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We describe an algorithm that constructs a list of all topological types of holomorphic actions of a finite group on a compact Riemann surface $C$ of genus at least $g \geq 2$ with $C/G \cong \mathbb{P}^1$., v2: corrected definition of Hurwitz equivalence using the braid group

We prove a sharp Schwarz type inequality for the Weierstrass- Enneper representation of the minimal surfaces. It states the following. If $F:\mathbf{D}\to \Sigma$ is a conformal harmonic parameterization of a minimal disk $\Sigma$, where $\mathbf{D}$ is the unit disk and $|\Sigma|=\pi R^2$, then $|F_x(z)|(1-|z|^2)\le R$. If for some $z$ the previous inequality is equality, then the surface is an affine disk, and $F$ is linear up to a M\"obius transformation of the unit disk., Comment: 6 pages

We consider the Weil-Petersson gradient vector field of renormalized volume on the deformation space of convex cocompact hyperbolic structures on (relatively) acylindrical manifolds. In this paper we prove the conjecture that the flow has a global attracting fixed point at the structure $M_{\rm geod}$ the unique structure with minimum convex core volume., Comment: 36 pages, 7 figures. Comments are welcome!

Examples by Poletsky and the author and by Zwonek show the existence nowhere extendable holomorphic functions with the property that the pluripolar hull of their graphs is much larger than the graph of the respective functions and contains multiple sheets. We will explain this phenomenon by fine analytic continuation of the function over part of a Cantor-type set involved. This gives more information on the hull, and allows for weakening and effectiveness of the conditions in the original examples., Comment: 12 pages. In version 2 quite some annoying typo's and mistakes were corrected. In version 3 a serious gap has been filled

The extension of the Hardy-Knopp-Carleman inequality to several classes of means was the subject of numerous papers. In the class of Gini means the Hardy property was characterized in 2015 by the second author. The precise value of the associated Hardy constant was only established for concave Gini means by the authors in 2016. The determination Hardy constant for nonconcave Gini means is still an open problem. The main goal of this paper is to establish sharper upper bounds for the Hardy constant in this case. The method is to construct a homogeneous and concave quasideviation mean which majorizes the nonconcave Gini mean and for which the Hardy constant can be computed.

We classify generic unfoldings of germs of antiholomorphic diffeomorphisms with a parabolic point of codimension 1 (i.e. a double fixed point) under conjugacy. These generic unfolding depend on one real parameter. The classification is done by assigning to each such germ a weak and a strong modulus, which are unfoldings of the modulus assigned to the antiholomorphic parabolic point. The weak and the strong moduli are unfoldings of the \'Ecalle-Voronin modulus of the second iterate of the germ which is a real unfolding of a holomorphic parabolic point. A preparation of the unfolding allows to identify one real analytic canonical parameter and any conjugacy between two prepared generic unfoldings preserves the canonical parameter. We also solve the realisation problem by giving necessary and sufficient conditions for a strong modulus to be realized. This is done simultaneously with solving the probem of the existence of an antiholomorphic square root to a germ of generic analytic unfolding of a holomorphic parabolic germ. As a second application we establish the condition for the existence of a real analytic invariant curve., Comment: 43 pages, 18 figures

In this article, we establish sharp weak endpoint estimates for $p$-adic fractional Hardy operator. In addition, sharp weak bounds for p-adic Hardy operator on p-adic central Morrey space are also obtained., 9 pages

We study quasiconformal mappings of the unit disk that have planar extension with controlled distortion. For these mappings we prove a bound for the modulus of continuity of the inverse map, which somewhat surprisingly is almost as good as for global quasiconformal maps. Furthermore, we give examples which improve the known bounds for the three point property of generalized quasidisks. Finally, we establish optimal regularity of such maps when the image of the unit disk has cusp type singularities., Revision based on referee's comments

