Papers

Let $\Phi$ be a holomorphic IFS on a bounded domain in $\mathbb{C}$. Suppose that the following conditions hold: (1) the maps in $\Phi$ do not have a common fixed point; (2) there does not exist a regular real-analytic curve which is invariant under all of the maps in $\Phi$; (3) $\Phi$ is not holomorphically conjugate to a homothetic IFS; (4) $\Phi$ is exponentially separated. Under these assumptions, we show that the dimensions of the self-conformal measures associated to $\Phi$, as well as the Hausdorff dimension of the associated self-conformal set, attain their natural upper bounds. The proof combines recently developed methods from the dimension theory of stationary fractal measures with complex-analytic arguments.

We construct planar simply-connected domains whose torsion function has an arbitrary prescribed number of local maxima while the distance to the boundary has only one local maximum inside the domain. This disproves a conjecture of Magnanini asserting that the number of local maxima of the torsion function is bounded above by the number of local maxima of the distance function to the boundary. We also answer a question of Steinerberger concerning the eccentricity of level sets near the point where the torsion function attains its maximum in simply-connected domains.

Let $\Omega\subset\R^n$ be a domain and let $u\in W^{2,n}_{\loc}(\Omega)$ satisfy \[ \det D^2u\geq\delta>0\qquad\text{a.e. in }\Omega. \] Guerra and Tione asked whether the gradient mapping $Du$ must be open and discrete. We give a negative answer in every dimension $n\geq4$, in a form that isolates the sharp regularity mechanism. First, a local Pogorelov model is chosen to solve the exact equation $\det D^2u=1$ a.e. It is convex, belongs to $C^{1,1-2/n}\cap W^{2,n}_{\loc}$, and its Hessian is positive definite off a line. Nevertheless, its gradient collapses that line to one point. In fact the gradient is neither open nor discrete, its branch set is exactly the collapsed line, and its Jacobian equals one a.e. We then develop a $k$-dimensional version of the construction. If $1\leq k<n/2$, there are convex potentials with a $k$-dimensional flat contact set, uniformly positive and bounded Hessian determinant, and exact critical exponent \[ p_{n,k}=\frac{n(n-k)}{2k}. \] Their Hessians belong to $L^p_{\loc}$ precisely for $p<p_{n,k}$, lie in the weak endpoint space $L^{p_{n,k},\infty}_{\loc}$, and fail to belong to any finite-index Lorentz endpoint. This matches the critical minimum-set theorem of Collins and Mooney. At the regularity required in the question, the construction produces branch sets of every integer dimension $k<n/3$. We also give a topological reformulation of the problem. In the convex branch, gradient fibers are exactly contact sets with supporting affine functions, and openness and discreteness are equivalent to strict convexity. For a general gradient under the hypotheses above, critical Sobolev mapping theory already supplies continuity and sense preservation. The question asks whether the monotone factor in the Eilenberg–Whyburn monotone–light factorization is trivial.

We study properties of the calculus $\varphi\mapsto\varphi(A,B)$ for self-adjoint operators with commutator $[A,B]$ in the Schaten–von Neumann class $\boldsymbol{S}_p$. It turns out that this calculus defined on the Besov class $B_{\infty,1}^1({\Bbb R}^2)$ in the case $p\le2$ admits a commutator Lipschitz estimate and is multiplicative modulo $\boldsymbol{S}_p$. On the other hand in the case $p>2$ there is no commutator Lipschitz estimate. There is no commutator Lipschitz estimate in the operator norm as well.

In this manuscript, we establish sharp bounds of the third-order Toeplitz determinant and a particular case of the generalized Zalcman functional for a class of holomorphic mappings defined on the unit ball in a complex Banach space. The obtained estimates yield corresponding bounds for several subclasses of starlike mappings as special cases and also provide higher-dimensional extensions of certain known results from the classical one-dimensional theory.

