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This paper investigates a kind of degenerated circle packings in hyperbolic background geometry. A main problem is whether a prescribed total geodesic curvature data can be realized by a degenerated circle packing or not. We fully characterize the sufficient and necessary conditions and show the uniqueness. Furthermore, we introduce the combinatoral Ricci flow to find the desired degenerated circle packed surface, analougus to the methods of Chow-Luo and Takatsu., Comment: 36 pages, 9 figures

In this paper, we study the topological properties of the Gromov boundary of the fine curve graph of an orientable finite-type surface of genus at least 2. This graph consisting of topological curves has much richer dynamics than the classical curve graph. Using the techniques introduced by Wright [Wri23], we show that this boundary is (path) connected and that the spheres in non-separating fine curve graph are connected., Comment: 16 pages. New version specifies the topology of curves, corrects some minor errors and typos. Comments are welcome!

The Gardiner-Masur compactification of Teichm\"uller space is homeomorphic to the horofunction compactification of the Teichm\"uller metric. Let $\xi$ and $\eta$ be a pair of boundary points in the Gardiner-Masur compactification that fill up the surface. We show that there is a unique Teichm\"uller geodesic which is optimal for the horofunctions corresponding to $\xi$ and $\eta$. In particular, when $\xi$ and $\eta$ are Busemann points that fill up the surface, the geodesic converges to $\xi$ in forward direction and to $\eta$ in backward direction. As an application, we show that if $\mathbf{G}_n$ is a sequence of Teichm\"uller geodesics passing through $X_n$ and $Y_n$ such that $X_n \to \xi$ and $Y_n \to \eta$, then $\mathbf{G}_n$ converges to a unique Teichm\"uller geodesic., Comment: 25 pages, 4 figures. We have expanded the introduction, added Section 6 and an appendix

We revisit the volume Green's function integral equation for modelling light scattering with discretization strategies as well as numerical integration recipes borrowed from finite element method. The merits of introducing finite element techniques into Green's function integral equation are apparent. Firstly, the finite element discretization provides a much better geometric approximation of the scatters, compared with that of the conventional discretization method using staircase approximation. Secondly, the accuracy of numerical integral inside one element associated with all different types of Green's function integral equations can be greatly improved by introducing quadrature rules. Within the standard framework of Green's function integral equation, we seamlessly introduce finite element techniques into the Green's function integral equation by introducing the auxiliary variables that confines the singular integrand inside each element, leading to a better and more flexible approximation of the geometry of the scatters and a more accurate numerical integral. We then illustrate the advantages of our finite element based Green's function integral equation method via a few concrete examples in modelling light scattering by optically large and complex scatters., Comment: 12 pages, 5 figures

We study the geometry of horospheres in Teichm\"uller space of Riemann surfaces of genus g with n punctures, where $3g-3+n\geq 2$. We show that every $C^1$-diffeomorphism of Teichm\"uller space to itself that preserves horospheres is an element of the extended mapping class group. Using the relation between horospheres and metric balls, we obtain a new proof of Royden's Theorem that the isometry group of the Teichm\"uller metric is the extended mapping class group., Comment: 25 pages. Accepted for publication in ANNALES DE L'INSTITUT FOURIER