Discrete Differential Geometry Lab


Hao Chen (陳浩)

I'm a post-doc at the Discrete Differential Geometry Lab of the Georg-August-Universität Göttingen. I'm funded by DFG individual grant under the project "Defects in Triply Periodic Minimal Surfaces" (Projekt number 398759432).



Research interests

As a discrete geometer, I'm interested in polytopes, sphere packings, geometric representation of graphs and so on, preferably in non-Euclidean and high dimensional spaces.
I recently developed interests in differential geometry, mainly on minimal surfaces and constant mean curvature surfaces.



Contact

Georg-August-Universität Göttingen
Institut für Numerische und Angewandte Mathematik
Lotzestr. 16-18
D-37083 Göttingen, Germany
Phone: +49 (0)551 39-12464
E-mail: h.chen@math.uni-goettingen.de
E-mail: hao.chen.math@gmail.com




Publication (see my arXiv, Google Scholar, ORCID or MathSciNet pages for a more complete list)

Preprints
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A new deformation family of Schwarz' D surface
arXiv PDF with Matthias Weber
An orthorhombic deformation family of Schwarz' H surfaces
arXiv PDF with Matthias Weber
Summary: These two papers show, respectively, two new 2-dimensional families of triply periodic minimal surfaces of genus three, which can be seen as orthorhombic deformations of Schwarz' D and H surfaces. These families are exceptional since they do not belong to Meeks' 5-dimensional family, yet the 1-dimensional "intersections" with Meeks family exhibit singularities in the moduli space of triply periodic minimal surfaces of genus three. Picture on the left compares a new surface with the classical Schwarz tD surface. See Matthias blog post for a gentle introduction.

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Weakly Inscribed Polyhedra
arXiv PDF with Jean-Marc Schlenker
Abstract: We study convex polyhedra in the projective 3-space with all their vertices on a sphere. We do not require, in particular, that the polyhedra lie in the interior of the sphere, hence the term "weakly inscribed". Such polyhedra can be interpreted as ideal polyhedra, if we regard the projective space as a combination of the hyperbolic space and the de Sitter space, with the sphere as the common ideal boundary. We have three main results: (1) the 1-skeleta of weakly inscribed polyhedra are characterized in a purely combinatorial way, (2) the exterior dihedral angles are characterized by linear programming, and (3) we also describe the hyperbolic-de Sitter structure induced on the boundary of weakly inscribed polyhedra.

Articles
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Competition brings out the best: modelling the frustration between curvature energy and chain stretching energy of lyotropic liquid crystals in bicontinuous cubic phases
arXiv PDF with Chenyu Jin
Abstract: It is commonly considered that the frustration between the curvature energy and the chain stretching energy plays an important role in the formation of lyotropic liquid crystals in bicontinuous cubic phases. Theoretic and numeric calculations were performed for two extreme cases: parallel surfaces eliminate the variance of the chain length; constant mean curvature surfaces eliminate the variance of the mean curvature. We have implemented a model with Brakke's Surface Evolver which allows a competition between the two variances. The result shows a compromise of the two limiting geometries. With data from real systems, we are able to recover the gyroid–diamond–primitive phase sequence which was observed in experiments.

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Minimal Twin Surfaces
arXiv PDF
Abstract: We report some minimal surfaces that can be seen as copies of a triply periodic minimal surface (TPMS) related by reflections in parallel mirrors. We call them minimal twin surfaces for the resemblance with twin crystal. Brakke’s Surface Evolver is employed to construct twinnings of various classical TPMS, including Schwarz’ Primitive (P) and Diamond (D) surfaces, their rhombohedral deformations (rPD), and Schoen’s Gyroid (G) surface. Our numerical results provide strong evidences for the mathematical existence of D twins and G twins, which are recently observed in experiment by material scientists. For rPD twins, we develop a good understanding, by noticing examples previously constructed by [Traizet 08] and [Fujimori and Weber 09]. Our knowledge on G twins is, by contrast, very limited. Nevertheless, our experiments lead to new cubic polyhedral models for the D and G surfaces, inspired by which we speculate new TPMS deformations in the framework of Traizet.

