In this paper we study the rigidity of proper holomorphic maps $f: \Omega \rightarrow \Omega^{\prime}$ between irreducible bounded symmetric domains $\Omega$ and $\Omega^{\prime}$ with small rank differences: 2ドル \leq \operatorname{rank}\left(\Omega^{\prime}\right)\lt 2 \operatorname{rank}(\Omega)-1$ . More precisely, if either $\Omega$ and $\Omega^{\prime}$ of the same type or $\Omega$ is of type III and $\Omega^{\prime}$ is of type I, then up to automorphisms, $f$ is of the form $f=\imath \circ F$ , where $F=F_1 \times F_2: \Omega \rightarrow \Omega_1^{\prime} \times \Omega_2^{\prime}$ . Here $\Omega_1^{\prime}, \Omega_2^{\prime}$ are bounded symmetric domains, the map $F_1: \Omega \rightarrow \Omega_1^{\prime}$ is a standard embedding, $F_2: \Omega \rightarrow \Omega_2^{\prime}$ , and $\imath: \Omega_1^{\prime} \times \Omega_2^{\prime} \rightarrow \Omega^{\prime}$ is a totally geodesic holomorphic isometric embedding. Moreover we show that, under the rank condition above, there exists no proper holomorphic map $f: \Omega \rightarrow \Omega^{\prime}$ if $\Omega$ is of type I and $\Omega^{\prime}$ is of type III, or $\Omega$ is of type II and $\Omega^{\prime}$ is either of type I or III. By considering boundary values of proper holomorphic maps on maximal boundary components of $\Omega$ , we construct rational maps between moduli spaces of subgrassmannians of compact duals of $\Omega$ and $\Omega^{\prime}$ , and induced CR maps between CR hypersurfaces of mixed signature, thereby forcing the moduli map to satisfy strong local differential-geometric constraints (or that such moduli maps do not exist), and complete the proofs from rigidity results on geometric substructures modeled on certain admissible pairs of rational homogeneous spaces of Picard number 1.

We show that every complete non-compact three-manifold with non-negatively pinched Ricci curvature admits a complete Ricci flow solution for all positive time, with scale-invariant curvature decay and preservation of pinching. Combining with recent work of Lott and Deruelle-Schulze-Simon gives a proof of Hamilton's pinching conjecture without additional hypotheses.

On a compact connected Riemann surface $X$ of genus $g(X) \geq 2$ , we study the limiting behavior of the solutions $h_t$ of Hitchin's equations associated with a generic family of stable $\mathrm{SU}(1,2)$ -Higgs bundles of the form ( $L, F, t \beta, t \gamma$ ), as $t \rightarrow \infty$ . The corresponding spectral data is equivalent to a Hecke modification of $V= L^{-2} K_X \oplus L K_X$ , where $K_X$ is the holomorphic cotangent bundle of $X$ . This realizes $F$ as a rank-two subsheaf of $V$ , isomorphic to $V$ over $X-D$ , where $D$ is the ramification locus of the corresponding spectral cover. We show by a gluing construction that on compact sets in $X-D, h_t$ converges in an appropriate sense to a metric on $F$ , singular at $D$ , solving decoupled Hitchin's equations. The limit is characterized, via the Hecke modification, by harmonic metrics on $L$ and on $K_X$ with certain parabolic weights at $D$ . We give rules to determine these parabolic weights.

Let $M$ be a compact oriented 3-manifold with non-empty boundary consisting of surfaces of genii $\gt 1$ such that the interior of $M$ is hyperbolizable. We show that for each spherical cone-metric $d$ on $\partial M$ such that all cone-angles are greater than 2ドル \pi$ and the lengths of all closed geodesics that are contractible in $M$ are greater than 2ドル \pi$ there exists a unique strictly polyhedral hyperbolic metric on $M$ such that $d$ is the induced dual metric on $\partial M$ .

The classical Prym construction associates to a smooth, genus $g$ complex curve $X$ equipped with a nonzero cohomology class $\theta \in H^1(X, \mathbb{Z} / 2 \mathbb{Z})$ , a principally polarized abelian variety (PPAV) Prym $(X, \theta)$ . Denote the moduli space of pairs $(X, \theta)$ by $\mathcal{R}_g$ , and let $\mathcal{A}_h$ be the moduli space of PPAVs of dimension $h$ . The Prym construction globalizes to a holomorphic map of complex orbifolds Prym: $\mathcal{R}_g \rightarrow \mathcal{A}_{g-1}$ . For $g \geq 4$ and $h \leq g-1$ , we show that Prym is the unique nonconstant holomorphic map of complex orbifolds $F$ : $\mathcal{R}_g \rightarrow \mathcal{A}_h$ . This solves a conjecture of Farb. A main component in our proof is a classification of homomorphisms $\pi_1^{\text {arb }}\left(\mathcal{R}_g\right) \rightarrow \mathrm{Sp}(2 h, \mathbb{Z})$ for $h \leq g-1$ . This is achieved using arguments from geometric group theory and low-dimensional topology.