For a linear subvariety $M$ of a stratum of meromorphic differentials, we investigate its closure in the multi-scale compactification constructed by Bainbridge-Chen-Gendron-Grushevsky-M\"oller. We prove various restrictions on the type of defining linear equations in period coordinates for $M$ near its boundary, and prove that the closure is locally a toric variety. As applications, we give a fundamentally new proof of a generalization of the cylinder deformation theorem of Wright to the case of meromorphic strata, and construct a smooth compactification of the Hurwitz space of covers of the Riemann sphere., Comment: v2: updated version with minor edits, accepted to Geometry&Topology

We give an alternate proof to the following generalization of the uniformization theorem by Bonk and Kleiner. Any linearly locally connected and Ahlfors 2-regular closed metric surface is quasisymmetrically equivalent to a model surface of the same topology. Moreover, we show that this is also true for surfaces as above with non-empty boundary and that the corresponding map can be chosen in a canonical way. Our proof is based on a local argument involving the existence of quasisymmetric parametrizations for metric discs as shown in a paper of Lytchak and Wenger.

We investigate the meromorphic quasi-modular forms and their $L$-functions. We study the space of meromorphic quasi-modular forms. Then we define their $L$-functions by using the technique of regularized integral. Moreover, we give an explicit formula for the $L$-functions. As an application, we obtain some vanishing results of special $L$-values of meromorphic quasi-modular forms.

In this article, we mainly consider the existence of solutions and global Mittag-Leffler stability of delayed fractional-order coupled reaction-diffusion neural networks without strong connectedness. Using the Leary-Schauder's fixed point theorem and the Lyapunov method, some criteria for the existence of solutions and global Mittag-Leffler stability are given. Finally, the correctness of the theory is verified by a numerical example.

A subset $E$ of a metric space $X$ is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into $\mathbb{R}^n$ for some $n$, sending $E$ to a starlike set. A subset $E\subset X$ is said to be recursively starlike-equivalent if it can be expressed as a finite nested union of closed subsets $\{E_i\}_{i=0}^{N+1}$ such that $E_{i}/E_{i+1}\subset X/E_{i+1}$ is starlike-equivalent for each $i$ and $E_{N+1}$ is a point. A decomposition $\mathcal{D}$ of a metric space $X$ is said to be recursively starlike-equivalent, if there exists $N\geq 0$ such that each element of $\mathcal{D}$ is recursively starlike-equivalent of filtration length $N$. We prove that any null, recursively starlike-equivalent decomposition $\mathcal{D}$ of a compact metric space $X$ shrinks, that is, the quotient map $X\to X/\mathcal{D}$ is the limit of a sequence of homeomorphisms. This is a strong generalisation of results of Denman-Starbird and Freedman and is applicable to the proof of Freedman's celebrated disc embedding theorem. The latter leads to a multitude of foundational results for topological $4$-manifolds, including the $4$-dimensional Poincar\'{e} conjecture., Comment: 11 pages, 2 figures

In this paper we study the field of definition of abelian subvarieties $B\subset A_{\overline{K}}$ for an abelian variety $A$ over a field $K$ of characteristic $0$. We show that, provided that no isotypic component of $A_{\overline{K}}$ is simple, there are infinitely many abelian subvarieties of $A_{\overline{K}}$ with field of definition $K_A$, the field of definition of the endomorphisms of $A_{\overline{K}}$. This result combined with earlier work of R\'emond gives an explicit maximum for the minimal degree of a field extension over which an abelian subvariety of $A_{\overline{K}}$ is defined with varying $A$ of fixed dimension and $K$ of characteristic $0$., Comment: Comments welcome

A bosonic Laplacian, which is a generalization of Laplacian, is constructed as a second-order conformally invariant differential operator acting on functions taking values in irreducible representations of the special orthogonal group, hence of the spin group. In this paper, we firstly introduce some properties for homogeneous polynomial null solutions to bosonic Laplacians, which give us some important results, such as an orthogonal decomposition of the space of polynomials in terms of homogeneous polynomial null solutions to bosonic Laplacians, etc. This work helps us to introduce Bergman spaces related to bosonic Laplacians, named as bosonic Bergman spaces, in higher spin spaces. Reproducing kernels for bosonic Bergman spaces in the unit ball and a description of bosonic Bergman projection are given as well. At the end, we investigate bosonic Hardy spaces, which are considered as generalizations of harmonic Hardy spaces. Analogs of some well-known results for harmonic Hardy spaces are provided here. For instance, connections to certain complex Borel measure spaces, growth estimates for functions in the bosonic Hardy spaces, etc.