We establish a new local-to-global rigidity phenomenon for the Bergman metric. Namely, under natural geometric hypotheses, a local conformal identification of Bergman metrics determines the underlying complex manifold globally, up to the unavoidable ambiguity of removing Bergman-negligible subsets. More precisely, let $\Omega\subseteq\mathbb C^n$ be a bounded domain with a complete Bergman metric, and suppose that the Bergman metric of a complex manifold $M$ is locally conformal, via a holomorphic map $f$, to that of $\Omega$. We prove that the given local map $f$ extends to a biholomorphism $F\colon M\to D$ onto a subdomain $D\subseteq\Omega$ in two complementary settings. If $M$ is Stein, then $\Omega\setminus D$ is a closed pluripolar set. If $M$ is a bounded domain and $\Omega$ satisfies a natural symmetry condition expressed in terms of its automorphism orbits, then $\Omega\setminus D$ is Bergman-negligible. In particular, this applies when $\Omega$ is a bounded homogeneous domain and yields a characterization, up to Bergman-negligible sets, of bounded domains with locally symmetric Bergman metrics. The latter answers a question raised by Loi–Palmieri and Zimmer. A key ingredient in the proof is a new Calabi-type extension theorem tailored to Bergman metrics.

We study holomorphic maps between complex manifolds that preserve the Bergman metric up to a positive constant. In the equal dimensional case, either the source is a Stein manifold and the target is a bounded domain in the complex Euclidean space, or the source is a bounded pseudoconvex domain and the target is a complex manifold, holomorphic Bergman isometry is a biholomorphism onto the complement of a relatively closed locally pluripolar set, and the constant is necessarily one. These include the bounded domains in the complex Euclidean space as special cases. Some applications to curvature-preserving maps are obtained. We further prove the rigidity result for local holomorphic Bergman isometries into Cartesian product of strongly pseudoconvex domains, with applications to finite analytic and modular correspondences.

Let $d\ge 1$ be an integer. We study $d$-polynomial convexity of smooth Jordan arcs in terms of their total absolute curvature $\T(K)$. We prove that every $\cC^2$-smooth Jordan arc $K\subset\C$ satisfying $\T(K)\le \frac{d-1}{d}\pi$ is $d$-polynomially convex. This bound is sharp: for every $\tau>\frac{d-1}{d}\pi$, there exists a smooth Jordan arc $K\subset\C$ such that $\T(K)<\tau$ and $K$ is not $d$-polynomially convex. We also show that, for $0<\alpha<\pi$, the circular arc $A_\alpha=\{e^{it}:|t|\le \alpha\}$ is $d$-polynomially convex if and only if $\alpha\le \frac{d-1}{d}\pi$.

Let $\D=\{z\in\C:|z|<1\}$, and let $\mathcal A$ be the class of analytic functions normalized by $f(0)=0$ and $f'(0)=1$. For an admissible Ma–Minda generator $\varphi$, write $\Sstar{\varphi}=\{f\in\mathcal A:zf'(z)/f(z)\prec\varphi(z)\}$. Given two generators $\varphi_1$ and $\varphi_2$, we study the largest $R\in(0,1]$ for which $f(rz)/r\in\Sstar{\varphi_2}$ whenever $f\in\Sstar{\varphi_1}$ and $0<r\le R$. We present the Geometric Function Atlas, a software system that records these directed radius problems and coefficient problems by their exact generators, parameter domains, normalizations, and sharpness statements. This representation identifies the same class across alternative names and transliterations while keeping the two directions of an inclusion problem distinct. The coefficient engine recovers all 216 Fekete–Szegő values predicted by the general Ma–Minda formula across 36 registered classes. The directed-radius atlas contains 702 ordered comparisons; omitting direction merges unequal constants in 253 of the 262 class-pair families represented in both directions. Using boundary contact, analytic majorants, and explicit Ma–Minda extremals, we prove nineteen exact sharp inclusion radii. In particular, the sine-to-modified-sigmoid radius is $\arcsin((e-1)/(e+1))$, improving the published sufficient radius $\operatorname{arsinh}((e-1)/(e+1))$ by 7.45%. For the crescent and exponential classes, the reciprocal sharp radii are $\sin1$ and $\log(1+\sqrt2)$; the latter corrects a published constant. The Python package, exact certificates, and registry records accompany the paper.

Let $(M,\omega)$ be a compact Hermitian manifold, and let $\Gamma$ be a symmetric convex cone. We develop a quantitative regularization method for $\Gamma$-admissible functions. As an application, we prove Hölder continuity for every pluripotential solution of complex $m$-Hessian equations whose right-hand sides belong to $L^p$, for $p>\frac{n}{m}$. These results extend to a more general class of complex Hessian equations satisfying the structural condition used by Guo–Phong–Tong.

Let $\sigma$ be a Poisson-finite measure on the real line. If its logarithmic integral diverges, then the functions from the classical Hardy space $H^2$ with compactly supported Fourier transform are dense in $L^2(\mathbb{R}, \sigma)$. We give a quantitative version of this result, adapting the ideas from the 2020 paper by Borichev, Kononova and Sodin, where a similar question was studied in the setting of polynomial approximation.

