We study a genus-three hyperelliptic holomorphic null-curve family in $\mathbb{C}^4$, modelled on the algebraic data of the classical Schwarz P/D family, whose real parts define minimal immersions into $\mathbb{R}^4$. For the order-four automorphism $\omega\mapsto i\omega$ of the underlying Schwarz curve, we compute explicitly its action on the four holomorphic Weierstrass $1$-forms and derive the resulting identities for all real period vectors. Consequently, for every lattice invariant under the induced target rotation, torus-period closure can be checked on one representative from each orbit of a symmetry-stable homology generating set. We also identify precisely when the additional parameter produces a nondegenerate codimension-two deformation. The paper does not claim the construction of a new embedded periodic minimal surface in $\mathbb{R}^4$; rather, it provides an explicit symmetry reduction of the period problem associated with this family.
In this paper, we develop methods for constructing $\mathbb{Z}/2$-harmonic 1-forms in dimension three by varying the background metric. On $\mathbb{R}^3$, we construct an example for which the complement of the smooth locus in the singular set is a Cantor set. We realize every finite graph with positive even valence at...
For every admissible non-maximal Toledo invariant $t$, we construct a locus of irreducible representations of a closed genus-$g$ surface group into $\mathrm{PU}_{2,n+1}$ with Toledo invariant $t$ whose orbit maps are quasi-isometric embeddings. These loci have real codimension $10(g-1)$ or $10(g-1)+2(n-1)$ in the chara...
In this paper we study the global geometry and classification of complete minimal surfaces with embedded ends and finite total curvature in the product space $\mathbb{H}^2 \times \mathbb{R}$. We describe the structure of embedded ends of such surfaces, constraining the combinatorics of their asymptotic polygons at infi...
Let $X$ be a genus $g\geq 0$ compact Riemann surface that admits an antiholomorphic involution $\iota$ with non empty fixed-point set, and let $D = \{p_1,\dots,p_k\}\subset \text{Fix}(\iota)$. At each puncture, we prescribe a Scherk map associated to a given real polynomial quadratic differential with even degree and n...
We develop a framework for studying discrete subgroups of $\mathsf{PGL}(d+1,\mathbb{R})$ via the paracomplex hyperbolic space $\mathbb{H}_\tau^d$, a rank-$1$ pseudo-Riemannian symmetric space. We characterize projective transverse, relatively Anosov, and Anosov subgroups in terms of properly discontinuous, geometricall...
Nearly $\mathrm{G}_2$ manifolds are positive Einstein seven-dimensional manifolds whose associated cone metric has special holonomy equal to, or contained in, $\mathrm{Spin}(7)$. We study nearly parallel $\mathrm{G}_2$-structures invariant under a cohomogeneity-one action of $\mathrm{SO}(4)$, and construct the first in...
S. Salamon, Ragini Singhal· 0 citations
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