In this paper we determine the automorphism groups of the profinite braid groups with four or more strings in terms of the profinite Grothendieck-Teichm\"uller group.

The notion of a subnexus based on ${\mathcal{N}}$-function (briefly, ${\mathcal{N}}$-subnexus) is introduced, and related properties are investigated. Also, the notions of ${\mathcal{N}}$-subnexus of type $(\alpha, \beta)$, where $(\alpha, \beta)$ is $(\in, \in)$, $(\in, q)$, $(\in, \in\! \vee \, {q})$, $(q, \in)$, $(q,q)$, $(q, \in\! \vee \, {q})$, $(\overline{\in}, \overline{\in})$ and $(\overline{\in}, \overline{\in} \vee \overline{q})$, are introduced, and their basic properties are investigated. Conditions for an ${\mathcal{N}}$-structure to be an ${\mathcal{N}}$-subnexus of type $(q, \in\! \vee \, {q})$ are given, and characterizations of ${\mathcal{N}}$-subnexus of type $(\in, \in\! \vee \, {q})$ and $(\overline{\in}, \overline{\in} \vee \overline{q})$ are provided. Homomorphic image and preimage of ${\mathcal{N}}$-subnexus are discussed.

We give improvements and generalizations of both the classical Hardy inequality and the geometric Hardy inequality based on the divergence theorem. Especially, our improved Hardy type inequality derives both two Hardy type inequalities with best constants. Besides, we improve two Rellich type inequalities by using the improved Hardy type inequality., 30 pages

Invariants for Riemann surfaces covered by the disc and for hyperbolic manifolds in general involving minimizing the measure of the image over the homotopy and homology classes of closed curves and maps of the $k$-sphere into the manifold are investigated. The invariants are monotonic under holomorphic mappings and strictly monotonic under certain circumstances. Applications to holomorphic maps of annular regions in $\mathbb{C}$ and tubular neighborhoods of compact totally real submanifolds in general in $\mathbb{C}^n$, $n \geq 2$, are given. The contractibility of a hyperbolic domain with contracting holomorphic mapping is explained., Comment: 18 pages, 1 diagram

The main purpose of this article is to present a localization of Forelli's theorem for the functions holomorphic along a standard suspension of linear discs. This generalizes one of the main results of \cite{CK21} and the original Forelli's theorem., 10 pages, 0 figure. The article is to appear in Complex Variables and Elliptic Equations

In this paper, we give a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh--Sibony measure) in terms of potential theory. This requires the theory of logarithmic potentials in the presence of an external field, which, in our case, is explicitly determined by the choice of a set of generators. Along the way, we establish the continuity of the logarithmic potential for the Dinh--Sibony measure, which might be of independent interest. We then use the $F$-functional of Mhaskar and Saff to discuss bounds on the capacity and diameter of the Julia sets of such semigroups., 25 pages; added some remarks in Section 1 and an additional reference; to appear in Journal d'Analyse Math\'{e}matique

This paper deals with a limiting case motivated by contact geometry. The limiting case of a tensorial characterization of contact hypersurfaces in Kahler manifolds leads to Hopf hypersurfaces whose maximal complex subbundle of the tangent bundle is integrable. It is known that in non-flat complex space forms and in complex quadrics such real hypersurfaces do not exist, but the existence problem in other irreducible Kahler manifolds is open. In this paper we construct explicitly a one-parameter family of homogeneous Hopf hypersurfaces, whose maximal complex subbundle of the tangent bundle is integrable, in a Hermitian symmetric space of non-compact type and rank two. These are the first known examples of such real hypersurfaces in irreducible Kahler manifolds., Comment: 34 pages; change of title; minor changes in text; to appear in Annali di Matematica Pura ed Applicata

