Papers

Motivated by the principle that positivity of the Beauville–Bogomolov–Fujiki (BBF) form controls the geometry of hyperkähler manifolds, we study BBF positivity from the perspective of bimeromorphic geometry, focusing on its relation to the Moishezonness of hyperfujiki manifolds (natural bimeromorphic analogues of hyperkähler manifolds). We prove that any Moishezon hyperfujiki manifold carries a big line bundle with positive BBF square. We also prove that the existence of a BBF-positive integral $(1,1)$-class implies Moishezonness for hyperfujiki $4$-folds and, more generally, for hyperfujiki $2n$-folds under a weak Kähler minimal-model condition. The proof mainly uses the theory of primitive symplectic varieties developed by B. Bakker–C. Lehn, a construction of small bimeromorphic models for certain $K$-trivial manifolds by I. Biswas–J. Cao–S. Dumitrescu–H. Guenancia, and the theory of pull-backs of reflexive differential forms established by S. Kebekus–C. Schnell. As an application, we give a Hodge-theoretic description of the Moishezon locus in certain smooth families by combining this criterion with the period theory developed by B. Anthes–A. Cattaneo–S. Rollenske–A. Tomassini.

We construct a real meromorphic function $F$ on $\mathbb C$ such that $F^{-1}(\{0,1,\infty\})\subset\mathbb R$, while $F$ is not of bounded type in either half-plane. More strongly, for every $a\in\widehat{\mathbb C}\setminus\{0,1,\infty\}$, the $a$-point divisor in either half-plane fails the Blaschke condition. Thus the construction provides an independent negative answer to a question going back to Nevanlinna's 1925 work that had remained open for over a century. Postcomposition gives the analogous counterexample for any prescribed triple of distinct values in the Riemann sphere. The core construction and proof were generated during an autonomous run of GPT-5.6 Sol Ultra.

We identify the critical boundary operator-norm profile of finite-prime composition operators on the Hardy–Hilbert space \(\mathcal H^2\) of Dirichlet series. For \[ \varphi_{\delta,\boldsymbol\rho}(s) = \frac12+\delta + \delta\sum_{j=1}^d\rho_jp_j^{-s}, \qquad \boldsymbol\rho\in B_d, \] the renormalized positive coefficient operators converge uniformly in operator norm, with \(O(\delta)\) error, to an explicit multivariate weighted Hankel operator \(\mathcal H_{\boldsymbol\rho}\); consequently, \[ 2\delta\|C_{\varphi_{\delta,\boldsymbol\rho}}\|^2 = \|\mathcal H_{\boldsymbol\rho}\| + O(\delta) \] uniformly over \(B_d\). We show that the limiting operator admits the total-degree reduction \[ \mathcal H_{\boldsymbol\rho} \simeq D_{\boldsymbol\rho} H_{R_{\boldsymbol\rho}/2} D_{\boldsymbol\rho}\oplus\mathbf{0}, \] where the diagonal factors are convolution-collision norms of the normalized prime weights. This structure, together with the affine comparison principle of Brevig and Perfekt, yields an explicit concentration inequality for \(\|\mathcal H_{\boldsymbol\rho}\|\), identifies the one-prime configurations as the exact equality cases in the limiting norm estimate, and gives a quantitative deficit away from them. For fixed \(\sigma>\frac12\), we also obtain a second-order expansion of the squared norm and fully finite-dimensional approximations with explicit total-degree and Dirichlet-sum truncation errors. Together, these results show that a single coefficient-operator structure governs the singular boundary profile, the fixed-\(\sigma\) perturbative regime, and certified finite-dimensional approximation.

The Serre problem asked if a locally trivial holomorphic fibration with Stein base and Stein fiber is itself Stein. \v Skoda constructed the first counterexample for this problem. Later, Cœuré and Loeb constructed another remarkable counterexample, with bounded domain of holomorphy as fiber. The total space of their fibration has global holomorphic functions which separate points and provide local coordinates. However, it does not admit a Stein envelope of holomorphy. In this paper, we prove that the Cœuré–Loeb fibration can be realized as an open subset of a Stein space, more precisely it is the complement of an analytic set in a Stein space.