The empirical transfer function estimate (ETFE) is a widely used method for system identification of linear time-invariant (LTI) systems in engineering disciplines such as acoustics, audio engineering, seismography, and tomography. However, ETFE suffers from numerical limitations when the excitation signal is band-limited or vanishes at certain frequencies, which is a common physical constraint of the excitation in practice. In such cases, division in the frequency domain becomes heavily ill-conditioned, and small measurement disturbances or numerical inaccuracies can degrade the solution. This paper presents L-ETFE, a generalization of ETFE based on Laguerre-Fourier expansions that addresses these limitations. After a suitable transformation, the method can yield a well-conditioned circulant problem even when the original ETFE system is ill-conditioned. Solving this problem via classical ETFE yields the discrete Laguerre-Fourier coefficients of the system's transfer function. The desired impulse response (IR) of the system-to-be-identified can then be recovered by a subsequent transformation pipeline. We derive novel and efficient algorithms for performing these transformations and analyse the conditioning of the transformed problem, explicitly characterizing its dependence on the input and a parameter used in the Laguerre-Fourier expansion. We evaluate the method on two simulated discrete-time LTI systems of varying complexity. The experiments demonstrate accurate IR recovery for spectral-zero and band-limited excitation, where standard ETFE fails.

We study the zeros of functions in the Dirichlet space. Using extremal functions, we produce a necessary and sufficient condition for a sequence of points in the unit disk to be a zero set of the classical Dirichlet space. This Shapiro-Shields type condition involves kernels of the harmonic Dirichlet space associated with measures depending on the zero sequence.

The aim of this article is two-fold. First, we obtain a normal form for automorphisms of the escaping sets of Hénon maps in a neighborhood of the attracting fixed point at infinity. Building on this local description, we characterize the global automorphisms of $\mathbb{C}^2$ that preserve the escaping sets. The analytic structure of the escaping sets, as established by Hubbard–Oberste-Vorth (\cite{HOV}), plays a crucial role in the proof of these results. In a similar spirit, we investigate the analytic structure of the super-level sets of the Green's functions associated with Hénon maps and give a description of the automorphisms of these domains.

The Union Problem, which has its genesis in the classical Levi problem, asks for a classification of complex manifolds $M$ that can be exhausted by an increasing union of submanifolds $M_j \subset M$ which are all biholomorphic to a fixed domain in $\mathbb C^n$. We explore a quasiconformal variant of this question and seek to classify $n$-Riemannian manifolds $M$ such that each $M_j$ is quasiconformally equivalent to a bounded domain in $\mathbb R^n$. It turns out that this is possible when these quasiconformal equivalences have uniformly bounded dilatations. Using Kiernan's quasiconformal Schwarz lemma when $n=2$ and Ferrand's conformal capacity when $n \geq 3$, we classify a class of $n$-Riemannian manifolds $M$ such that each $M_j$ is $K_j$-quasiconformally equivalent to $\Omega \setminus A$, where $\sup K_j < \infty$ and $\Omega \subset \mathbb R^n$ is a $C^2$-smoothly bounded domain and $A \subset \Omega$ is at most finite. As a consequence, we obtain that Gehring's example of a bounded domain in $\mathbb R^n$ which has $C^1$-smooth boundary everywhere except at a point and is known to be quasiconformally inequivalent to the unit ball in $\mathbb R^n$, possesses the additional property that it cannot even be exhausted by quasiconformal images of the unit ball with uniformly bounded dilatations.

We compare two natural Hardy quantizations carried by the unit cosphere bundle of the round two-sphere. Through $S^2\simeq CP^1$, the cosphere bundle is the unit circle bundle of the canonical bundle $O(-2)$, and its Hardy space assembles the section spaces $H^0(CP^1,O(2\ell))$. Through the real-analytic round metric, the imaginary-time exponential map identifies the same cosphere bundle with every Grauert-tube boundary carrying the adapted complex structure, whose Reeb flow is the geodesic flow. Both Hardy spaces are realized in $L^2(SO(3))$, are multiplicity-free under the left action, and select one line in each Peter-Weyl multiplicity space. We compute the normalized overlap of these lines in closed form as $\frac{\sqrt{\binom{2\ell}{\ell}}}{2^\ell}\frac{\sinh^\ell\tau}{\sqrt{P_\ell(\cosh 2\tau)}}$, with $P_\ell$ the Legendre polynomial. The normalized kernel vectors define an explicit equivariant unitary between the two Hardy spaces, and the squared overlaps are the eigenvalues of the positive trace-class operator $\Pi_h\Pi_\tau\Pi_h$. We derive four exact trace series with the Hardy projectors inserted and show that their sum differs from the full flat trace by a distribution whose Abel regularization has cubic growth at every geodesic period. The eigenvalues also give a genus-zero Fredholm determinant of order zero, and truncating the complete large-$\ell$ expansion at any fixed order gives a finite polylogarithmic expression. A Bargmann-Fock calculation identifies the $\ell$-independent prefactor $(2\sinh 2\tau)^{1/4}e^{-\tau/2}$ in the large-$\ell$ asymptotic of the overlap with the Gaussian matrix coefficient of a metaplectic operator comparing the two contact planes.