A real hypersurface in $\mathbb{C}^2$ is said to be Reinhardt if it is invariant under the standard $\mathbb{T}^2$-action on $\mathbb{C}^2$. Its CR geometry can be described in terms of the curvature function of its ``generating curve'', i.e., the logarithmic image of the hypersurface in the plane $\mathbb{R}^2$. We give a sharp upper bound for the first positive eigenvalue of the Kohn Laplacian associated to a natural pseudohermitian structure on a compact and strictly pseudoconvex Reinhardt real hypersurface having closed generating curve (which amounts to the $\mathbb{T}^2$-action being free). Our bound is expressed in terms of the $L^2$-norm of the curvature function of the generating curve and is attained if and only if the curve is a circle., 11 pages

In this paper, by introducing the notion of "\textit{distributive constant}" of a family of hypersurfaces with respect to a projective variety, we prove a second main theorem in Nevanlinna theory for meromorphic mappings with arbitrary families of hypersurfaces in projective varieties. Our second main theorem generalizes and improves previous results for meromorphic mappings with hypersurfaces, in particular for algebraically degenerate mappings and for the families of hypersurfaces in subgeneral position. The analogous results for the holomorphic curves with finite growth index from a complex disc into a project variety, and for meromorphic mappings on a complete K\"{a}hler manifold are also given. For the last aim, we will prove a Schmidt's subspace theorem for an arbitrary families of homogeneous polynomials, which is the counterpart in Number theory of our second main theorem. Our these results are generalizations and improvements of all previous results., Comment: Some typos are cleared. arXiv admin note: text overlap with arXiv:1712.05698

Eremenko and Lyubich proved that an entire function whose set of singular values is bounded is expanding at points where its image has large modulus. These expansion properties have been at the centre of the subsequent study of this class of functions, now called the Eremenko-Lyubich class. We improve the estimate of Eremenko and Lyubich, and show that the new estimate is asymptotically optimal. As a corollary, we obtain an elementary proof that functions in the Eremenko-Lyubich class have lower order at least $1/2$., 5 pages. V2: Author accepted manuscript; minor corrections from V1. To appear in Bull. London Math. Soc

We study representations of inner functions on the bidisc from a fractional linear transformation point of view, and provide sufficient conditions, in terms of colligation matrices, for the existence of two-variable inner functions. Here the sufficient conditions are not necessary in general, and we prove a weak converse for rational inner functions that admit one variable factorization. We present a complete classification of de Branges-Rovnyak kernels on the bidisc (which equally works in the setting of polydisc and the open unit ball of $\mathbb{C}^n$, $n \geq 1$). We also classify, in terms of Agler kernels, two-variable Schur functions that admit one variable factor., Comment: 25 pages, minor revision. To appear in Computational Methods and Function Theory (CMFT)

We compute the hybrid limit (in the sense of Boucksom-Jonsson) of the family of K\"ahler-Einstein volume forms on a degeneration of canonically polarized manifolds. The limit measure is a weighted sum of Dirac masses at divisorial valuations, determined by the natural algebro-geometric limit of the family. We also make some remarks on the non-archimedean Monge-Amp\`ere operator and hybrid continuity of K\"ahler-Einstein potentials in this context.

In this paper, we obtain the criterion of boundedness of maximal operators associated with smooth hypersurfaces. Also we compute the exact value of the boundedness index of such operators associated with arbitrary convex analytic hypersurfaces in the case where the height of a hypersurface in the sense of A. N. Varchenko is greater than 2. Moreover, we obtain the exact value of the boundedness index for degenerated smooth hypersurfaces, i.e., for hypersurfaces satisfying conditions of the classical Hartman-Nirenberg theorem. The obtained results justify the Stein-Iosevich-Sawyer hypothesis for arbitrary convex analytic hypersurfaces as well as for smooth degenerated hypersurfaces. Also we discuss some related problems of the theory of oscillatory integrals.

