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Kingshook Biswas

Publications and source records attributed to Kingshook Biswas.

At least 19 recordsLinked to original sources
arXiv2026

On the exponential convergence of Kobayashi geodesics in strongly convex domains

In this paper, we have proved a quantitative version of the approaching geodesic property for certain convex domains. We have proved that that if $\Omega \subset \mathbb{C}^{d}$ is a bounded strongly convex domain with $\mathcal{C}^3$ boundary and $\gamma_{1}, \gamma_{2}:[0, \infty) \to \Omega$ are two geodesic rays such that $\gamma_{1}(\infty)=\gamma_{2}(\infty)=\xi \in \partial \Omega$. Then if the images of $\gamma_{1}$ and $\gamma_{2}$ are contained in the same complex geodesic, then there exists $T\in \mathbb{R}$ \[ \lim_{t \to \infty} \frac{1}{t} \log K_{\Omega}\big(\gamma_{1}(t), \gamma_{2}(t+T)\big) = -2, \] otherwise \[ \lim_{t \to \infty} \frac{1}{t} \log K_{\Omega}\big(\gamma_{1}(t), \gamma_{2}(t+T)\big) = -1. \] Furthermore, using this property we provided a characterization of strongly pseudoconvex domain via a biholomorphic invariant function namely generalized squeezing function. We have proved that: For every $\alpha>0$ there exists $\epsilon(d,\alpha)>0$ such that the following holds: if $\Omega \subset \mathbb{C}^d$ is a bounded convex domain with $\mathcal{C}^{2,\alpha}$-boundary and \[ T_{\Omega}^{D}(z)\geq 1-\epsilon \] outside a compact subset of $\Omega$, where $D \Subset \mathbb{C}^{d}$ is a balanced strongly convex domain with $\mathcal{C}^{3}$ boundary and $T_{\Omega}^{D}$ is the squeezing function of $\Omega$ with respect to the domain $D$ then $\Omega$ is strongly pseudoconvex. We also establish exponential convergence of a certain family of quasi-geodesics in the unit ball of $\mathbb{C}^{d}$. We further show that the study of this family of quasi-geodesics provides a useful tool that allows the exponential convergence property of geodesics to be transferred from local subdomains to the ambient domain, as well as in the reverse direction.

math.CV↗
arXiv2025

Gromov-Hausdorff convergence of maximal Gromov hyperbolic spaces and their boundaries

The relation between negatively curved spaces and their boundaries is important for various rigidity problems. In \cite{biswas2024quasi}, the class of Gromov hyperbolic spaces called maximal Gromov hyperbolic spaces was introduced, and the boundary functor $X \mapsto \partial X$ was shown to give an equivalence of categories between maximal Gromov hyperbolic spaces (with morphisms being isometries) and a class of compact quasi-metric spaces called quasi-metric antipodal spaces (with morphisms being Moebius homeomorphisms). The proof of this equivalence involved the construction of a filling functor $Z \mapsto {\mathcal M}(Z)$, associating to any quasi-metric antipodal space $Z$ a maximal Gromov hyperbolic space ${\mathcal M}(Z)$. We study the ``continuity" properties of the boundary and filling functors. We show that convergence of a sequence of quasi-metric antipodal spaces (in a certain sense called ``almost-isometric convergence") implies convergence (in the Gromov-Hausdorff sense) of the associated maximal Gromov hyperbolic spaces. Conversely, we show that convergence of maximal Gromov hyperbolic spaces together with a natural hypothesis of ``equicontinuity" on the boundaries implies convergence of boundaries. We use this to show that Gromov-Hausdorff convergence of a sequence of proper, geodesically complete CAT(-1) spaces implies Gromov-Hausdorff convergence of their boundaries equipped with visual metrics. We also show that convergence of maximal Gromov hyperbolic spaces to a maximal Gromov hyperbolic space with finite boundary implies convergence of boundaries.