For a real polynomial $F$ and $\omega\geq 0$, set $A_\omega(z)=(F(z+i\omega)+F(z-i\omega))/2$ and $\Omega_F=\{\omega\geq0:A_\omega\text{ has only real zeros}\}$. We present an explicit rational even polynomial of degree eight for which $6/25$ and $12/25$ belong to $\Omega_F$, while $3/10$ does not. Exact Sturm certificates give respectively eight, four, and eight distinct real zeros. Consequently $\Omega_F$ is neither an interval nor an up-set. All zeros of $F$ lie in the strip $|\operatorname{Im}z|\leq11/25$, and the classical strip-contraction theorem gives the eventual tail $[11/25,\infty)\subset\Omega_F$. We also include a direct elementary proof of that tail and a standard-library exact verifier.

We study two regularity problems on Hardy-type analytic tent spaces $\mathcal{AT}^p_{q,\alpha}$ on the unit disk: fractional integration and randomization of Taylor coefficients. For fractional integration, we characterize completely the boundedness and compactness of the Hadamard, Flett, and Riemann–Liouville operators between analytic tent spaces, and obtain parallel results for analytic Triebel–Lizorkin spaces. In particular, the case $t=0$ yields a complete solution to the corresponding embedding problem for analytic tent spaces. For randomization, we characterize completely when the random Taylor series $\mathcal{R}f$ belongs almost surely to an analytic tent space whenever $f\in \mathcal{AT}^p_{q,\alpha}$, and we also obtain the Triebel–Lizorkin counterpart. As part of the proof, we identify the random symbol space associated with $\mathcal{AT}^p_{q,\alpha}$ and solve the embedding problem from analytic tent spaces into mixed norm spaces. These results extend classical theorems of Hardy–Littlewood and Littlewood, as well as their later analogues for Bergman and mixed norm spaces.

Let $\mu$ be a positive Borel measure on $\mathbb C^n$. We prove that, for every $0<p<\infty$, the Toeplitz operator $T_\mu^{\mathrm{ph}}$ induced by $\mu$ on the pluriharmonic Fock space belongs to the Schatten class $S_p$ if and only if the local mass function $z\mapsto\mu(B(z,r))$ belongs to $L^p(\mathbb C^n)$ for one, or equivalently every, $r>0$. For $n\geq2$, this resolves a conjecture of Jaguzovi'c and Vujadinovi'c, and the range $0<p<1$ is new in every dimension. Writing $T_\mu$ for the corresponding holomorphic Toeplitz operator, we obtain the sharp estimates $$\|T_\mu\|_{S_p}^p \leq \|T_\mu^{\mathrm{ph}}\|_{S_p}^p \leq 2^{\max\{1,p\}} \|T_\mu\|_{S_p}^p. $$ We also prove a sharp comparison with constant two in every symmetrically normed ideal and an exact trace formula, using positivity and a $2\times2$ block decomposition whose diagonal blocks are $T_\mu$ and an antiunitary copy of its compression to the functions orthogonal to constants. We then consider generalized Fock weights satisfying $m \, dd^c|z|^2\le dd^c\phi\le M \, dd^c|z|^2$. For the canonical holomorphic and antiholomorphic direct sum norm, the same Schatten and symmetrically normed ideal estimates hold. For the norm inherited from $L^2(\mathbb C^n,e^{-2\phi}dV)$, the local mass criterion also holds whenever $e^{-2\phi}$ is comparable to a generalized Fock weight invariant under the scalar circle action. Without further assumptions, the local mass criterion can fail for the inherited norm: in one complex dimension, we construct a weight of the form $\phi(z)=|z|^2/2+\operatorname{Re}q(z)$, with $q$ entire, and a finite positive measure whose local masses belong to every $L^p$, although the Toeplitz form on the inherited pluriharmonic space is unbounded.

