Given an Enriques surface $T$, its universal $\mathrm{K3}$ cover $f : S \to T$, and a genus $g$ linear system $\lvert C \rvert$ on $T$, we construct the relative Prym variety $P_H = \textrm{Prym}_{v, H} (\mathcal{D/C})$, where $\mathcal{C} \to \lvert C \rvert$ and $\mathcal{D} \to \lvert f*C \rvert$ are the universal families, $v$ is the Mukai vector $(0, [D], 2-2g)$, and $H$ is a polarization on $S$. The relative Prym variety is a $(2g-2)$-dimensional possibly singular variety, whose smooth locus is endowed with a hyperkähler structure. This variety is constructed as the closure of the fixed locus of a symplectic birational involution defined on the moduli space $M_{v,H} (S)$. There is a natural Lagrangian fibration $\eta : P_H \to \lvert C \rvert$ that makes the regular locus of $P_H$ into an integrable system whose general fiber is a $(g-1)$-dimensional (principally polarized) Prym variety, which in most cases is not the Jacobian of a curve. We prove that if $\lvert C \rvert$ is a hyperelliptic linear system, then $P_H$ admits a symplectic resolution which is birational to a hyperkähler manifold of $K3^{[g-1]}$-type, while if $\lvert C \rvert$ is not hyperelliptic, then $P_H$ admits no symplectic resolution. We also prove that any resolution of $P_H$ is simply connected and, when $g$ is odd, any resolution of $P_H$ has $h^{2,0}$-Hodge number equal to one.

Li-Zinger’s hyperplane property for reduced genus one GW-invariants of quintics states that the genus one GW-invariants of the quintic threefold is the sum of its reduced genus one GW-invariants and $1/12$ times its genus zero GW-invariants. We apply the theory of GW-invariants of stable maps with fields to give an algebro-geometric proof of this hyperplane property.

We consider contracting and expanding curvature flows in $\mathbb{S}^{n+1}$. When the flow hypersurfaces are strictly convex we establish a relation between the contracting hypersurfaces and the expanding hypersurfaces which is given by the Gauß map. The contracting hypersurfaces shrink to a point $x_0$ while the expanding hypersurfaces converge to the equator of the hemisphere $\mathcal{H}(-x_0)$. After rescaling, by the same scale factor, the rescaled hypersurfaces converge to the unit spheres with centers $x_0$ resp. $-x_0$ exponentially fast in $C^{\infty} (\mathbb{S}^n)$.

In Rigidity for quasi-Möbius group actions, M. Bonk and B. Kleiner proved a rigidity theorem for expanding quasi-Möbius group actions on Ahlfors $n$-regular metric spaces with topological dimension $n$. This led naturally to a rigidity result for quasi-convex geometric actions on $\textrm{CAT}(-1)$-spaces that can be seen as a metric analog to the “entropy rigidity” theorems of U. Hamenstädt and M. Bourdon. Building on the ideas developed in Rigidity for quasi-Möbius group actions, we establish a rigidity theorem for certain expanding quasi-Möbius group actions on spaces with different metric and topological dimensions. This is motivated by a corresponding entropy rigidity result in the coarse geometric setting.

Using results of Colding-Minicozzi and an extension due to Meeks, we prove that a sequence of properly embedded minimal disks in a 3-ball must have a subsequence whose curvature blow-up set lies in a union of disjoint $C^1$curves.