We study the ring generated by the Chern classes of tautological line bundles on the moduli space of parabolic bundles of arbitrary rank on a Riemann surface. We show the Poincaré duals to these Chern classes have simple geometric representatives. We use this construction to show that the ring generated by these Chern classes vanishes below the dimension of the moduli space, in analogy with the Newstead–Ramanan conjecture for stable bundles.

Let $M$ be a compact Riemann surface, $\alpha j \gt -1$, and $h(x)$ a positive $C^2$ function of $M$. In this paper, we consider the following mean field equation: \[ \Delta u (x) + \rho \left( \frac{h(x) e^{u(x)}}{\int_{M} h (x) e^{u(x)}} - \frac{1}{\lvert M \rvert} \right) = 4\pi \sum^{d}_{j=1} \alpha_j \left( \delta_{q_j} - \frac{1}{\lvert M \rvert} \right) \textrm{ in } M \textrm{.} \] We prove that for $\alpha_j \in \mathbb{N}$ and any $\rho \gt \rho_0$, the equation has one solution at least if the Euler characteristic $\chi (M) \leq 0$, where $\rho_0 = \underset{M}{\max} ( 2K - \Delta \mathrm{ln} h + N^*)$, $K$ is the Gaussian curvature, and $N^* = 4\pi \sum^{d}_{j=1} \alpha_j$. This result was proved in [10] when $\alpha_j = 0$. Our proof relies on the bubbling analysis if one of the blowup points is at the vortex $q_j$. In the case where $\alpha_j \notin \mathbb{N}$, the sharp estimate of solutions near $q_j$ has been obtained in [11]. However, if $\alpha_j \in \mathbb{N}$, then the phenomena of non-simple blowup might occur. One of our contributions in part 1 is to obtain the sharp estimate for the non-simple blowup phenomena.

We introduce coordinates on the moduli spaces of maximal globally hyperbolic constant curvature 3d spacetimes with cusped Cauchy surfaces $S$. They are derived from the parametrization of the moduli spaces by the bundle of measured geodesic laminations over Teichmüller space of $S$ and can be viewed as analytic continuations of the shear coordinates on Teichmüller space. In terms of these coordinates, the gravitational symplectic structure takes a particularly simple form, which resembles the Weil–Petersson symplectic structure in shear coordinates, and is closely related to the cotangent bundle of Teichmüller space. We then consider the mapping class group action on the moduli spaces and show that it preserves the gravitational symplectic structure. This defines three distinct mapping class group actions on the cotangent bundle of Teichmüller space, corresponding to different values of the curvature.

In this work we prove the existence of embedded closed minimal hypersurfaces in non-compact manifolds containing a bounded open subset with smooth and strictly mean-concave boundary and a natural behavior on the geometry at infinity. For doing this, we develop a modified min-max theory for the area functional following Almgren and Pitts’ setting, to produce minimal hypersurfaces with intersecting properties. In particular, we prove that any strictly mean-concave region of a compact Riemannian manifold without boundary intersects a closed minimal hypersurface.

In this paper, we construct a completion of the moduli space for polarized Calabi–Yau manifolds by using Ricci-flat Kähler–Einstein metrics and the Gromov–Hausdorff topology, which parameterizes certain Calabi–Yau varieties. We then study the algebro-geometric properties and the Weil–Petersson geometry of such completion. We show that the completion can be exhausted by sequences of quasi-projective varieties, and new points added have finite Weil–Petersson distance to the interior.