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We revisit the phenomenon where, for certain domains $D$, if the squeezing function $s_D$ extends continuously to a point $p\in \partial{D}$ with value $1$, then $\partial{D}$ is strongly pseudoconvex around $p$. In $\mathbb{C}^2$, we present weaker conditions under which the latter conclusion is obtained. In another direction, we show that there are bounded domains $D\Subset \mathbb{C}^n$, $n\geq 2$, that admit large $\partial{D}$-open subsets $\mathscr{O}\subset \partial{D}$ such that $s_D\to 0$ approaching any point in $\mathscr{O}$. This is impossible for planar domains. We pose a few questions related to these phenomena. But the core result of this paper identifies a new family of holomorphic homogeneous regular domains. We show via a family of examples how abundant domains satisfying the conditions of this result are., Comment: 25 pages; added some expository remarks; corrected a typo in the statement of Lemma 5.1; to appear in Internat. J. Math

We study two notions of topological entropy of correspondences introduced by Friedland and Dinh-Sibony. Upper bounds are known for both. We identify a class of holomorphic correspondences whose entropy in the sense of Dinh-Sibony equals the known upper bound. This provides an exact computation of the entropy for rational semigroups. We also explore a connection between these two notions of entropy., Comment: 18 pages; added references; added a new expository section (Subsection 1.1) and further exposition in Section 5; to appear in Ann. Acad. Sci. Fennicae

Let $X$ and $Y$ be compact connected complex manifolds of the same dimension with $b_2(X)= b_2(Y)$. We prove that any surjective holomorphic map of degree one from $X$ to $Y$ is a biholomorphism. A version of this was established by the first two authors, but under an extra assumption that $\dim H^1(X {\mathcal O}_X)\,=\,\dim H^1(Y {\mathcal O}_Y)$. We show that this condition is actually automatically satisfied., Complex Variables and Elliptic Equations (to appear)

We derive optimal estimates for the Bergman kernel and the Bergman metric for certain model domains in $\mathbb{C}^2$ near boundary points that are of infinite type. Being unbounded models, these domains obey certain geometric constraints -- some of them necessary for a non-trivial Bergman space. However, these are mild constraints: unlike most earlier works on this subject, we are able to make estimates for non-convex pseudoconvex models as well. In fact, the domains we can analyse range from being mildly infinite-type to very flat at infinite-type boundary points., Comment: 16 pages; corrected a typo in v2 in the proof of Lemma 4.1; the final authenticated version of this is available online with the DOI mentioned here

This paper is motivated by Brolin’s theorem. The phenomenon we wish to demonstrate is as follows: if F is a holomorphic correspondence on , then (under certain conditions) F admits a measure such that, for any point z drawn from a ‘large’ open subset of , is the weak-limit of the normalized sums of point masses carried by the pre-images of z under the iterates of F. Let denote the transpose of F. Under the condition , where denotes the topological degree, the above phenomenon was established by Dinh and Sibony. We show that the support of this is disjoint from the normality set of F. There are many interesting correspondences on for which . Examples are the correspondences introduced by Bullett and collaborators. When , equidistribution cannot be expected to the full extent of Brolin’s theorem. However, we prove that when F admits a repeller, equidistribution in the above sense holds true.

We begin by giving an example of a smoothly bounded convex domain that has complex geodesics that do not extend continuously up to $\partial\mathbb{D}$. This example suggests that continuity at the boundary of the complex geodesics of a convex domain $\Omega\Subset \mathbb{C}^n$, $n\geq 2$, is affected by the extent to which $\partial\Omega$ curves or bends at each boundary point. We provide a sufficient condition to this effect (on $\mathcal{C}^1$-smoothly bounded convex domains), which admits domains having boundary points at which the boundary is infinitely flat. Along the way, we establish a Hardy--Littlewood-type lemma that might be of independent interest., Comment: 10 pages; to appear in Ann. Acad. Sci. Fennicae. Math

