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On the exponential convergence of Kobayashi geodesics in strongly convex domains
Kingshook Biswas, Sanjoy Chatterjee
https://arxiv.org/abs/2607.17259 https://arxiv.org/pdf/2607.17259 https://arxiv.org/html/2607.17259
arXiv:2607.17259v1 Announce Type: new
Abstract: In this paper, we have proved a qualitative version of the approaching geodesic property for certain convex domains. More precisely, 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 geodesics such that $\gamma_{1}(\infty)=\gamma_{2}(\infty)=\xi \in \partial \Omega$. Then there exists $A\big(\gamma_{1}(0), \gamma_{2}(0) \big)>0$ and $T \in \mathbb{R}$ such that
$$K_{\Omega}(\gamma_{1}(t), \gamma_{2}(t T))\leq Ae^{-\frac{t}{2}} ~~~\,\hspace{1em} \forall t \geq 0.$$ 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 $\alpha>2$ 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.
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Maximal subextension of $m$-subharmonic functions
Hichame Amal, Sa\"id Asserda, Ayoub El-Gasmi
https://arxiv.org/abs/2607.19132 https://arxiv.org/pdf/2607.19132 https://arxiv.org/html/2607.19132
arXiv:2607.19132v1 Announce Type: new
Abstract: In this paper, we prove that given a quasi-$m$-hyperconvex domain $\Omega \subset X$ in a compact K\"ahler manifold $(X, \omega)$, and a function $\varphi $ in the weighted energy class $\mathcal{E}_\chi^m(\Omega, \omega)$ with respect to a convex weight function $\chi : \mathbb{R} \to \mathbb{R}$, then there exists a maximal $\omega$-$m$-subharmonic subextension $\tilde{\varphi}$ to $X$ that preserves the weighted energy and satisfies a good control properties for its Hessian measure $ \mathbf{1}_\Omega H_m(\tilde{\varphi}) \leq \mathbf{1}_\Omega H_m(\varphi) $. In the last part, we study the particular case where $(X,\omega)=(\mathbb{P}^n,\omega_{FS}).$
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Ergodic Optimization with Linear Constraints
Shengwen Guo, Kevin McGoff
https://arxiv.org/abs/2608.02435 https://arxiv.org/pdf/2608.02435 https://arxiv.org/html/2608.02435
arXiv:2608.02435v1 Announce Type: new
Abstract: Let $T : X \to X$ be a continuous map of a compact metrizable space, and let $\phi : X \to \mathbb{R}$ be a continuous function. The ergodic optimization problem is to maximize the integral $\int \phi \, d\mu$ as $\mu$ ranges over all $T$-invariant Borel probability measures on $X$. In this paper we consider a constrained version of the ergodic optimization problem. Given a `constraint set' $\mathcal{C}\subset C(X)$, let $M_\mathcal{C}(X,T)$ be the set of $T$-invariant Borel probability measures $\mu$ on $X$ such that $\int g \, d\mu = 0$ for all $g \in \mathcal{C}$. We investigate the problem of maximizing the integral $\int \phi \, d\mu$ over the constrained set $M_\mathcal{C}(X,T)$. We address basic properties of this optimization problem, beginning with nonemptiness of $M_\mathcal{C}(X,T)$ and existence of optimal solutions. Additionally, we establish the generic and prevalent uniqueness of optimal measures, we provide a realization result, and we give a characterization of the dual problem. This framework provides a common generalization of several previously considered optimization problems in dynamical systems and optimal transport.
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Visualization of Complex Projective Curves
Seth Dutter
https://arxiv.org/abs/2608.04323 https://arxiv.org/pdf/2608.04323 https://arxiv.org/html/2608.04323
arXiv:2608.04323v1 Announce Type: new
Abstract: We introduce a nonlinear map $\alpha:\mathbb{C}^2\rightarrow\mathbb{R}^3$ with the purpose of visualizing curves. Basic properties of $\alpha$ are proved, including preservation of orthogonality, recovery of the magnitudes of vectors in the preimage, and continuous extension of $\alpha$ to $\widetilde{\alpha}:\mathbb{P}^2_\mathbb{C}\rightarrow\mathbb{R}^3$. For plane curves $Z\subset\mathbb{P}^2_\mathbb{C}$, it is proved that $\widetilde{\alpha}(Z)$ is the union of boundaries of star-shaped domains. Methods are established to descend finite-order automorphisms of smooth projective curves to rotations of their images in $\mathbb{R}^3$. Efficient techniques for creating meshes and ray-traced images of $\widetilde{\alpha}(Z)$ are developed.
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