We prove the convergence of the normalized Fubini-Study measures and the logarithms of the Bergman kernels of various Bergman spaces of holomorphic and weakly holomorphic sections associated to a singular Hermitian holomorphic line bundle on an algebraic curve. Using this, we study the asymptotic distribution of the zeros of random sequences of sections in these spaces., 21 pages; Dedicated to Professor Ahmed Zeriahi on the occasion of his retirement

We study metric spheres (Z,dZ) obtained by gluing two hemispheres of S2 along an orientation-preserving homeomorphism g:S1→S1, where dZ is the canonical distance that is locally isometric to S2 off the seam. We show that if (Z,dZ) is quasiconformally equivalent to S2, in the geometric sense, then g is a welding homeomorphism with conformally removable welding curves. We also show that g is bi-Lipschitz if and only if (Z,dZ) has a 1-quasiconformal parametrization whose Jacobian is comparable to the Jacobian of a quasiconformal mapping h:S2→S2. Furthermore, we show that if g−1 is absolutely continuous and g admits a homeomorphic extension with exponentially integrable distortion, then (Z,dZ) is quasiconformally equivalent to S2. peerReviewed

application/pdf, We study here a proof of the Riemann Hypothesis., 論文(Article)

The Geometric Shafarevich Conjecture and the Theorem of de Franchis state the finiteness of the number of certain holomorphic objects on closed or punctured Riemann surfaces. The analog of these kind of theorems for Riemann surfaces of second kind is an estimate of the number of irreducible holomorphic objects up to homotopy (or isotopy, respectively). This analog can be interpreted as a quantitatve statement on the limitation for Gromov's Oka principle. For any finite open Riemann surface $X$ (maybe, of second kind) we give an effective upper bound for the number of irreducible holomorphic mappings up to homotopy from $X$ to the twice punctured complex plane, and an effective upper bound for the number of irreducible holomorphic torus bundles up to isotopy on such a Riemann surface. The bound depends on a conformal invariant of the Riemann surface. If $X_{\sigma}$ is the $\sigma$-neighbourhood of a skeleton of an open Riemann surface with finitely generated fundamental group, then the number of irreducible holomorphic mappings up to homotopy from $X_{\sigma}$ to the twice punctured complex plane grows exponentially in $\frac{1}{\sigma}$., Comment: 51 pages, 4 figures

In this paper, we prove a uniform approximation theorem with interpolation for complete conformal minimal surfaces with finite total curvature in the Euclidean space $\mathbb{R}^n$ $(n\ge 3)$. As application, we obtain a Mittag-Leffler type theorem for complete conformal minimal immersions $M\to\mathbb{R}^n$ on any open Riemann surface $M$., To appear in Analysis & PDE

In the present paper, we investigate and introduce several properties of certain families of analytic functions in the open unit disc, which are defined by q -analogue of Mittag-Leffler function associated with conic domain. A number of coefficient estimates of the functions in these classes have been obtained. Sufficient conditions for the functions belong to these classes are also considered.

We present different constructions of abstract boundaries for bounded complete (Kobayashi) hyperbolic domains in ${\mathbb C}^d$, $d \geq 1$. These constructions essentially come from the geometric theory of metric spaces. We also present, as an application, some extension results concerning biholomorphic maps.

We revisit construction of the Atiyah-Hitchin manifold in the generalized Legendre transform approach. This is originally studied by Ivanov and Rocek and is subsequently investigated more by Ionas, in the latter of which the explicit forms of the K\"ahler potential and the K\"ahler metric are calculated. There is a difference between the former and the latter. In the generalized Legendre transform approach, a K\"ahler potential is constructed from the contour integration of one function with holomorphic coordinates. The choice of the contour in the latter is different from the former's one, whose difference may yield a discrepancy in the K\"ahler potential and eventually in the K\"ahler metric. We show that the former only gives the real K\"ahler potential, which is consistent with its definition, while the latter yields the complex one. We derive the K\"ahler potential and the metric for the Atiyah-Hitchin manifold in terms of holomorphic coordinates for the contour considered by Ivanov and Ro\v{c}ek for the first time., Comment: 30 pages, 3 figures. Typos corrected. References added. To appear in PTEP