We introduce two new arithmetic invariants associated with Hermitian line bundles on arithmetic varieties, which play the role of the Bergman distortion function in the arithmetic setting. As a first step, we show that these two functions are asymptotically equivalent. This leads to the introduction of an intrinsic measure $ \widehat{\mu}^{\sup}_{\mathrm{eq}},$ which we show to be a natural arithmetic analogue of the equilibrium measure. We prove that $\widehat{\mu}^{\sup}_{\mathrm{eq}}$ serves as a detector of arithmetic positivity. In particular, we determine it for weakly nef Hermitian line bundles and, in full generality, in the toric case. Moreover, $\widehat{\mu}^{\sup}_{\mathrm{eq}}$ governs the arithmetic volume function and its variational properties. More precisely, we establish weak integral representations of the arithmetic volume and of its first variation in terms of $\widehat{\mu}^{\sup}_{\mathrm{eq}}$. These formulas provide a measure-theoretic interpretation of the volume derivative of Yuan–Zhang and reveal a structural parallel with variational formulas in complex pluripotential theory. Our approach differs fundamentally from previous treatments based on Harder–Narasimhan filtrations or arithmetic Okounkov bodies. It places the arithmetic volume at the center of the theory and suggests that asymptotic volume invariants, rather than heights, provide the natural analytic framework for problems in equidistribution and arithmetic dynamics.

The primary objective of this paper is to systematically generalize this phenomenon by replacing the standard unit disk with a family of nested, internally tangent shifted disks $\Omega_\gamma$ parameterized by $\gamma \in [0, 1)$, defined by$$\Omega_\gamma = \left\{ z \in \mathbb{C} : \left| z + \frac{\gamma}{1 - \gamma} \right| < \frac{1}{1 - \gamma},\; \gamma\in [0, 1) \right\}.$$ By exploiting the geometric characteristics of $\Omega_\gamma$ and evaluating the limiting behavior as $\gamma \to 1^-$, we establish a novel framework to determine the Bohr radius for the unbounded half-plane $\mathbb{H}_1 = \{z \in \mathbb{C} : \text{Re}(z) < 1\}$. Furthermore, we prove several sharp variations of the Bohr inequality within these domains, including refined and improved formulations for unimodular bounded analytic functions. The results obtained herein not only extend classical radius problems to unbounded regions but also illuminate the delicate interplay between domain deformation and coefficient estimates.

Let $\mathbb{H}=\{z\in\mathbb{C}:\operatorname{Im}z>0\}$, and let $N(\mathbb{H})$ denote the Nevanlinna class in $\mathbb{H}$, consisting of meromorphic functions representable as quotients of two bounded analytic functions in $\mathbb{H}$. We construct a nonconstant meromorphic function $F$ on $\mathbb{C}$ such that $F^{-1}(\{0,1,\infty\})\subset\mathbb{R}$ and $F|_{\mathbb{H}}\notin N(\mathbb{H})$. Thus, omitting three distinct values of the Riemann sphere in a half-plane does not force a meromorphic function on $\mathbb{C}$ to be of bounded type there. This provides a counterexample to Nevanlinna's century-old half-plane problem.

We study the relation between the geometric properties of a chord-arc curve and its conformal welding. Let $h$ be the conformal welding of a closed quasicircle $\Gamma$. By Jones's theorem, the pull-back operator $C_h u=u\circ h$ is bounded on BMO if and only if $h$ corresponds to the welding of a Bishop-Jones quasicircle, equivalently, $h$ is strongly quasisymmetric. Let $A_h$ denote the analytic projection of $C_h$. We prove that $A_h$ is a bounded isomorphism on BMOA if and only if $\Gamma$ is a chord-arc curve. More strongly, the same characterization holds if invertibility is replaced by Fredholmness. This gives an intrinsic conformal-welding characterization of chord-arc curves and a complete geometric answer to the invertibility problem posed by Semmes in the 1980s. Furthermore, we establish an exact correspondence between the inverse of $A_h$ and the classical Faber integral operator, showing that for a rectifiable quasicircle, the Faber operator is a bounded isomorphism on BMOA if and only if the curve satisfies the chord-arc condition.