math.MG↗
arXiv2024

Polyhedral structure of maximal Gromov hyperbolic spaces with finite boundary

The boundary $\partial X$ of a boundary continuous Gromov hyperbolic space $X$ carries a natural Moebius structure on the boundary. For a proper, geodesically complete, boundary continuous Gromov hyperbolic space $X$, the boundary $\partial X$ equipped with its cross-ratio is a particular kind of quasi-metric space, called a quasi-metric antipodal space. Given a quasi-metric antipodal space $Z$, one may consider the family of all hyperbolic fillings of $Z$. In \cite{biswas2024quasi} it was shown that this family has a unique upper bound $\mathcal{M}(Z)$ (with respect to a natural partial order on hyperbolic fillings of $Z$), which can be described explicitly in terms of the cross-ratio on $Z$. As shown in \cite{biswas2024quasi}, the spaces $\mathcal{M}(Z)$ constitute a natural class of spaces called maximal Gromov hyperbolic spaces. A natural problem is to describe explicitly the maximal Gromov hyperbolic spaces $X$ whose boundary $\partial X$ is finite. We show that for a maximal Gromov hyperbolic space $X$ with boundary $\partial X$ of cardinality $n$, the space $X$ is isometric to a finite polyhedral complex embedded in $(\mathbb{R}^n, ||\cdot||_{\infty})$ with cells of dimension at most $n/2$, given by attaching $n$ half-lines to vertices of a compact polyhedral complex. In particular the geometry at infinity of $X$ is trivial. The combinatorics of the polyhedral complex is determined by certain relations $R \subset \partial X \times \partial X$ on the boundary $\partial X$, called antipodal relations. In \cite{biswas2024quasi} it was shown that maximal Gromov hyperbolic spaces are injective metric spaces. We give a shorter, simpler proof of this fact in the case of spaces with finite boundary. We also consider the space of deformations of a maximal Gromov hyperbolic space with finite boundary, and define an associated Teichmuller space.

math.MG↗
arXiv2024

Estimates of the Poisson kernel on negatively curved Hadamard manifolds

Let $M$ be an $n$-dimensional Hadamard manifold of pinched negative curvature $-b^2 \leq K_M \leq -a^2$. The solution of the Dirichlet problem at infinity for $M$ leads to the construction of a family of mutually absolutely continuous probability measures $\{\mu_x\}_{x \in M}$ called the harmonic measures. Fixing a basepoint $o \in M$, the Poisson kernel of $M$ is the function $P : M \times \partial M \to (0, \infty)$ defined by \begin{equation*} P(x, \xi) = \frac{d\mu_x}{d\mu_o}(\xi) \ , \ x \in M, \xi \in \partial M. \end{equation*} We prove the following global upper and lower bounds for the Poisson kernel: \begin{equation*} \frac{1}{C}\: e^{-2K{(o|\xi)}_x}\: e^{a d(x, o)} \le P(x,\xi) \le C\: e^{2K{(x|\xi)}_o}\: e^{-a d(x,o)} \:, \end{equation*} for some positive constants $C \geq 1, K > 0$ depending solely on $a, b$ and $n$. The above estimates may be viewed as a generalization of the well-known formula for the Poisson kernel in terms of Busemann functions for the special case of Gromov hyperbolic harmonic manifolds. These estimates do not follow directly from known estimates on Green's functions or harmonic measures. Instead we use techniques due to Anderson-Schoen for estimating positive harmonic functions in cones. As applications, we obtain quantitative estimates for the convergence $\mu_x \to \delta_{\xi}$ as $x \in M \to \xi \in \partial M$, and for the convergence of harmonic measures on finite spheres to the harmonic measures on the boundary at infinity as the radius of the spheres tends to infinity.

math.DG↗
arXiv2023

Restricted mean value property on Riemannian manifolds

A well studied classical problem is the harmonicity of functions satisfying the restricted mean-value property (RMVP). While this has so far been studied mainly for domains in $\mathbb{R}^n$, we consider this problem in the general setting of domains in Riemannian manifolds, and obtain results generalizing classical results of Fenton. We also obtain a result for complete, simply connected Riemannian manifolds of pinched negative curvature where there is no restriction on the radius function in the RMVP.