The classes $\mathcal D _{\mathcal Q}$ of flat relative Mittag-Leffler modules are sandwiched between the class $\mathcal F \mathcal M$ of all flat (absolute) Mittag-Leffler modules, and the class $\mathcal F$ of all flat modules. Building on the works of Angeleri H\" ugel, Herbera, and \v Saroch, we give a characterization of flat relative Mittag-Leffler modules in terms of their local structure, and show that Enochs' Conjecture holds for all the classes $\mathcal D _{\mathcal Q}$. In the final section, we apply these results to the particular setting of f-projective modules., Comment: 9 pages

Let $f$ be a univalent self-map of the unit disc. We introduce a technique, that we call {\sl semigroup-fication}, which allows to construct a continuous semigroup $(\phi_t)$ of holomorphic self-maps of the unit disc whose time one map $\phi_1$ is, in a sense, very close to $f$. The semigrup-fication of $f$ is of the same type as $f$ (elliptic, hyperbolic, parabolic of positive step or parabolic of zero step) and there is a one-to-one correspondence between the set of boundary regular fixed points of $f$ with a given multiplier and the corresponding set for $\phi_1$. Moreover, in case $f$ (and hence $\phi_1$) has no interior fixed points, the slope of the orbits converging to the Denjoy-Wolff point is the same. The construction is based on holomorphic models, localization techniques and Gromov hyperbolicity. As an application of this construction, we prove that in the non-elliptic case, the orbits of $f$ converge non-tangentially to the Denjoy-Wolff point if and only if the Koenigs domain of $f$ is "almost symmetric" with respect to vertical lines.

Under a mild technical assumption, we prove a necessary and sufficient condition for a totally real compacdt set in $\mathbb{C}^n$ to be rationally convex. This generalizes a classical result of Duval-Sibony

We prove an improved version of Poincar\'e-Hardy inequality in suitable subspaces of the Sobolev space on the hyperbolic space via Bessel pairs. As a consequence, we obtain a new Hardy type inequality with an improved constant (than the usual Hardy constant). Furthermore, we derive a new kind of improved Caffarelli-Kohn-Nirenberg inequality on the hyperbolic space., Comment: 14 pages

A Hopf manifold is a quotient of $C^n\backslash 0$ by the cyclic group generated by a holomorphic contraction. Hopf manifolds are diffeomorphic to $S^1\times S^{2n-1}$ and hence do not admit Kahler metrics. It is known that Hopf manifolds defined by linear contractions (called linear Hopf manifolds) have locally conformally Kahler (LCK) metrics. In this paper we prove that the Hopf manifolds defined by non-linear holomorphic contractions admit holomorphic embeddings into linear Hopf manifolds, and, moreover they admit LCK metrics., 11 pages, Latex (no change in the article, but the title in the submission was wrong)

Munteanu defined the canonical connection associated to a strongly pseudoconvex complex Finsler manifold $(M,F)$. We first prove that the holomorphic sectional curvature tensors of the canonical connection coincide with those of the Chern-Finsler connection associated to $F$ if and only if $F$ is a K\"ahler-Finsler metric. We also investigate the relationship of the Ricci curvatures (resp. scalar curvatures) of these two connections when $M$ is compact. As an application, two characterizations of balanced complex Finsler metrics are given. Next, we obtain a sufficient and necessary condition for a balanced complex Finsler metric to be K\"ahler-Finsler. Finally, we investigate conformal transformations of a balanced complex Finsler metric., Comment: I misquoted the definition of canonical connection?