We investigate a form of visibility introduced recently by Bharali and Zimmer -- and shown to be possessed by a class of domains called Goldilocks domains. The range of theorems established for these domains stem from this form of visibility together with certain quantitative estimates that define Goldilocks domains. We show that some of the theorems alluded to follow merely from the latter notion of visibility. We call those domains that possess this property visibility domains with respect to the Kobayashi distance. We provide a sufficient condition for a domain in $\mathbb{C}^n$ to be a visibility domain. A part of this paper is devoted to constructing a family of domains that are visibility domains with respect to the Kobayashi distance but are not Goldilocks domains. Our notion of visibility is reminiscent of uniform visibility in the context of CAT(0) spaces. However, this is an imperfect analogy because, given a bounded domain $\Omega$ in $\mathbb{C}^n$, $n\geq 2$, it is, in general, not even known whether the metric space $(\Omega,{\sf k}_{\Omega})$ (where ${\sf k}_{\Omega}$ is the Kobayashi distance) is a geodesic space. Yet, with just this weak property, we establish two Wolff--Denjoy-type theorems., Comment: 35 pages; added a reference; corrected a typo and an estimate in (4.3); final version of this is available at the Ann. Scuola Norm. Sup. Pisa Cl. Sci. website with the DOI mentioned below

In this paper we introduce a new class of domains in complex Euclidean space, called Goldilocks domains, and study their complex geometry. These domains are defined in terms of a lower bound on how fast the Kobayashi metric grows and an upper bound on how fast the Kobayashi distance grows as one approaches the boundary. Strongly pseudoconvex domains and weakly pseudoconvex domains of finite type always satisfy this Goldilocks condition, but we also present families of Goldilocks domains that have low boundary regularity or have boundary points of infinite type. We will show that the Kobayashi metric on these domains behaves, in some sense, like a negatively curved Riemannian metric. In particular, it satisfies a visibility condition in the sense of Eberlein and O'Neill. This behavior allows us to prove a variety of results concerning boundary extension of maps and to establish Wolff-Denjoy theorems for a wide collection of domains., 36 pages. v2: minor changes, final version to appear in Advances in Mathematics

We give a solution to Pick's interpolation problem on the unit polydisc in $\mathbb{C}^n$, $n\geq 2$, by characterizing all interpolation data that admit a $\mathbb{D}$-valued interpolant, in terms of a family of positive-definite kernels parametrized by a class of polynomials. This uses a duality approach that has been associated with Pick interpolation, together with some approximation theory. Furthermore, we use duality methods to understand the set of points on the $n$-torus at which the boundary values of a given solution to an extremal interpolation problem are not unimodular., Comment: 20 pages

In this paper, we present a new technique for studying the dynamics of a finitely generated rational semigroup. Such a semigroup can be associated naturally to a certain holomorphic correspondence on $\mathbb{P}^1$. Then, results on the iterative dynamics of such a correspondence can be applied to the study of the rational semigroup. We focus on a certain invariant measure for the aforementioned correspondence---known as the equilibrium measure. This confers some advantages over many of the known techniques for studying the dynamics of rational semigroups. We use the equilibrium measure to analyse the distribution of repelling fixed points of a finitely generated rational semigroup, and to derive a sharp bound for the Hausdorff dimension of the Julia set of such a semigroup., Comment: 20 pages; minor revisions, Theorem 1.6 rephrased more simply; final version to be published in Internat. J. Math

We show that a map between complex-analytic manifolds, at least one of which is in the Fujiki class, is a biholomorphism under a natural condition on the second cohomologies. We use this to establish that, with mild restrictions, a certain relation of "domination" introduced by Gromov is in fact a partial order., 6 pages; significantly restructured first & last sections; corrected statement of Result 3.4 (formerly Result 4.4); added references; to appear in Complex Manifolds

We introduce a family of domains --- which we call the $\mu_{1,n}$-quotients --- associated with an aspect of $\mu$-synthesis. We show that the natural association that the symmetrized polydisc has with the corresponding spectral unit ball is also exhibited by the $\mu_{1,n}$-quotient and its associated unit "$\mu_E$-ball". Here, $\mu_E$ is the structured singular value for the case $E = \{[w]\oplus(z I_{n-1})\in \mathbb{C}^{n\times n}: z,w\in \mathbb{C}\}$, n = 2, 3, 4,... Specifically: we show that, for such an $E$, the Nevanlinna-Pick interpolation problem with matricial data in a unit "$\mu_E$-ball", and in general position in a precise sense, is equivalent to a Nevanlinna-Pick interpolation problem for the associated $\mu_{1,n}$-quotient. Along the way, we present some characterizations for the $\mu_{1,n}$-quotients., Comment: 15 pages; added Remark 3.6 and an additional reference; to appear in Integral Eqns. Operator Theory

