The behavior and the location of singular points of a solution to Painlevé VI equation could encode important geometric properties. For example, Hitchin’s formula indicates that singular points of algebraic solutions are exactly the zeros of Eisenstein series of weight one. In this paper, we study the problem: How many singular points of a solution $\lambda (t)$ to the Painlevé VI equation with parameter $(\frac{1}{8}, \frac{-1}{8}, \frac{1}{8}, \frac{3}{8})$ might have in $\mathbb{C} \setminus \lbrace 0, 1\rbrace$? Here $t_0 \in \mathbb{C} \setminus \lbrace 0, 1\rbrace$ is called a singular point of $\lambda (t)$ if $\lambda (t_0) \in \lbrace 0, 1, t_0, \infty \rbrace$. Based on Hitchin’s formula, we explore the connection of this problem with Green function and the Eisenstein series of weight one. Among other things, we prove:

(i) There are only three solutions which have no singular points in $\mathbb{C} \setminus \lbrace 0, 1\rbrace$. (ii) For a special type of solutions (called real solutions here), any branch of a solution has at most two singular points (in particular, at most one pole) in $\mathbb{C} \setminus \lbrace 0, 1\rbrace$}. (iii) Any Riccati solution has singular points in $\mathbb{C} \setminus \lbrace 0, 1\rbrace$. (iv) For each $N \geq 5$ and $N \neq 6$, we calculate the number of the real $j$-values of zeros of the Eisenstein series $\mathfrak{E}^N_1 (\tau ; k_1, k_2)$ of weight one, where $(k_1, k_2)$ runs over ${[0, N-1]}^2$ with $\mathrm{gcd}(k_1, k_2, N) = 1$.

The geometry of the critical points of the Green function on a flat torus $E_{\tau}$, as $\tau$ varies in the moduli $\mathcal{M}_1$, plays a fundamental role in our analysis of the Painlevé VI equation. In particular, the conjectures raised in “Elliptic functions, Green functions and the mean field equations on tori” [C.-S. Lin and C.-L. Wang, Annals of Math. 172 (2010), no. 2, 911–954] on the shape of the domain $\Omega_5 \subset \mathcal{M}_1$, which consists of tori whose Green function has extra pair of critical points, are completely solved here.

We carry out “exotic gluings” à la Carlotto–Schoen for asymptotically hyperbolic general relativistic initial data sets. In particular, we obtain a direct construction of non-trivial initial data sets which are exactly hyperbolic in large regions extending to conformal infinity.

We consider the family of constant curvature fiber metrics for a Lefschetz fibration with regular fibers of genus greater than one. A result of Obitsu and Wolpert is refined by showing that on an appropriate resolution of the total space, constructed by iterated blow-up, this family is log-smooth, i.e., polyhomogeneous with integral powers but possible multiplicities, at the preimage of the singular fibers in terms of parameters of size comparable to the logarithm of the length of the shrinking geodesic.

The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this paper, we prove the stronger statement that it is even inextendible as a Lorentzian manifold with a continuous metric. To capture the obstruction to continuous extensions through the curvature singularity, we introduce the notion of the spacelike diameter of a globally hyperbolic region of a Lorentzian manifold with a merely continuous metric and give a sufficient condition for the spacelike diameter to be finite. The investigation of low-regularity inextendibility criteria is motivated by the strong cosmic censorship conjecture.