math.DG↗
arXiv2021

Quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces

Hyperbolic fillings of metric spaces are a well-known tool for proving results on extending quasi-Moebius maps between boundaries of Gromov hyperbolic spaces to quasi-isometries between the spaces. For CAT(-1) spaces, and more generally boundary continuous Gromov hyperbolic spaces, one can refine the quasi-Moebius structure on the boundary to a Moebius structure. It is then natural to ask whether there exists a functorial hyperbolic filling of the boundary by a boundary continuous Gromov hyperbolic space with an identification between boundaries which is not just quasi-Moebius, but in fact Moebius. We give a positive answer to this question for a large class of boundaries satisfying one crucial hypothesis, the {\it antipodal property}. This gives a class of compact spaces called {\it quasi-metric antipodal spaces}. For any such space $Z$, we give a functorial construction of a boundary continuous Gromov hyperbolic space $\mathcal{M}(Z)$ together with a Moebius identification of its boundary with $Z$. The space $\mathcal{M}(Z)$ is maximal amongst all fillings of $Z$. These spaces $\mathcal{M}(Z)$ give in fact all examples of a natural class of spaces called {\it maximal Gromov hyperbolic spaces}. We prove an equivalence of categories between quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces. This is part of a more general equivalence we prove between the larger categories of certain spaces called {\it antipodal spaces} and {\it maximal Gromov product spaces}. We prove that the injective hull of a Gromov product space $X$ is isometric to the maximal Gromov product space $\mathcal{M}(Z)$, where $Z$ is the boundary of $X$. We also show that a Gromov product space is injective if and only if it is maximal.

math.MG↗
arXiv2019

On Sullivan's construction of eigenfunctions via exit times of Brownian motion

The purpose of this note is to give details for an argument of Sullivan to construct eigenfunctions of the Laplacian on a Riemannian manifold using exit times of Brownian motion \cite{sullivanpos}. Let $X$ be a complete, simply connected Riemannian manifold of pinched negative sectional curvature. Let $\lambda_1 = \lambda_1(X) < 0$ be the supremum of the spectrum of the Laplacian on $L^2(X)$, and let $D \subset X$ be a bounded domain in $X$ with smooth boundary. Let $(B_t)_{t \geq 0}$ be Brownian motion on $X$ and let $\tau = \tau_D$ be the first exit time of Brownian motion from $D$. For each $\lambda \in \mathbb{C}$ with $\hbox{Re } \ \lambda > \lambda_1$ and $x \in D$, we show that for any continuous function $\phi : \partial D \to \mathbb{C}$, the function $$ h(x) = \mathbb{E}_x(e^{-\lambda \tau} \phi(B_{\tau})) \ , \ x \in D, $$ is an eigenfunction of the Laplacian on $D$ with eigenvalue $\lambda$ and boundary value $\phi$.

math.DG↗
arXiv2019

The Fourier transform on harmonic manifolds of purely exponential volume growth

Let $X$ be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of harmonic manifolds except for the flat spaces. Denote by $h > 0$ the mean curvature of horospheres in $X$, and set $\rho = h/2$. Fixing a basepoint $o \in X$, for $\xi \in \partial X$, denote by $B_{\xi}$ the Busemann function at $\xi$ such that $B_{\xi}(o) = 0$. then for $\lambda \in \C$ the function $e^{(i\lambda - \rho)B_{\xi}}$ is an eigenfunction of the Laplace-Beltrami operator with eigenvalue $-(\lambda^2 + \rho^2)$. For a function $f$ on $X$, we define the Fourier transform of $f$ by $$\tilde{f}(\lambda, \xi) := \int_X f(x) e^{(-i\lambda - \rho)B_{\xi}(x)} dvol(x)$$ for all $\lambda \in \C, \xi \in \partial X$ for which the integral converges. We prove a Fourier inversion formula $$f(x) = C_0 \int_{0}^{\infty} \int_{\partial X} \tilde{f}(\lambda, \xi) e^{(i\lambda - \rho)B_{\xi}(x)} d\lambda_o(\xi) |c(\lambda)|^{-2} d\lambda$$ for $f \in C^{\infty}_c(X)$, where $c$ is a certain function on $\mathbb{R} - \{0\}$, $\lambda_o$ is the visibility measure on $\partial X$ with respect to the basepoint $o \in X$ and $C_0 > 0$ is a constant. We also prove a Plancherel theorem, and a version of the Kunze-Stein phenomenon.