Let O be a bounded, weakly pseudoconvex domain in C2, having smooth boundary. A(O) is the algebra of all functions holomorphic in O and continuous up to the boundary. A smooth curve C ? ?O is said to be complex-tangential if Tp(C) lies in the maximal complex subspace of Tp(?O) for each p in C. We show that if C is complex-tangential and ?O is of constant type along C, then every compact subset of C is a peak-interpolation set for A(O). Furthermore, we show that if ?O is real-analytic and C is an arbitrary real-analytic, complex-tangential curve in ?O, compact subsets of C are peak-interpolation sets for A(O).

Let Ω \Omega be a bounded, weakly convex domain in C n {\mathbb {C}}^n , n ≥ 2 n\geq 2 , having real-analytic boundary. A ( Ω ) A(\Omega ) is the algebra of all functions holomorphic in Ω \Omega and continuous up to the boundary. A submanifold M ⊂ ∂ Ω \boldsymbol {M}\subset \partial \Omega is said to be complex-tangential if T p ( M ) T_p(\boldsymbol {M}) lies in the maximal complex subspace of T p ( ∂ Ω ) T_p(\partial \Omega ) for each p ∈ M p\in \boldsymbol {M} . We show that for real-analytic submanifolds M ⊂ ∂ Ω \boldsymbol {M}\subset \partial \Omega , if M \boldsymbol {M} is complex-tangential, then every compact subset of M \boldsymbol {M} is a peak-interpolation set for A ( Ω ) A(\Omega ) .

With the shift in focus towards unconventional oil and gas as the future of the oil and gas industry, it has become essential to optimize performance of wells in these formations and ensure economic feasibility in these unpredictable scenarios. Down Spacing of wells is a procedure by which we reduce the space between the laterals of two wells to maximize recovery from the formation. In conventional reservoirs, due to the homogeneity in petro-physical parameters of the formation, down spacing is usually not economic as the lower number of wells would also be sufficient to achieve optimum productivity, furthermore, the wells will have a tendency to cannibalize each other's production. However, in unconventional reservoirs like shale, down spacing becomes essential to maximize reserves and the productivity from them as poor permeability hinders majority of the pores to be reached by a relatively lower number of wells. The presented paper, achieves to find the optimum down spacing distance that can be achieved between two fractured lateral wells in Eagle Ford Shale Play. The paper analyses the performance of group of wells with different well spacing in the Eagle Ford formation. The data used is from the public forum. The EUR/well is calculated using decline curve analysis. Fekete-Harmony has been used for calculation of EUR and then a simplistic cash flow analysis had been conducted to analyse the feasibility for each presented scenario. This paper also has a group of analog wells created in Fekete Harmony with drilling and completion techniques similar to the ones widely used in Eagle Ford Formation. Real world Eagle Ford geological and petrophysical parameters are incorporated into the Harmony model for the analysis of these wells. This paper emphasizes on the need for downspacing in the shale oil reservoirs for optimum production. It also discusses the advantages and disadvantages of downspacing as well as looks into the economic side of this process.

In this paper, we generalize Chirka's theorem on the extension of functions holomorphic in a neighbourhood of S U (oD x D) where D is the open unit disc in C and S is the graph of a continuous D-valued function on D to higher dimensions, for certain classes of graphs S C D x Dn, n > 1. In particular, we show that Chirka's extension theorem generalizes to configurations in Cn+1, n > 1, involving graphs of (non-holomorphic) polynomial maps with small coefficients. 1. THE MAIN THEOREM This paper is motivated by an article by Chirka [1] (also see [2]) in which he proves the following result (in what follows, D will denote the open unit disc in C, while Dr will denote the open disc of radius r, centered at 0 C C): Theorem 1.1 (Chirka). Let 0: D -> C be a continuous function having sup cIq5(z)I 1. Rosay [3] showed that the theorem fails in general for higher dimensions. A natural question that arises is whether holomorphic extension to Dn+1r n > 1, occurs when the component functions of the Dn -valued map defining our graph are small in some appropriate sense (for instance, when the graph is a sufficiently small perturbation of a holomorphic graph). We are able to answer Chirka's question in the affirmative for the class of graphs described in Theorem 1.3 below. Before stating that theorem, however, we state the following proposition, which is a special case of Theorem 1.3. We highlight this as a separate proposition because of the clarity of its statement. Received by the editors May 1, 2000. 2000 Mathematics Subject Classification. Primary 32D15.