We describe the asymptotics of Chebyshev polynomials on an analytic Jordan arc in the plane. This gives an affirmative answer to a conjecture of Christiansen-Simon-Zinchenko, based on predictions of Widom from 1969. The proof combines weighted Faber polynomials with extremal signatures, discrete orthogonal polynomials and a Marcinkiewicz-Zygmund sampling inequality, and yields Szegő-Widom asymptotics for the Chebyshev polynomials themselves.

In this paper, we introduce the notion of a distinguished variety in the domain $\mathbf{\Theta}_n$. One of the main results of the paper is a determinantal representation for every distinguished variety in $\mathbf{\Theta}_n$. We also show that the closure of every distinguished variety is polynomially convex. Furthermore, we obtain a dilation and a functional model for a class of pure $\mathbf{\Theta}_n$-contractions. Finally, we show that for a $\mathbf{\Theta}_n$-contraction $\mathbf{T}=(T_1,\dots,T_n)$ such that $T_n^*$ is a pure contraction, there exists an algebraic variety in $\mathbf{\Theta}_n$ for which the von Neumann inequality holds on the intersection of the closure of the variety with the distinguished boundary of $\mathbf{\Theta}_n$.

In this paper, we establish new fixed point theorems for multivalued mappings satisfying Hardy-Rogers type contractive conditions in complete $b$-metric spaces. Motivated by the constant-coefficient framework of Chaib et al. (Fixed Point Theory 27 (2026), 209-228), we develop a distance-dependent formulation in which the contractive coefficients are nonnegative functions of the interpoint distance. We introduce a piecewise contraction factor corresponding to the two cases determined by the relative sizes of the cross terms in the Hardy-Rogers inequality, thereby providing a formulation adapted to the structure of the contractive condition. Under appropriate assumptions on the coefficient functions, we establish fixed point existence through either $h$-upper semicontinuity or asymptotic limsup conditions. The resulting framework yields corresponding fixed point theorems for multivalued Reich, Kannan, and Chatterjea-type contractions. We further investigate strict fixed points and obtain conditions under which the fixed point and strict fixed point sets coincide and consist of a unique point. In addition, we establish a fixed point transfer result for suitable finite iterates, showing that the contractive condition may be imposed on $T^p$ rather than on the original mapping, provided that the latter has a strict fixed point. Corresponding single-valued consequences are also derived. Several examples illustrate the applicability of the results and demonstrate situations in which the proposed conditions apply beyond the corresponding existing formulations.

For $k\ge 2$, let $M_k=[z^k-w^k]$ be the principal homogeneous submodule of the Hardy space over the bidisk. We determine Yang's complete sequence of numerical invariants and prove $$ \Sigma_0(M_k)=\frac{\pi^2}{6},\qquad \Sigma_j(M_k)=\Sigma_{\lceil j/k\rceil}([z-w]),\quad j\ge1. $$ The proof exploits a residue-class decomposition of the Toeplitz matrices associated with the graded wandering spaces. Consequently, Yang's monotonicity conjecture holds for the family $\{M_k:k\ge2\}$. We also show that the nonzero spectral data of the core operator are independent of $k$, whereas the numerical invariant sequence recovers $k$ from the length of its constant blocks. Thus the higher numerical invariants detect module-theoretic information invisible to the core spectrum.

The objective of this work is to determine optimal local restrictions on the coefficients of integer and Gaussian integer continued fractions that imply convergence. We identify all minimal restrictions involving words of length two in the integer case, and we identify all reversible minimal restrictions of length two in the Gaussian integer case. In the integer setting, our classification is equivalent to a classification of minimal unavoidable words of length two in Conway–Coxeter quiddity sequences. We also construct a canonical set of restrictions of infinite cardinality that is strictly stronger than every finite set of restrictions.