Let $f\colon M^{2n}\to\mathbb{R}^{2n+4}$ be an isometric immersion of a Kaehler manifold of complex dimension $n\geq 5$ into Euclidean space with complex rank at least $5$ everywhere. Our main result is that, along each connected component of an open dense subset of $M^{2n}$, either $f$ is holomorphic in $\mathbb{R}^{2n+4}\cong\mathbb{C}^{n+2}$ or it is in a unique way a composition $f=F\circ h$ of isometric immersions. In the latter case, we have that $h\colon M^{2n}\to N^{2n+2}$ is holomorphic and $F\colon N^{2n+2}\to\mathbb{R}^{2n+4}$ belongs to the class, by now quite well understood, of non-holomorphic Kaehler submanifold in codimension two. Moreover, the submanifold $F$ is minimal if and only if $f$ is minimal., Version 2

This paper considers composition operators on Zen spaces (a class of weighted Bergman spaces of the right half-plane related to weighted function spaces on the positive half-line by means of the Laplace transform). Generalizations are given to work of Kucik on norms and essential norms, to work of Schroderus on (essential) spectra, and to work by Arvanitidis and the authors on semigroups of composition operators. The results are illustrated by consideration of the Hardy--Bergman space; that is, the intersection of the Hardy and Bergman Hilbert spaces on the half-plane., 14 pages

In this paper, we study the existence of constant holomorphic d-scalar curvature and the prescribing holomorphic d-scalar curvature problem on closed, connected almost Hermitian manifolds of dimension $n\geq6$. In addition, we obtain an application and a variation formula for the associated conformal invariant., 20 pages, SCIENCE CHINA Mathematics, (2023), free access on Just Accepted: http://engine.scichina.com/doi/10.1007/s11425-022-2089-1

We define a subclass of quasiregular curves, called signed quasiregular curves, which contains holomorphic curves and quasiregular mappings. As our main result, we prove a growth theorem of Bonk-Heinonen type for signed quasiregular curves. To obtain our main result, we prove that signed quasiregular curves satisfy a weak reverse H\"older inequality and that this weak reverse H\"older inequality implies the main result. We also obtain higher integrability for signed quasiregular curves. Further, we prove a cohomological value distribution result for signed quasiregular curves by using our main result and equidistribution.

We develop a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Amp\`ere equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel \cite{GL21a} we have shown how this method allows one to obtain new and efficient proofs of several fundamental results in K\"ahler geometry. In \cite{GL21b} we have studied the behavior of Monge-Amp\`ere volumes on hermitian manifolds. We extend here the techniques of \cite{GL21a} to the hermitian setting and use the bounds established in \cite{GL21b}, producing new relative a priori estimates, as well as several existence results for degenerate complex Monge-Amp\`ere equations on compact hermitian manifolds.

Extending a recent theory developed on compact K\"ahler manifolds by Guedj-Lu-Zeriahi and the author, we define and study pluripotential solutions to degenerate parabolic complex Monge-Amp\`ere equations on compact Hermitian manifolds. Under natural assumptions on the Cauchy boundary data, we show that the pluripotential solution is semi-concave in time and continuous in space and that such a solution is unique. We also establish a partial regularity of such solutions under some extra assumptions of the densities and apply it to prove the existence and uniqueness of the weak Chern-Ricci flow on complex compact varieties with log terminal singularities., Comment: 26 pages. arXiv admin note: text overlap with arXiv:1810.02121 by other authors

For a classical link, Milnor defined a family of isotopy invariants, called Milnor $\overline{\mu}$-invariants. Recently, Chrisman extended Milnor $\overline{\mu}$-invariants to welded links by a topological approach. The aim of this paper is to show that Milnor $\overline{\mu}$-invariants can be extended to welded links by a combinatorial approach. The proof contains an alternative proof for the invariance of the original $\overline{\mu}$-invariants of classical links., Comment: 25 pages; v2: the title changed, Section 1 rewritten, remarks and references added; to appear in Michigan Mathematical Journal

We introduce meromorphic nearby cycle functors and study their functorial properties. Moreover we apply them to monodromies of meromorphic functions in various situations. Combinatorial descriptions of their reduced Hodge spectra and Jordan normal forms will be obtained., Comment: 37 pages, to appear in Revista Matematica Complutense