This paper gives examples and counterexamples in Oka theory: two about deleting closed sets from complex Euclidean space, one about blowing up along a connected Oka center, and one about deforming continuous maps to regular maps. Two positive results show that $\mathbb{C}^3\setminus S$ is Oka for every closed set $S\subset\mathbb{R}^3$, and that the complement of the closed Hartogs triangle in $\mathbb{C}^2$ is Oka. In contrast, for every $n\ge3$ there is a proper holomorphic embedding $\mathbb{C}\hookrightarrow\mathbb{C}^n$ whose image $A$ is a closed connected smooth curve biholomorphic to $\mathbb{C}$, but whose blow-up $Bl_A\mathbb{C}^n$ is Brody volume hyperbolic and hence not Oka. Finally, for every $n\geq 2$, there exist a smooth connected affine algebraic variety $X$ and a continuous map $X\to\mathbb{C}^n\setminus\{0\}$ that is not homotopic to any regular map $X\to\mathbb{C}^n\setminus\{0\}$; equivalently, $\mathbb{C}^n\setminus\{0\}$ fails the algebraic basic Oka property (aBOP).

We discuss estimates for the maximal operator associated with the non-linear Fourier transform of an $L^2$-function on the half-line. For potentials with bounded dyadic $L^1$ masses we prove weak-type maximal and maximal fluctuation estimates on the sets where the spectral densities are bounded away from zero and three classical maximal functions of the spectral data are uniformly small; such sets exhaust almost all of the real line as the parameters of their definition relax.

We give the first known examples of Oka Calabi–Yau manifolds in any dimension: every smooth hypersurface of multidegree $(2,\ldots,2)$ in a product of projective lines is Oka. Building on this example, we prove that in every local Kuranishi family of K3 surfaces the points with Oka fibers form a dense $G_\delta$ subset of the base, and the sublocus of Oka fibers with vanishing Néron–Severi group is likewise a dense $G_\delta$ subset. We also obtain projective and nonprojective Oka Kummer surfaces and prove an analogous density statement in the Kummer period domain. We also prove the existence of Oka Enriques surfaces and Oka irreducible hyperkähler fourfolds.

Results on holomorphic mappings of finitely connected planar domains are often proved using Koebe's famous circle mapping theorem. A prominent example is Julia's bound on the size of the automorphism group of finitely connected planar domains of connectivity $k\geq3$ and the sharp bound obtained later by Heins. The purpose of this article is to explore whether it is possible to give a proof of these results, and several other related results, without using Koebe's theorem. The main result is a rigidity theorem for proper holomorphic maps between finitely connected domains of connectivity higher than $2$ that immediately yields a bound on the number of proper holomorphic mappings between two such domains in terms of their connectivities. Using the same ideas, we also provide a new elementary proof of the planar Riemann–Hurwitz formula that does not use the Euler characteristic. Most of our proofs rely only on the Riemann mapping theorem, the Schwarz reflection principle, and the argument principle.

Let $\Omega\subset\mathbb C$ be a bounded John domain and set $\delta(z)=\text{dist}(z,\partial\Omega)$. For $1<p<\infty$ and $\alpha>\text{dim}_A(\partial\Omega)-2$, we establish a norm equivalence between $\int_\Omega |g|^p\delta^\alpha\,dA$ and $\int_\Omega |g'|^p\delta^{\alpha+p}\,dA$ for analytic functions $g$ on $\Omega$, with a point-evaluation term fixing the additive constant. The estimate of the derivative term is local and holds on every proper planar domain, whereas the converse follows from a distance-weighted Poincaré inequality on John domains. Taking $\alpha=mp-2$ yields the corresponding comparison between the $m$-th and $(m+1)$-st derivatives. For $m\ge2$ the boundary-dimension condition is automatic, so the only dimension-sensitive case is the comparison between $\int_\Omega |f'|^p\delta^{p-2}\,dA$ and $\int_\Omega |f''|^p\delta^{2p-2}\,dA$ when $1<p<2$. We show that this restriction is sharp within the class of quasidisks by using self-similar Rohde snowflakes. We also construct, for every $s>1$, an inward-cusp $s$-John domain on which the comparison fails, showing that the ordinary John condition cannot in general be weakened.

We prove measure-rigidity results for regular polynomial endomorphisms of~$\mathbb{C}^k$. For maps of the same degree whose leading homogeneous parts differ by an invertible linear map on the target, equality of equilibrium measures is equivalent to postcomposition by an affine symmetry of the Julia set. We then show that, for $f$ in a nonempty Zariski open subset of the parameter space, $\mu_g=\mu_f$ implies $g=f$, with no hypothesis relating the leading terms; in dimension two this holds across all degrees, with $g$ an iterate of~$f$. We also establish finiteness results for maps with a prescribed equilibrium measure under essentially necessary hypotheses. Our approach rests on the stable manifold structure near the hyperplane at infinity. Equality of equilibrium measures yields a Zariski-dense family of common local stable-manifold germs, whose intrinsic first-order jets lead to a divisibility incidence problem on a Grassmannian. We resolve this problem by analyzing the power-map fiber and applying a generic stabilizer argument. In dimension two, applications include a characterization in terms of preperiodic sets, a Tits-type alternative, results on iterated centralizers, and an arithmetic non-density theorem.