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Selectively Balancing Unit Vectors
arXiv PDF with Aart Blokhuis
Abstract: A set U of unit vectors is selectively balancing if one can find two disjoint subsets U+ and U-, not both empty, such that the Euclidean distance between the sum of U+ and the sum of U- is smaller than 1. We prove that the minimum number of unit vectors that guarantee a selectively balancing set in ℝ n is asymptotically 1/2nlogn.

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Scribability problems for polytopes
arXiv PDF with Arnau Padrol
Abstract: In this paper we study various scribability problems for polytopes. We begin with the classical k-scribability problem proposed by Steiner and generalized by Schulte, which asks about the existence of d-polytopes that cannot be realized with all k-faces tangent to a sphere. We answer this problem for stacked and cyclic polytopes for all values of d and k. We then continue with the weak scribability problem proposed by Gr\"unbaum and Shephard, for which we complete the work of Schulte by presenting non weakly circumscribable 3-polytopes. Finally, we propose new (i,j)-scribability problems, in a strong and a weak version, which generalize the classical ones. They ask about the existence of d-polytopes that can not be realized with all their i-faces "avoiding" the sphere and all their j-faces "cutting" the sphere. We provide such examples for all the cases where j−i≤d−3.

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Ball packings with high chromatic numbers from strongly regular graphs
arXiv PDF
Abstract: Inspired by Bondarenko's counter-example to Borsuk's conjecture, we notice some strongly regular graphs that provide examples of ball packings whose chromatic numbers are significantly higher than the dimensions. In particular, from generalized quadrangles we obtain unit ball packings in dimension q3−q2+q with chromatic number q3+1, where q is a prime power. This improves the previous lower bound for the chromatic number of ball packings. (Fig. on the left from Wikipedia)

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Lorentzian Coxeter systems and Boyd-Maxwell ball packings
arXiv PDF with Jean-Philippe Labbé
Even More Infinite Ball Packings from Lorentzian Root Systems
arXiv PDF
Summary: These two papers generalizes the infinite ball packings generated by Coxeter groups proposed by Boyd (1974) and Maxwell (1983). It is motivated by recent studies on infinite root systems. It turns out that the accumulation points of the roots in the projective space leave a pattern of spheres on the light cone. We first noticed a connection between this pattern and the Boyd-Maxwell packings, then extend this connection to a more general notion of root system. In particular, we enumerate all the Coxeter groups of "level 2", which are all the Coxeter groups that generate a ball packing.

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Limit Directions for Lorentzian Coxeter Systems
arXiv PDF with Jean-Philippe Labbé
Abstract: Every Coxeter group admits a geometric representation as a group generated by reflections in a real vector space. In the projective representation space, limit directions are limits of injective sequences in the orbit of some base point. Limit roots are limit directions that can be obtained starting from simple roots. In this article, we study the limit directions arising from any point when the representation space is a Lorentz space. In particular, we characterize the light-like limit directions using eigenvectors of infinite-order elements. This provides a spectral perspective on limit roots, allowing for efficient computations. Moreover, we describe the space-like limit directions in terms of the projective Coxeter arrangement.

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Apollonian Ball Packings and Stacked Polytopes
arXiv PDF
Abstract: We investigate in this paper the relation between Apollonian d-ball packings and stacked (d+1)-polytopes for dimension d≥3 . For d=3, the relation is fully described: we prove that the 1-skeleton of a stacked 4-polytope is the tangency graph of an Apollonian 3-ball packing if and only if there is no six 4-cliques sharing a 3-clique. For higher dimension, we have some partial results. (Fig. on the left from Wikipedia)

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Minimum vertex covers and the spectrum of the normalized Laplacian on trees
arXiv PDF with Jürgen Jost
Abstract: We show that, in the graph spectrum of the normalized graph Laplacian on trees, the eigenvalue 1 and eigenvalues near 1 are strongly related to minimum vertex covers. In particular, for the eigenvalue 1, its multiplicity is related to the size of a minimum vertex cover, and zero entries of its eigenvectors correspond to vertices in minimum vertex covers; while for eigenvalues near 1, their distance to 1 can be estimated from minimum vertex covers; and for the largest eigenvalue smaller than 1, the sign graphs of its eigenvectors take vertices in a minimum vertex cover as representatives.