math.DG↗
arXiv2019

The Ramificant Determinant

We give an introduction to the transalgebraic theory of simply connected log-Riemann surfaces with a finite number of infinite ramification points (transalgebraic curves of genus $0$). We define the base vector space of transcendental functions and establish by elementary means some transcendental properties. We introduce the Ramificant Determinant constructed with transcendental periods and we give a closed-form formula that gives the main applications to transalgebraic curves. We prove an Abel-like Theorem and a Torelli-like Theorem. Transposing to the transalgebraic curve the base vector space of transcendental functions, they generate the structural ring from which the points of the transalgebraic curve can be recovered algebraically, including infinite ramification points.

math.CV↗
arXiv2018

Moebius rigidity for simply connected, negatively curved surfaces

Let $X, Y$ be complete, simply connected Riemannian surfaces with pinched negative curvature $-b^2 \leq K \leq -1$. We show that if $f : \partial X \to \partial Y$ is a Moebius homeomorphism between the boundaries at infinity of $X, Y$, then $f$ extends to an isometry $F : X \to Y$. This can be viewed as a generalization of Otal's marked length spectrum rigidity theorem for closed, negatively curved surfaces, in the sense that Otal's theorem asserts that if $X, Y$ admit properly discontinuous, cocompact, free actions by groups of isometries and the boundary map $f$ is Moebius and equivariant with respect to these actions then it extends to an isometry. In our case there are no cocompactness or equivariance assumptions, indeed the isometry groups of $X, Y$ may be trivial.

math.DG↗
arXiv2018

Moebius rigidity for compact deformations of negatively curved manifolds

Let $(X, g_0)$ be a complete, simply connected Riemannian manifold with sectional curvatures $K_{g_0}$ satisfying $-b^2 \leq K_{g_0} \leq -1$ for some $b \geq 1$. Let $g_1$ be a Riemannian metric on $X$ such that $g_1 = g_0$ outside a compact in $X$, and with sectional curvatures $K_{g_1}$ satisfying $K_{g_1} \leq -1$. The identity map $id : (X, g_0) \to (X, g_1)$ is bi-Lipschitz, and hence induces a homeomorphism between the boundaries at infinity of $(X, g_0)$ and $(X, g_1)$, which we denote by $\hat{id}_{g_0, g_1} : \partial_{g_0} X \to \partial_{g_1} X$. We show that if the boundary map $\hat{id}_{g_0, g_1}$ is Moebius (i.e. preserves cross-ratios), then it extends to an isometry $F : (X, g_0) \to (X, g_1)$.

