal image. Normal-families arguments again yield extremal embeddings, which in many planar examples can be described explicitly (Gumenyuk et al., 2020).

2. Invariant metrics, volumes, and geometric interpretation

The geometric content of the squeezing function is encoded by comparison inequalities with intrinsic metrics and measures. For the Carathéodory and Kobayashi metrics one has

$z$4

so a lower bound on $z$5 forces quantitative equivalence of these two metrics (Deng et al., 2013).

A parallel comparison holds at the level of invariant volume forms. Near the proof of the h-extendible rigidity theorem, the relevant inequalities are

$z$6

where $z$7 and $z$8 denote the Kobayashi–Eisenman and Carathéodory–Eisenman volumes. These inequalities are strong enough to convert asymptotic information about $z$9 into rigidity statements for scaling limits (Nikolov, 2017).

More general measure comparisons also occur. If $s_D$0 and $s_D$1 are any two among the Carathéodory measure, Eisenman–Kobayashi measure, and the volume forms of $s_D$2 or $s_D$3, then

$s_D$4

On bounded pseudoconvex domains there are also explicit squeezing-controlled comparisons with the Bergman and Kähler–Einstein metrics (Deng et al., 2013).

These inequalities explain why HHR is structurally significant. If $s_D$5, then the Carathéodory, Kobayashi, and Bergman metrics are complete and mutually equivalent; in the HHR setting one also obtains pseudoconvexity of the domain (Deng et al., 2011). A plausible implication is that the squeezing function serves as a quantitative proxy for how far a domain is from the holomorphic geometry of the ball, not merely at the level of embeddings but also at the level of intrinsic metric comparability.

3. Boundary asymptotics, finite type, and rigidity

A principal theme is the behavior of $s_D$6 as $s_D$7 approaches $s_D$8. For bounded domains with $s_D$9 strongly pseudoconvex boundary one has

$S_\Omega$0

Equivalent formulations were also obtained for bounded domains with $S_\Omega$1 strongly convex boundary and for domains near globally strongly convex boundary points (Kim et al., 2013). Quantitative estimates sharpen this: if $S_\Omega$2 is bounded strictly pseudoconvex with boundary of class $S_\Omega$3, $S_\Omega$4, then

$S_\Omega$5

while for $S_\Omega$6,

$S_\Omega$7

with $S_\Omega$8 a defining function normalized by $S_\Omega$9 on $\sigma_D$0 (Fornaess et al., 2014).

In one variable, boundary regularity can be weakened. If $\sigma_D$1 has a Dini-smooth boundary point $\sigma_D$2, then

$\sigma_D$3

where $\sigma_D$4. If $\sigma_D$5 is only $\sigma_D$6-smooth, then for every $\sigma_D$7,

$\sigma_D$8

(Nikolov et al., 2016).

The strongest several-variable rigidity result in the data concerns h-extendible boundary points. A boundary point $\sigma_D$9 of a $(0,1]$0-smooth pseudoconvex domain is h-extendible if it is of finite type in the sense of D’Angelo and Catlin’s multitype equals D’Angelo’s multitype at $(0,1]$1. Equivalently, in suitable local holomorphic coordinates $(0,1]$2 sending $(0,1]$3, the defining function has the normal form

$(0,1]$4

where $(0,1]$5 is a non-pluriharmonic weighted homogeneous polynomial. Typical examples include strictly pseudoconvex points, convex finite-type points, and points where the Levi form has corank at most one (Nikolov, 2017).

Nikolov’s theorem states that if $(0,1]$6 is bounded pseudoconvex with $(0,1]$7 boundary, $(0,1]$8 is h-extendible, and there exists a nontangential sequence $(0,1]$9 with

$1$0

then $1$1 is strictly pseudoconvex (Nikolov, 2017). The proof passes through a weighted model domain

$1$2

uses scaling maps adapted to Catlin’s multitype, derives asymptotics for $1$3 and $1$4 after multiplication by a suitable power of $1$5, and then forces

$1$6

at the model point $1$7. Tautness, the Carathéodory–Cartan–Kaup–Wu theorem, and a theorem of Coupet–Pinchuk then imply that the weighted homogeneous model is biholomorphic to the ball only in the strictly pseudoconvex case (Nikolov, 2017).