We prove that a proper holomorphic map between two bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism. Our methods allow us to give a single, all-encompassing argument that unifies the various special cases in which this result is known. We discuss an application of these methods to domains having noncompact automorphism groups that are not assumed to act transitively., Comment: 19 pages; typos corrected; missing hypothesis added to the statement of Lemma 4.2; to appear in Pacific J. Math

In the study of holomorphic maps, the term "rigidity" refers to certain types of results that give us very specific information about a general class of holomorphic maps owing to the geometry of their domains or target spaces. Under this theme, we begin by studying when, given two compact connected complex manifolds $X$ and $Y$, a degree-one holomorphic map $f: Y\to X$ is a biholomorphism. Given that the real manifolds underlying $X$ and $Y$ are diffeomorphic, we provide a condition under which $f$ is a biholomorphism. Using this result, we deduce a rigidity result for holomorphic self-maps of the total space of a holomorphic fiber space. Lastly, we consider products $X=X_1\times X_2$ and $Y=Y_1\times Y_2$ of compact connected complex manifolds. When $X_1$ is a Riemann surface of genus $\geq 2$, we show that any non-constant holomorphic map $F:Y\to X$ is of a special form., Comment: 7 pages; expanded Remark 1.2; provided an explanation for the notation in Section 3; to appear in Internat. J. Math

The Levi geometry at weakly pseudoconvex boundary points of domains in C^n, n \geq 3, is sufficiently complicated that there are no universal model domains with which to compare a general domain. Good models may be constructed by bumping outward a pseudoconvex, finite-type \Omega \subset C^3 in such a way that: i) pseudoconvexity is preserved, ii) the (locally) larger domain has a simpler defining function, and iii) the lowest possible orders of contact of the bumped domain with \bdy\Omega, at the site of the bumping, are realised. When \Omega \subset C^n, n\geq 3, it is, in general, hard to meet the last two requirements. Such well-controlled bumping is possible when \Omega is h-extendible/semiregular. We examine a family of domains in C^3 that is strictly larger than the family of h-extendible/semiregular domains and construct explicit models for these domains by bumping., Comment: 28 pages; typos corrected; Remarks 2.6 & 2.7 added; clearer proof of Prop. 4.2 given; to appear in IMRN

The initial motivation for this paper is to discuss a more concrete approach to an approximation theorem of Axler and Shields, which says that the uniform algebra on the closed unit disc closure(D) generated by z and h --- where h is a nowhere-holomorphic harmonic function on D that is continuous up to the boundary --- equals the algebra of continuous functions on closure(D). The abstract tools used by Axler and Shields make harmonicity of h an essential condition for their result. We use the concepts of plurisubharmonicity and polynomial convexity to show that, in fact, the same conclusion is reached if h is replaced by h+R, where R is a non-harmonic perturbation whose Laplacian is "small" in a certain sense., 6 pages; added an important remark on page 2 and a reference; to appear in Proc. Amer. Math. Soc

We study diagonal estimates for the Bergman kernels of certain model domains in $\mathbb{C}^2$ near boundary points that are of infinite type. To do so, we need a mild structural condition on the defining functions of interest that facilitates optimal upper and lower bounds. This is a mild condition; unlike earlier studies of this sort, we are able to make estimates for non-convex pseudoconvex domains as well. This condition quantifies, in some sense, how flat a domain is at an infinite-type boundary point. In this scheme of quantification, the model domains considered below range -- roughly speaking -- from being ``mildly infinite-type'' to very flat at the infinite-type points., Significant revisions made; simpler estimates; very mild strengthening of the hypotheses on Theorem 1.2 to get much stronger conclusions than in ver.1. To appear in Math. Ann

