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Anjali Bhatnagar

Publications and source records attributed to Anjali Bhatnagar.

6 recordsLinked to original sources
arXiv2026

The invariant Szeg\H{o} metric on Egg domains

We study the Fefferman--Szeg\H{o} metric on egg domains \[ \mathcal D_{2m}=\{(z,w)\in\mathbb C^2: |z|^2+|w|^{2m}<1\},\qquad\qquad\qquad m\in\mathbb Z^+. \] Our first main result establishes the existence of the Fefferman--Szeg\H{o} kernel on $\mathcal{D}_{2m}$ by verifying that the Fefferman weight lies in the Muckenhoupt class $A_2(\partial\mathcal{D}_{2m})$. We then derive an explicit closed-form expression for this kernel, demonstrate that its blowup occurs precisely on the boundary diagonal, and determine its boundary asymptotic behaviour. Using this kernel, we compute the associated Fefferman--Szeg\H{o} metric and its Ricci curvature. As applications, we prove several rigidity results: the metric is K\"ahler--Einstein if and only if $m=1$; proportionality to the Bergman metric or to some complete K\"ahler metric $g_m^{\mathcal D_{2m}}$ is also equivalent to $m=1$. Finally, we establish the vanishing of the $L^2$-cohomology outside the middle dimension for the Fefferman--Szeg\H{o} metric.

math.CV↗
arXiv2026

The Invariant Szeg\H{o} metric on strongly pseudoconvex domains

The Fefferman--Szeg\H{o} metric \(g_{\operatorname{FS}}^\Omega\) on a \(C^\infty\)-smooth bounded strongly pseudoconvex domain \(\Omega\subset\mathbb C^n\) is an invariant metric defined via the Fefferman surface measure. For this metric, we first establish the vanishing of its \(L^2\)-Dolbeault cohomology outside the middle degree: \(\dim H^{p,q}_2(\Omega)=0\) if \(p+q\ne n\), while \(\dim H^{p,q}_2(\Omega)=\infty\) if \(p+q=n\). We also prove that the metric has \(C^\infty\)-bounded geometry. Using this analytic property, we obtain several rigidity results. In particular, if the Fefferman--Szeg\H{o} metric is a gradient Kahler--Ricci soliton, then \(\Omega\) is biholomorphic to the unit ball \(\mathbb B^n\). Moreover, if the metric has constant scalar curvature, then it is Einstein, and again \(\Omega\) is biholomorphic to \(\mathbb B^n\). We also give a Ramadanov-type criterion in terms of the Fefferman--Szeg\H{o} invariant function. Finally, in dimension \(n=2\), assuming the existence of a Kahler immersion into a finite-dimensional ball that maps boundary to boundary transversally, we show that the logarithmic term of the Fefferman--Szeg\H{o} kernel vanishes to infinite order. Consequently, the boundary is locally spherical; if, in addition, \(\Omega\) is simply connected, then \(\Omega\) is biholomorphic to \(\mathbb B^2\).

math.CV↗
arXiv2025

On the Boundary Behaviour of Invariants and Curvatures of the Kobayashi--Fuks Metric in Strictly Pseudoconvex Domains

The purpose of this article is to investigate the boundary behaviour of the Kobayashi--Fuks metric and several associated invariants on strictly pseudoconvex domains in the paradigm of scaling. This approach allows us to examine more invariants, such as the canonical invariant, holomorphic sectional curvature, and Ricci curvature of this metric, in a manner that extends and refines some existing analysis.

math.CV↗
arXiv2025

On the geodesics of the Szeg\"o metric

We explore the existence of closed geodesics and geodesic spirals for the Szeg\"o metric in a $C^{\infty}$-smoothly bounded strongly pseudoconvex domain $\Omega\subset\mathbb{C}^n$, which is not simply connected for $n \geq 2$.

math.CV↗
arXiv2024

Some remarks on the Carath\'eodory and Szeg\"o metrics on planar domains

We study several intrinsic properties of the Carath\'eodory and Szeg\"o metrics on finitely connected planar domains. Among them are the existence of closed geodesics and geodesic spirals, boundary behaviour of Gaussian curvatures, and $L^2$-cohomology. A formula for the Szeg\"o metric in terms of the Weierstrass $\wp$-function is obtained. Variations of these metrics and their Gaussian curvatures on planar annuli are also studied. Consequently, we obtain optimal universal upper bounds for their Gaussian curvatures and show that no universal lower bounds exist for their Gaussian curvatures. Moreover, it follows that there are domains where the Gaussian curvature of the Szeg\"o metric assumes both negative and positive values. Lastly, it is also observed that there is no universal upper bound for the ratio of the Szeg\"o and Carath\'eodory metrics.

math.CV↗