Let $\Omega \subset \mathbb{C}^{n}$ and $\xi \in \ell^{1}$. The $\xi$-Bergman kernel $K_{\xi, \Omega}$, introduced by Bao and Guan, generalizes the classical Bergman kernel by replacing the point evaluation functional with a functional determined by sequence $\xi$. While this kernel inherits several important extremal and plurisubharmonic properties, it is intrinsically an on-diagonal object and therefore lacks the two-variable reproducing kernel structure that lies at the heart of the classical Bergman theory. The purpose of this paper is to associate a canonical Hermitian positive-definite kernel with the $\xi$-Bergman kernel and to investigate its analytic and geometric properties. Our construction is based on the family of Riesz representatives corresponding to the $\xi$-evaluation functionals. More precisely, we introduce a Hermitian kernel obtained as the Gram kernel of these representatives and show that it is positive definite and for $z \in \Omega$ satisfies \[ B_{\xi,\Omega}(z,z)=K_{\xi,\Omega}(z), \] thereby recovering the $\xi$-Bergman kernel as its diagonal restriction. As a consequence, we prove that the $\xi$-Bergman kernel is real analytic on $\Omega$. We also establish biholomorphic transformation laws, and obtain a representation of the $\xi$-Bergman kernel in terms of derivatives of the classical Bergman kernel. Furthermore, we obtain explicit formulas for the $\xi$-Bergman kernel on the upper half-plane $\mathbb{H}$ corresponding to several classes of sequences $\xi$, establish corresponding $\xi$-Lu Qi-Keng results, and derive precise boundary asymptotics. These examples illustrate how the choice of the differential functional influences both the zero set and the boundary growth of the associated kernel.

We show that for every conformal map $\phi:\mathbb{D}\to\Omega\subsetneq\mathbb{C}$ from the unit disk onto a simply connected proper domain $\Omega$, the length of $\phi^{-1}(\Omega\cap L)$ is at most $\pi^2$ for every line $L$.

We disprove the conjecture of Gau, Wang and Wu that the numerical range of a finite-dimensional partial isometry, when circular, must be centred at the origin. More precisely, we prove that there is an \(\varepsilon>0\) such that, for every \(a\in(0,\varepsilon)\), one can find a rank-four partial isometry \(V_a\in\M_6(\R)\) satisfying $$ W(V_a)=\{\zeta\in\C:|\zeta-a|\leq 3/4\}. $$

We show that every diffeomorphism of $\mathbb{R}^n$ for $n \ge 2$ can be approximated by an automorphism of $\mathbb{C}^n$ in the Whitney $C^k$-topology for any positive integer $k$ using the notion of density property. More precisely we find sufficient conditions for this holomorphic approximation of diffeomorphisms to hold and prove that the split real forms of most linear algebraic groups satisfy these conditions. In the same manner we also show holomorphic approximations of volume-preserving diffeomorphisms for the split real forms of linear algebraic groups equipped with the left-invariant volume form.

In this paper we prove the Corona theorem for slice hyperholomorphic functions in a general way that applies to functions with values in a Clifford algebra $\mathbb R_n$ for any $n\geq 2$. This general proof is based on a matrix version of Corona's theorem and then exploiting some symmetries of the matrices that allow to reconstruct the values of the functions. The crucial idea is to tie together the algebra of the functions, the Bezout identities and the Carleson-type conditions. The results we obtain rest on a new way of writing the slice product which improves the standard approach and which holds in general dimension.

Lewy's theorem says that a one-to-one harmonic mapping between plane domains has nonvanishing Jacobian. In dimensions at least three this statement is false for general harmonic homeomorphisms. We prove a four-dimensional Lewy theorem under the additional complex-analytic assumption of pluriharmonicity. More precisely, if \[ F:\Omega\subset \mathbb C^2\to \mathbb C^2 \] is a \(C^2\) pluriharmonic mapping which is one-to-one in a neighborhood of a point \(p\), then the real Jacobian of \(F\) is nonzero at \(p\). The proof is local. If the Jacobian vanished, a nontrivial real linear projection of \(F\) would be the real part of a holomorphic function \(f\) with \(df(p)=0\). The level hypersurface of this projection would therefore be a real analytic germ of the form \[ \{\operatorname{Re} f=0\}. \] We prove that such a germ cannot be locally flat at a critical point of \(f\). The obstruction is topological: local flatness forces the local homology of the hypersurface germ to agree with that of a real hyperplane, and hence forces every sufficiently small admissible link to have the integral homology, in particular the Euler characteristic, of \(S^2\). In the reduced case the Milnor open book of the plane curve singularity \(f^{-1}(0)\) gives a contradictory Euler-characteristic formula, while in the nonreduced case the local normal form produces more than two local complementary components. We then prove a separate bi-Lipschitz criterion for harmonic quasiconformal mappings: a harmonic quasiconformal homeomorphism from the unit ball onto a bounded \(C^1\)-Dini domain is bi-Lipschitz, provided it is already a local \(C^1\)-diffeomorphism.