We provide some conditions for the graph of a Hoelder-continuous function on \bar{D}, where \bar{D} is a closed disc in the complex plane, to be polynomially convex. Almost all sufficient conditions known to date --- provided the function (say F) is smooth --- arise from versions of the Weierstrass Approximation Theorem on \bar{D}. These conditions often fail to yield any conclusion if rank_R(DF) is not maximal on a sufficiently large subset of \bar{D}. We bypass this difficulty by introducing a technique that relies on the interplay of certain plurisubharmonic functions. This technique also allows us to make some observations on the polynomial hull of a graph in C^2 at an isolated complex tangency., Comment: 11 pages; typos corrected; to appear in Internat. J. Math

Let S be a smooth real surface in C^2 and let p\in S be a point at which the tangent plane is a complex line. How does one determine whether or not S is locally polynomially convex at such a p --- i.e. at a CR singularity ? Even when the order of contact of T_p(S) with S at p equals 2, no clean characterisation exists; difficulties are posed by parabolic points. Hence, we study non-parabolic CR singularities. We show that the presence or absence of Bishop discs around certain non-parabolic CR singularities is completely determined by a Maslov-type index. This result subsumes all known facts about Bishop discs around order-two, non-parabolic CR singularities. Sufficient conditions for Bishop discs have earlier been investigated at CR singularities having high order of contact with T_p(S). These results relied upon a subharmonicity condition, which fails in many simple cases. Hence, we look beyond potential theory and refine certain ideas going back to Bishop., Comment: 20 pages; all theorems restated to admit surfaces of lower regularity; minor errors in Step 4 of Thm. 1.5 corrected; to appear in Math. Zeitschrift

We present several results associated to a holomorphic-interpolation problem for the spectral unit ball \Omega_n, n\geq 2. We begin by showing that a known necessary condition for the existence of a $\mathcal{O}(D;\Omega_n)$-interpolant (D here being the unit disc in the complex plane), given that the matricial data are non-derogatory, is not sufficient. We provide next a new necessary condition for the solvability of the two-point interpolation problem -- one which is not restricted only to non-derogatory data, and which incorporates the Jordan structure of the prescribed data. We then use some of the ideas used in deducing the latter result to prove a Schwarz-type lemma for holomorphic self-maps of \Omega_n, n\geq 2., Comment: Added a definition (Def.1.1); 2 of the 4 results herein are minor refinements of those in the author's preprint math.CV/0608177; to appear in Integral Eqns. Operator Theory

Let $\Omega$ be a bounded, weakly pseudoconvex domain in C^n, n > 1, with real-analytic boundary. A real-analytic submanifold $M \subset bd\Omega$ is called an analytic interpolation manifold if every real-analytic function on M extends to a function belonging to $\Cal{O}(\bar\Omega)$. We provide sufficient conditions for M to be an analytic interpolation manifold. We give examples showing that neither of these conditions can be relaxed, as well as examples of analytic interpolation manifolds lying entirely within the set of weakly pseudoconvex points of $bd\Omega$., Comment: Final version: corrected statement of Burns-Stout theorem and typos in Example 4.5; added Remark 1.5

We begin with the following question: given a closed disc $\bar{D}$ in the complex plane and a complex-valued function F in $C(\bar{D})$, is the uniform algebra on $\bar{D}$ generated by z and F equal to $C(\bar{D})$ ? When F is in $C^1(\bar{D})$, this question is complicated by the presence of points in the surface S:=graph(F) that have complex tangents. Such points are called CR singularities. Let $p\in S$ be a CR singularity at which the order of contact of the tangent plane with S is greater than 2; i.e. a degenerate CR singularity. We provide sufficient conditions for S to be locally polynomially convex at the degenerate singularity p. This is useful because it is essential to know whether S is locally polynomially convex at a CR singularity in order to answer the initial question. To this end, we also present a general theorem on the uniform algebra generated by z and F, which we use in our investigations. This result may be of independent interest because it is applicable even to non-smooth, complex-valued F., Comment: 17 pages; final version; restated Thm.1.2 using slightly clearer notation, corrected minor typos