The rigidity statement has sharp limitations. Fornæss–Wold constructed a $1$8-smooth non-strictly pseudoconvex example with $1$9, so mere $D\subset\mathbb C^n$0-smoothness is not enough for the h-extendible gap phenomenon (Nikolov, 2017). Later work also proved that $D\subset\mathbb C^n$1 along certain uniformly $D\subset\mathbb C^n$2-tangential and spherically $D\subset\mathbb C^n$3-tangential sequences approaching finite-type pseudoconvex points, while preserving the corollary that at an h-extendible boundary point the existence of a sequence with $D\subset\mathbb C^n$4 forces strong pseudoconvexity (Thu, 13 Sep 2025).

4. Exact computations and explicit models

The most complete explicit formulas are known in low-dimensional or highly symmetric settings.

For annuli

$D\subset\mathbb C^n$5

Ng, Tang, and Tsai proved

$D\subset\mathbb C^n$6

Their proof combines the Schottky–Klein prime function with Komatu’s Loewner differential equation on annuli and shows that the extremal configuration is a circularly slit disk whose slit has radius $D\subset\mathbb C^n$7 or $D\subset\mathbb C^n$8, depending on which boundary component is relevant (Ng et al., 2020). Gumenyuk and Roth later gave a potential-theoretic proof of the same formula and identified all extremal embeddings: up to post-rotation, the only extremals are $D\subset\mathbb C^n$9 and $z\in D$00 (Gumenyuk et al., 2020).

For doubly connected planar domains, conformal equivalence with an annulus reduces the problem to the same formula. If $z\in D$01 is conformal, then

$z\in D$02

(Ng et al., 2020). In this setting the extremal map is one of the two canonical circular-slit uniformizations, unless the two associated slit radii coincide (Solynin, 2021).

For finitely connected planar domains of connectivity at least $z\in D$03, the picture changes. The conjecture that the extremal map should always be one of the canonical circularly slit disk maps was disproved: for every $z\in D$04, there exists an $z\in D$05-connected planar domain $z\in D$06 and a point $z\in D$07 for which none of the canonical circular-slit maps is extremal (Gumenyuk et al., 2020).

For an infinitely connected domain of the form $z\in D$08, where $z\in D$09 accumulates only at $z\in D$10, one has the explicit formula

$z\in D$11

In this case the canonical disk automorphism sending $z\in D$12 to $z\in D$13 is extremal, and the boundary behavior reflects the accumulation of punctures near $z\in D$14 (Kumar, 2022).

Domain	Squeezing function	Source
$z\in D$15	$z\in D$16	(Ng et al., 2020)
$z\in D$17	$z\in D$18	(Ng et al., 2020)
$z\in D$19	$z\in D$20	(Deng et al., 2011)
$z\in D$21, $z\in D$22	$z\in D$23	(Kumar, 2022)

These formulas show that explicit computation is possible when the extremal embeddings are controlled by strong symmetry, one-dimensional potential theory, or removable-singularity arguments.

5. Uniform squeezing, symmetric domains, and product phenomena

A large part of the theory identifies classes of HHR domains. Every bounded convex domain in $z\in D$24 is uniformly squeezing; more generally, every convex Kobayashi-hyperbolic domain satisfies $z\in D$25 (Kim et al., 2013). The same conclusion holds for every non-degenerate $z\in D$26-convex domain: if $z\in D$27 is $z\in D$28-convex and contains no complex affine line, then there exists a constant $z\in D$29, depending only on $z\in D$30, such that $z\in D$31 for all $z\in D$32 (Nikolov et al., 2016).