math.DG↗
arXiv2018

Dynamics of $L^p$ multipliers on harmonic manifolds

Let $X$ be a complete, simply connected harmonic manifold with sectional curvatures $K$ satisfying $K \leq -1$. In \cite{biswas6}, a Fourier transform was defined for functions on $X$, and a Fourier inversion formula and Plancherel theorem were proved. We use the Fourier transform to investigate the dynamics on $L^p(X)$ for $p > 2$ of certain bounded linear operators $T : L^p(X) \to L^p(X)$ which we call "$L^p$-multipliers" in accordance with standard terminology. These operators are required to preserve the subspace of $L^p$ radial functions. A notion of convolution with radial functions was defined in \cite{biswas6}, and these operators are also required to be compatible with convolution in the sense that $$ T\phi * \psi = \phi * T\psi $$ for all radial $C^{\infty}_c$-functions $\phi, \psi$. They are also required to be compatible with translation of radial functions. Examples of $L^p$-multipliers are given by the operator of convolution with an $L^1$ radial function, or more generally convolution with a finite radial measure. In particular elements of the heat semigroup $e^{t\Delta}$ act as multipliers. Given $2 < p < \infty$, we show that for any $L^p$-multiplier $T$ which is not a scalar multiple of the identity, there is an open set of values of $\nu \in \mathbb{C}$ for which the operator $\frac{1}{\nu} T$ is chaotic on $L^p(X)$ in the sense of Devaney, i.e. topologically transitive and with periodic points dense. Moreover such operators are topologically mixing. We also show that there is a constant $c_p > 0$ such that for any $c \in \mathbb{C}$ with $\Re c > c_p$, the action of the shifted heat semigroup $e^{ct} e^{t\Delta}$ on $L^p(X)$ is chaotic. These results generalize the corresponding results for rank one symmetric spaces of noncompact type and negatively curved harmonic $NA$ groups (or Damek-Ricci spaces).

math.DS↗
arXiv2018

The Fourier transform on negatively curved harmonic manifolds

Let $X$ be a complete, simply connected harmonic manifold with sectional curvatures $K$ satisfying $K \leq -1$, and let $\partial X$ denote the boundary at infinity of $X$. Let $h > 0$ denote the mean curvature of horospheres in $X$, and let $\rho = h/2$. Fixing a basepoint $o \in X$, for $\xi \in \partial X$, let $B_{\xi}$ denote the Busemann function at $\xi$ such that $B_{\xi}(o) = 0$, then for $\lambda \in \mathbb{C}$ the function $e^{(i\lambda - \rho)B_{\xi}}$ is an eigenfunction of the Laplace-Beltrami operator with eigenvalue $-(\lambda^2 + \rho^2)$. For a function $f$ on $X$, we define the Fourier transform of $f$ by $$\tilde{f}(\lambda, \xi) := \int_X f(x) e^{(-i\lambda - \rho)B_{\xi}(x)} dvol(x)$$ for all $\lambda \in \mathbb{C}, \xi \in \partial X$ for which the integral converges. We prove a Fourier inversion formula $$f(x) = C_0 \int_{0}^{\infty} \int_{\partial X} \tilde{f}(\lambda, \xi) e^{(i\lambda - \rho)B_{\xi}(x)} d\lambda_o(\xi) |c(\lambda)|^{-2} d\lambda$$ for $f \in C^{\infty}_c(X)$, where $c$ is a certain function on $\mathbb{R} - \{0\}$, $\lambda_o$ is the visibility measure on $\partial X$ with respect to the basepoint $o \in X$ and $C_0 > 0$ is a constant. We also prove a Plancherel theorem. This generalizes the corresponding results for rank one symmetric spaces of noncompact type and negatively curved harmonic $NA$ groups (or Damek-Ricci spaces).

math.DG↗
arXiv2017

Hyperbolic $p$-barycenters, circumcenters, and Moebius maps

Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, and a family of probability measures $\{ μ_x \}_{x \in X}$ on $\partial X$, we describe a continuous family of extensions $\{\hat{f}_p : X \to Y \}_{1 \leq p \leq \infty}$ of $f$, called the hyperbolic $p$-barycenter maps of $f$. If all the measures $μ_x$ have full support then for $p = \infty$ the map $\hat{f}_{\infty}$ coincides with the circumcenter map $\hat{f}$ defined previously in \cite{biswas5}. We use this to show that if $X, Y$ are complete, simply connected manifolds with sectional curvatures $K$ satisfying $-b^2 \leq K \leq -1$, then the circumcenter maps of $f$ and $f^{-1}$ are $\sqrt{b}$-bi-Lipschitz homeomorphisms which are inverses of each other. It follows that closed negatively curved manifolds with the same marked length spectrum are bi-Lipschitz homeomorphic.

