For all open Riemann surface $\mathcal{N}$ and real number $\theta \in (0, \pi/2)$, we construct a conformal minimal immersion $X = (X_1,X_2,X_3) : \mathcal{N} \to \mathbb{R}^3$ such that $X_3+\tan(\theta)\left|X_1\right| : \mathcal{N} \to \mathbb{R}$ is positive and proper. Furthermore, $X$ can be chosen with an arbitrarily prescribed flux map.

Moreover, we produce properly immersed hyperbolic minimal surfaces with non-empty boundary in $\mathbb{R}^3$ lying above a negative sublinear graph.

The authors proved in "On the volume of the intersection of two geodesic balls," Differential Geom. Appl. 29 (2011), 567–576, that in a complete, connected, and simply connected Riemannian manifold, the volume of the intersection of two small geodesic balls depends only on the distance between the centers and the radii if and only if the space is harmonic. In this paper, we show that this proposition remains true, if the condition is restricted to balls of equal radii.

Using the degeneration formula for Donaldson-Thomas invariants, we proved formulae for blowing up a point, simple flops, and extremal transitions.

Motivated by results on the simplicial volume of locally symmetric spaces of finite volume, in this note, we observe that the simplicial volume of the moduli space $M_{g,n}$ is equal to $0$ if $g \ge 2$; $g = 1$, $n \ge 3$; or $g = 0$, $n \ge 6$; and the orbifold simplicial volume of $M_{g,n}$ is positive if $g = 1, n = 0, 1; g = 0, n = 4$. We also observe that the simplicial volume of the Deligne-Mumford compactification of $M_{g,n}$ is equal to $0$, and the simplicial volumes of the reductive Borel-Serre compactification of arithmetic locally symmetric spaces $\Gamma\backslash X$ and the Baily-Borel compactification of Hermitian arithmetic locally symmetric spaces $\Gamma\backslash X$ are also equal to $0$ if the $\mathbb{Q}$-rank of $\Gamma\backslash X$ is at least $3$ or if $\Gamma\backslash X$ is irreducible and of $\mathbb{Q}$-rank 2.

Let $S$ be a complex smooth projective surface and $L$ be a line bundle on $S$. Göttsche conjectured that for every integer $r$, the number of $r$-nodal curves in $\left| L\right|$ is a universal polynomial of four topological numbers when $L$ is sufficiently ample. We prove Göttsche’s conjecture using the algebraic cobordism group of line bundles on surfaces and degeneration of Hilbert schemes of points. In addition, we prove the Göttsche-Yau-Zaslow Formula which expresses the generating function of the numbers of nodal curves in terms of quasimodular forms and two unknown series.

We describe a new family of representations of $\pi_1(\Sigma)$ in $\mathrm{PU}(2,1)$, where $\Sigma$ is a hyperbolic Riemann surface with at least one deleted point. This family is obtained by a bending process associated to an ideal triangulation of $\Sigma$. We give an explicit description of this family by describing a coordinates system in the spirit of shear coordinates on the Teichmüller space. We identify within this family new examples of discrete, faithful, and type-preserving representations of $\pi_1(\Sigma)$. In turn, we obtain a 1-parameter family of embeddings of the Teichmüller space of $\Sigma$ in the $\mathrm{PU}(2,1)$-representation variety of $\pi_1(\Sigma)$. These results generalise to arbitrary $\Sigma$ the results obtained in "The punctured torus and Lagrangian triangle groups in $\mathrm{PU}(2,1)$," J. reine angew. Math., 602 (2007), 95–121, for the 1-punctured torus.