For a $n$-dimensional compact Kähler manifold $X$ with a pair of Kähler classes $(\alpha,\beta)$, we show that the algebraically defined $J$-null locus is equal to the analytically defined $J$-non-ample locus under the assumption of $J$-bigness(which is automatic for semistable pair up to dimension $3$ \cite{3d}), and boundary cone condition $c_{\alpha,\beta}\omega^{n-1}-(n-1)\omega^{n-2}\wedge\chi\geq 0$ for some Kähler form $\omega\in \alpha,\chi\in \beta$. The key tool is the generalized Khovanskii-Teissier inequality associated to $J$ equation formulated by Collins \cite{CT21} and its extension to singular Kähler space.

Let $\omega$ be a radial $\widehat{\mathcal D}$-weight and $u$ be a bounded function on the unit disk $\mathbb D$. We prove that the Toeplitz operator \(T_{\omega,u}\) is compact on \(A_\omega^2\) if and only if its Berezin transform vanishes at the boundary. Our approach is based on a polynomial frame for $A_\omega^2$ and a detailed localization analysis of the resulting infinite matrix representation of $T_{\omega,u}$. Even in the unweighted Bergman space \(A^2\), our argument is new and does not rely on the classical translation operators. We further show that this Axler–Zheng compactness characterization does not extend, in general, to products of Toeplitz operators on \(\widehat{\mathcal D}\)-weighted Bergman spaces, and hence to the corresponding Toeplitz algebra generated by bounded symbols. More precisely, we construct a radial log-subharmonic \(\widehat{\mathcal D}\)-weight \(\omega\) and bounded symbols \(u,v\) such that the product \(T_{\omega,v}T_{\omega,u}\) is noncompact, whereas its Berezin transform vanishes at the boundary.

We establish a Demailly-Kollár type continuity theorem for plurisubharmonic functions with respect to adapted measures on normal complex analytic klt pairs. As applications, we prove the equality of the analytic and divisorial versions of the alpha and delta invariants on compact normal Kähler klt pairs, thereby completing a program initiated by the second author. We further show that the two local alpha invariants introduced by Guedj and Trusiani for an isolated log terminal singularity coincide and that their common value is the $n$th root of Li's normalized volume. These identities have geometric consequences. The equality for the delta invariant yields a Yau-Tian-Donaldson type divisorial criterion for the solvability of twisted Kähler-Einstein equations in big cohomology classes, without a semipositivity assumption on the twist. The local alpha identity determines algebraically the critical exponent governing the existence of positively curved KE metrics near an isolated log terminal singularity, thus confirming a prediction of Guedj and Trusiani.

We use Morse Theory to prove that the Burkholder function is quasiconvex on symmetric matrices, which implies several sharp inequalities for the Beurling–Ahlfors transform when acting on real-valued functions. More generally, we prove that the separating functions introduced by Székelyhidi are quasiconvex on symmetric matrices.

We establish a dynamic generalization of Nevanlinna-Pick interpolation for functions invariant under the action of a finite Blaschke product $f$, reducing global bounded holomorphic extension on orbit spaces to structured block-kernel positivity. Furthermore, we demonstrate that membership in $H^\infty(\mathbb{D})$ is universally detectable via deformations by any finite Blaschke product. These results are proved via an underlying operator-theoretic lifting framework for covariant operator pairs in the spirit of Sarason. Finally, in the setting of non-commutative function theory, we show that despite the universal validity of the matrix-valued von Neumann inequality over the free polydisk, Arveson-type complete spectral set representations break down for non-commutative inner varieties.