math.DG↗
arXiv2017

Circumcenter extension of Moebius maps to CAT(-1) spaces

Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, we describe an extension $\hat{f} : X \to Y$ of $f$, called the circumcenter map of $f$, which is constructed using circumcenters of expanding sets. The extension $\hat{f}$ is shown to coincide with the $(1, \log 2)$-quasi-isometric extension constructed in [biswas3], and is locally $1/2$-Holder continuous. When $X,Y$ are complete, simply connected manifolds with sectional curvatures $K$ satisfying $-b^2 \leq K \leq -1$ for some $b \geq 1$ then the extension $\hat{f} : X \to Y$ is a $(1, (1 - \frac{1}{b})\log 2)$-quasi-isometry. Circumcenter extension of Moebius maps is natural with respect to composition with isometries.

math.DG↗
arXiv2016

A Torelli type theorem for exp-algebraic curves

An exp-algebraic curve consists of a compact Riemann surface $S$ together with $n$ equivalence classes of germs of meromorphic functions modulo germs of holomorphic functions, $\HH = \{ [h_1], \cdots, [h_n] \}$, with poles of orders $d_1, \cdots, d_n \geq 1$ at points $p_1, \cdots, p_n$. This data determines a space of functions $\OO_{\HH}$ (respectively, a space of $1$-forms $\Omega^0_{\HH}$) holomorphic on the punctured surface $S' = S - \{p_1, \cdots, p_n\}$ with exponential singularities at the points $p_1, \cdots, p_n$ of types $[h_1], \cdots, [h_n]$, i.e., near $p_i$ any $f \in \OO_{\HH}$ is of the form $f = ge^{h_i}$ for some germ of meromorphic function $g$ (respectively, any $\omega \in \Omega^0_{\HH}$ is of the form $\omega = \alpha e^{h_i}$ for some germ of meromorphic $1$-form). For any $\omega \in \Omega^0_{\HH}$ the completion of $S'$ with respect to the flat metric $|\omega|$ gives a space $S^* = S' \cup \RR$ obtained by adding a finite set $\RR$ of $\sum_i d_i$ points, and it is known that integration along curves produces a nondegenerate pairing of the relative homology $H_1(S^*, \RR ; \C)$ with the deRham cohomology group defined by $H^1_{dR}(S, \HH) := \Omega^0_{\HH}/d\OO_{\HH}$. There is a degree zero line bundle $L_{\HH}$ associated to an exp-algebraic curve, with a natural isomorphism between $\Omega^0_{\HH}$ and the space $W_{\HH}$ of meromorphic $L_{\HH}$-valued $1$-forms which are holomorphic on $S'$, so that $H_1(S^*, \RR ; \C)$ maps to a subspace $K_{\HH} \subset W^*_{\HH}$. We show that the exp-algebraic curve $(S, \HH)$ is determined uniquely by the pair $(L_{\HH},\, K_{\HH} \subset W^*_{\HH})$.

math.CV↗
arXiv2016

Algebraic deRham cohomology of log-Riemann surfaces of finite type

Log-Riemann surfaces of finite type are Riemann surfaces with finitely generated fundamental group equipped with a local diffeomorphism to C such that the surface has finitely many infinite order ramification points. We define and prove nondegeneracy of a period pairing for log-Riemann surfaces of finite type, given by pairing differentials with finitely many exponential singularities, of the form g exp(\int R_0) dz (where g, R_0 are meromorphic functions on a compact Riemann surface, with R_0 fixed) with closed curves and curves joining infinite order ramification points. As a consequence we show that the dimension of a cohomology group (given by differentials with exponential singularities of fixed type, modulo differentials of functions with exponential singularities of the same fixed type) is finite, equal to (2g + #R + (n-2)), where g is the genus of the compact Riemann surface, R is the set of infinite order ramification points, and n the number of exponential singularities.

math.CV↗