Other HHR classes arise from finite-type model geometry. If $z\in D$33 is a general ellipsoid that is a $z\in D$34-domain, then $z\in D$35 is holomorphically homogeneous regular. More generally, near a $z\in D$36-extreme boundary point one has a uniform positive lower bound for the squeezing function on weighted nontangential approach regions $z\in D$37 (Thu et al., 2020). A further family is provided by bounded, contractible, weakly linearly convex domains $z\in D$38, $z\in D$39, for which there exists a point $z\in D$40 such that every affine complex hyperplane through $z\in D$41 meets $z\in D$42 in a connected slice; then $z\in D$43 (Bharali et al., 2023).

Highly symmetric domains admit exact constants rather than merely lower bounds. If $z\in D$44 is any bounded symmetric domain of rank $z\in D$45, then homogeneity forces $z\in D$46 to be constant and

$z\in D$47

This applies to all irreducible Cartan domains and to arbitrary Cartesian products, with $z\in D$48 and

$z\in D$49

The proof uses the Polydisk Theorem and the realization of $z\in D$50 as the open unit ball of a positive Hermitian Jordan triple system, avoiding case-by-case analysis by rank (Bharali et al., 2023).

Products also enter through general inequalities. If $z\in D$51 and $z\in D$52, then

$z\in D$53

which yields explicit lower bounds for products such as $z\in D$54 once the annulus formula is known (Ng et al., 2020). Conversely, the squeezing function can obstruct product decompositions: if $z\in D$55 is bounded pseudoconvex and contractible, and there exist $z\in D$56 and $z\in D$57 with $z\in D$58 such that

$z\in D$59

then $z\in D$60 is not biholomorphic to any product of $z\in D$61 or more irreducible factors (Bharali et al., 2023).

6. Generalizations, dual invariants, and limitations

The classical squeezing function has several model-dependent generalizations. For a bounded, convex, balanced domain $z\in D$62, the generalized squeezing function $z\in D$63 replaces the target ball by $z\in D$64 and uses the Minkowski functional of $z\in D$65. If

$z\in D$66

then

$z\in D$67

Thus classical HHR and generalized HHR are equivalent at the level of positivity of the global infimum (Gupta et al., 2022).

A weighted version is the $z\in D$68-balanced squeezing function. If $z\in D$69 and $z\in D$70 is a bounded, convex, $z\in D$71-balanced model domain, then the corresponding invariant $z\in D$72 is again biholomorphically invariant, has extremal maps, and is continuous when $z\in D$73 is homogeneous. It is related to the Fridman invariant $z\in D$74 by

$z\in D$75

and if $z\in D$76 itself is convex and $z\in D$77-balanced, then at the origin

$z\in D$78

When $z\in D$79, this recovers the balanced-model comparison theory (Gupta et al., 2021).

For the classical ball model, the Fridman function is a dual invariant satisfying

$z\in D$80

for every bounded domain $z\in D$81 (Ng et al., 2020). In special planar cases the duality is exact. For $z\in D$82 with $z\in D$83 a sequence accumulating at $z\in D$84, Kumar proved

$z\in D$85

(Kumar, 2022).

Several limitations prevent a naive reading of the invariant. The squeezing function is not monotone under inclusion of domains (Solynin, 2021). In planar multiply connected domains of connectivity at least $z\in D$86, extremal images need not be circularly slit disks (Gumenyuk et al., 2020). In $z\in D$87, $z\in D$88, there exist bounded pseudoconvex domains with large $z\in D$89-open subsets on which $z\in D$90 along boundary approach; the corresponding phenomenon is impossible for planar domains (Bharali, 2021). This suggests that the boundary behavior of $z\in D$91 is markedly more rigid in one variable than in several variables.

Taken together, these developments position the squeezing function as a unifying invariant for comparing bounded domains with model balls, balanced targets, or weighted models; for detecting product structure and rigid boundary geometry; and for isolating the precise points at which holomorphic ball-likeness is forced, obstructed, or only partially visible.