The sewing operation is the fundamental algebraic structure of the (infinite-dimensional) Segal moduli spaces as the conformal cobordism category. We prove a universal property of local $\mathbb{R}_+$-extensions of the sewing operation, characterizing them by one real parameter: their central charge. Our result is an analogue of “Mumford's theorem,” that appeared in Segal's monograph [Seg88, Seg04], classifying complex extensions, also known as one-dimensional modular functors. Algebraically, the real extensions are characterized by real-valued cocycles on pairs of surfaces under sewing. We identify the disk-disk cocycle as the universal Liouville action, also known as loop Loewner energy. The real determinant line bundle encodes the trace anomaly of euclidean two-dimensional conformal field theories (CFT) and the restriction function of Schramm-Loewner evolution (SLE) loop measures. It serves as the guiding example in our setup and proof strategy, providing the universal $\mathbb{R}_+$-extension. Our results thus give a mathematical explanation for the appearance of the central charge in CFT and SLE, and prove the conjectured uniqueness of Brownian loop measures as the canonical restriction function for SLE.

We prove that a possibly unbounded Reinhardt domain in $\mathbb{C}^n$ with smooth boundary and a complete Bergman-Einstein metric is biholomorphic to the complex unit ball. In particular, no unbounded smooth Reinhardt domain admits a complete Bergman-Einstein metric.

We prove extremality results concerning specific quasiconformal mappings in the hyperbolic plane, namely, hyperbolic spiral and stretch maps.

Let \[ \mu(r)=\frac{\pi}{2}\frac{\Kc(r')}{\Kc(r)}, \qquad r'=\sqrt{1-r^2},\qquad 0<r<1, \] be the Grötzsch modulus function. Problems 3.38(a)–(b) in a survey of Vuorinen ask whether Newton's iteration for $\mu^{-1}(y)$, initialized by $x_0=1/\cosh y$, converges for every $y>\pi/2$, and whether it is strictly increasing when $y>\pi$. We prove both assertions. The main point is that $\mu$ has exactly one inflection point $a\in(1/2,1/\sqrt2)$. If the zero lies in the convex region, the Newton iterates increase to it from the left. If the zero lies in the concave region, the orbit crosses the inflection point, overshoots the zero at most once, and then decreases to the zero. For $y>\pi$ we obtain the stronger estimate $0<x_n<x_{n+1}<\mu^{-1}(y)<3-2\sqrt2<1.$

Let $D=C_1+C_2\subset\mathbb{P}^2$ be a simple-normal-crossing union of smooth plane curves of degrees $1\leqslant d_1\leqslant d_2$. We prove an effective Second Main Theorem on a nonempty Zariski-open set of ordered pairs whenever \[ d_1,d_2\geqslant3, \qquad\text{or}\qquad d_1=2,\ d_2\geqslant5, \qquad\text{or}\qquad d_1=1,\ d_2\geqslant8. \] After intersecting with the standard very-general algebraic-hyperbolicity locus, we obtain Kobayashi hyperbolicity and hyperbolic embedding of the complement. The proof uses logarithmic invariant two-jet differentials, the corrected Demailly–El Goul zero-locus argument, and a two-component adaptation of slanted vector fields with base twist $\mathcal{O}_{\mathbb{P}^2}(3)$. The adapted twist three is uniform in $(d_1,d_2)$: simultaneous cancellation of the two root equations requires third Taylor differences, but no seventh-order polarization. For a linear component the third-remainder space vanishes, and ten compatible mixed generators replace the generic Taylor package. A numerical phase diagram shows that exact finite computation is needed only for the six boundary pairs $(3,3),(3,4),(2,6),(2,5),(1,9),(1,8)$. Their key vanishings are certified by full-column-rank matrices over finite fields. For two cubics, a strengthened range reduces to twelve theorem-relevant degree–twist cases; two low-weight self-tests bring the computational total to $168$ character–eigenvalue blocks. The exact source, certificates, and reproduction instructions form an accompanying computational archive. In the two-cubic case the resulting explicit estimate is \[ T_f(r)\leqslant57\bigl(N_f^{[1]}(r,C_1)+N_f^{[1]}(r,C_2)\bigr) +o(T_f(r))\ \|. \] For a conic and a quintic one may take the explicit coefficient $45$, and for a line and an octic one may take $69$.

Motivated by Landis's conjecture on unique continuation at infinity for the Laplacian, we study the corresponding property for the Cauchy-Riemann operator. We prove that every weak solution of $\bar\partial u=Vu$ on a neighborhood of infinity, with $V\in L^\infty$, vanishes identically if it decays exponentially at a rate greater than $ 2\|V\|_{L^\infty}$. This conclusion is sharp both in the constant $2\|V\|_{L^\infty}$ and in the order of exponential decay required. More generally, we establish unique continuation at infinity for a broad class of radially decaying bounded potentials, with optimal decay rates determined by the decay of the potential. We also obtain related unique continuation results for $L^2$ potentials and for compactly supported potentials under weaker assumptions at infinity.

We consider families of pseudoconvex domains $X[s]\subset \mathbb C^n$ parametrized by complex manifolds $S$, and the maximal hermitian ellipsoids inscribed in each $X[s]$. The paper investigates how these maximal ellipsoids and their volume vary with $s$.

Let $\bar{V}$ be a connected smooth complex projective variety. Let $D \subset \bar{V}$ be a normal crossing divisor. Let $\mathbb{V}$ be a complex polarizable variation of Hodge structure on $V := \bar{V} \setminus D$. Suppose that the period map of $\mathbb{V}$ is immersive at a point of $V$. We prove that for every integer $p$ with $1 \leq p \leq \dim V$, the vector bundle $\Omega_{\bar{V}}^p(\log D)$ is L-big (i.e. the tautological line bundle on $\mathbb{P}\Omega_{\bar{V}}^p(\log D)$ is big). If the local monodromy is quasi-unipotent, we give a method to determine an $m \in \mathbb{N}$ such that if $p > m$, then $\Omega_{\bar{V}}^p(\log D)$ is moreover Viehweg-big. We give the optimal value of $m$ explicitly when $V$ is a locally symmetric variety. We prove that if $V$ is a finite étale cover of the fine moduli space of smooth quintic threefolds, the result holds for $m = 90$ (in this case $\dim V = 101$). The proof of the previous results is based on a study of an augmented base locus associated with $\Omega_{\bar{V}}^p(\log D)$. In the case of locally symmetric varieties, we introduce “higher degree characteristic subvarieties”, generalizing the characteristic subvariety defined by Mok in the case $p = 1$, and prove that they coincide with these augmented base loci.

For a $3$-dimensional compact Kähler manifold $X$ with a pair of Kähler class $(\alpha,\beta)$ which is $J$-semistable, we show that the $J$-null locus of $(\alpha,\beta)$ forms a subvariety of $X$. We also give an analyic characterization of the $J$-null locus using Kähler currents with analytic singularities analogous to the theorem of Collins-Tosatti \cite{CT15}. As an application, we show the smooth convergence of the $J$-flow off the $J$-null locus. It gives higer order partial regularity for weak solutions of semistable $J$-equations obtained in \cite{M26b}.

We develop an extension-by-approximation principle for holomorphic Runge embeddings of increasing union of Stein manifolds into Stein manifolds with density property. The basic hypothesis is the existence, on each stage of the exhaustion, of a Runge isotopy which compresses the stage and whose terminal map extends holomorphically to the next stage. The resulting global embedding of the union may be chosen with Runge image, and every Runge embedding of a fixed stage can be approximated uniformly on compact subsets by the Runge embeddings of the union. We apply this principle to domains that are invariant under positive time part of holomorphic $(R,+)$-actions, to Stein manifolds carrying a semicomplete holomorphic vector field with globally attracting fixed point. It also gives a Runge embedding of $(\mathbb{C}^n\setminus \{z\in\mathbb{C}^n: f(z)=0\})\times \mathbb{C}$ in $\mathbb{C}^{n+1}$, which generalizes previous result of Runge embedding of $(\mathbb{C}^*)^n\times\mathbb{C}$ into $\mathbb{C}^{n+1}$. We also construct Stein globalization of an injective holomorphic semigroup action to holomorphic $(R,+)$-action. Finally, the abstract Loewner range of a Herglotz vector field is shown to admit a same-dimensional Runge embedding whenever the initial domain admits a Runge embedding into a Stein domain with density property; this yields a corresponding solution of the Loewner PDE with values in $\mathbb{C}^n$. We also give an example of non-Runge complete hyperbolic domain which admits $\mathbb{C}^n$-valued solution of the Loewner PDE.

The main result of the paper shows that each connected polyhedral surface with a hyperbolic background metric, or a Euclidean background metric with uniformly bounded circumdisk radii, is discrete conformal to a complete constant-curvature Riemannian surface equipped with a nonempty closed discrete subset. We also prove a discrete Riemann mapping theorem. The proofs are based on the recent work on the discrete Schwarz lemma, the discrete Liouville theorem for polyhedral surfaces, and a Weyl-type realization theorem for hyperbolic surfaces.