arXiv research

Simplifies Wulff theorem for crystalline shapes using Minkowski Theory.

Novel framework improves randomized smoothing for various norms.

The paper proves a Wulff inequality for minimal submanifolds with boundary in Euclidean space.

The paper proves stability of Wulff shapes using anisotropic curvature functionals.

New proof of Wulff-Gage inequality with applications.

Study proves inequality for hypersurfaces and shows almost extremals are close to Wulff shape.

Local minimizers are convex and close to Wulff shapes.

In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class $C^1$. Moreover, applications of this result are given.

New theorem proves convex bodies with specific curvature measures are rescaled Wulff shapes.

We show that for elliptic parametric functionals whose Wulff shape is smooth and has strictly positive curvature, any surface with constant anisotropic mean curvature which is a topological sphere is a rescaling of the Wulff shape.

Researchers create crystallizations of lens spaces.

Sharp reverse affine isoperimetric inequalities for asymmetric Wulff shapes and their polars are established, along with the characterization of all extremals. These new inequalities have as special cases previously obtained simplex inequalities by Ball, Barthe and Lutwak, Yang, and Zhang. In particular, they provide t…

Machine learning predicts molecular crystal stability.

ShotgunCSP predicts crystal structures using machine learning, achieving high accuracy with minimal computation.

We study the stability of closed, not necessarily smooth, equilibrium surfaces of an anisotropic surface energy for which the Wulff shape is not necessarily smooth. We show that if the Cahn Hoffman field can be extended continuously to the whole surface and if the surface is stable, then the surface is, up to rescaling…

We constructed physically stable sp2 negatively curved cubic carbon structures which reticulate a Schwarz P-like surface. The method for constructing such crystal structures is based on the notion of the standard realization of abstract crystal lattices. In this paper, we expound on the mathematical method to construct…

The Machine Recognition of Crystallization Outcomes (MARCO) initiative has assembled roughly half a million annotated images of macromolecular crystallization experiments from various sources and setups. Here, state-of-the-art machine learning algorithms are trained and tested on different parts of this data set. We fi…

Optimal inequality for free boundary hypersurfaces in convex domains.

New minimal surfaces found from vortex crystals.

The paper introduces a new flow to converge to a Wulff shape from a smooth convex hypersurface.

Given a positive function F on S n satisfying an appropriate con-vexity assumption, we consider hypersurfaces for which a linear combination of some higher order anisotropic curvatures is constant. We define the varia-tional problem for which these hypersurfaces are critical points and we prove that, up to translations…

Minimal crystallizations of simply connected PL 4-manifolds are very natural objects. Many of their topological features are reflected in their combinatorial structure which, in addition, is preserved under the connected sum operation. We present a minimal crystallization of the standard PL K3 surface. In combination w…

The paper studies special crystallizations of 4-manifolds to minimize certain PL-invariants.

Study shows limits of volume-constrained sets are finite unions of Wulff shapes.

One-dimensional crystals have convex shapes under certain conditions.

The paper proves inequalities for star-shaped and $F$-mean convex hypersurfaces in $\mathbb{R}^{n+1}$.

We give a complete solution to the extremal topological combinatorial problem of finding the minimum number of tiles needed to construct a polyomino with $h$ holes. We denote this number by $g(h)$ and say that a polyomino is crystallized if it has $h$ holes and $g(h)$ tiles. We analyze structural properties of crystall…

CRYSPNet predicts crystal structures from chemical compositions.

We study a variational problem for piecewise-smooth hypersurfaces in the (n+1)-dimensional Euclidean space with an anisotropic energy. An anisotropic energy is the integral of an energy density that depends on the normal at each point over the considered hypersurface. The minimizer of such an energy among all closed hy…

The paper studies a curve flow preserving anisotropic length for convex curves, leading to a homothetic limit.

Study on anisotropic curvature flow for noncompact convex hypersurfaces.

A fundamental problem in applying machine learning techniques for chemical problems is to find suitable representations for molecular and crystal structures. While the structure representations based on atom connectivities are prevalent for molecules, two-dimensional descriptors are not suitable for describing molecula…

Given a positive function F on Sn which satisfies a convexity condition, we introduce the r-th anisotropic mean curvature Mr for hypersurfaces in Rn+1 which is a generalization of the usual r-th mean curvature Hr. We get integral formulas of Minkowski type for compact hypersurfaces in Rn+1. We give some new characteriz…

Study anisotropic flow for capillary hypersurfaces, proving new inequalities.

We prove a qualitative and a quantitative stability of the following rigidity theorem: an anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider $n \geq 2$, $p\in (1, \, +\infty)$ and $Σ$ an $n$-dimensional, closed hypersurface in $\mathbb{R}^{n+1}$, boundary of a convex, open set. We show that …

The article studies crystallizations of small covers over simple polytopes and finds unique crystallizations for the $n$-simplex.

We have defined weight of the pair $(\langle S \mid R \rangle, R)$ for a given presentation $\langle S \mid R \rangle$ of a group, where the number of generators is equal to the number of relations. We present an algorithm to construct crystallizations of 3-manifolds whose fundamental group has a presentation with two …

The paper solves a thermodynamics problem about crystal shape.

Compact hypersurfaces minimize area in convex cones with free boundary.

Consider a finite connected graph possibly with multiple edges and loops. In discrete geometric analysis, Kotani and Sunada constructed the crystal associated to the graph as a standard realization of the maximal abelian covering of the graph. As an application of what the author showed in an earlier paper with Seshadr…

The paper studies curvature measures and volume-preserving flows on convex bodies.

Imaging techniques are essential tools for inquiring a number of properties from different materials. Liquid crystals are often investigated via optical and image processing methods. In spite of that, considerably less attention has been paid to the problem of extracting physical properties of liquid crystals directly …

Paper extends Wente's result to anisotropic capillary surfaces in half-spaces.

We show how the smooth geometry of Calabi-Yau manifolds emerges from the thermodynamic limit of the statistical mechanical model of crystal melting defined in our previous paper arXiv:0811.2801. In particular, the thermodynamic partition function of molten crystals is shown to be equal to the classical limit of the par…

Minimal crystallizations bound for 3-manifolds with boundary.

For $d\geq 2$, the regular genus of a closed connected PL $d$-manifold $M$ is the least genus (resp., half of the genus) of an orientable (resp., a non-orientable) surface into which a crystallization of $M$ imbeds regularly. The regular genus of every orientable surface equals its genus, and the regular genus of every…

Machine learning predicts perovskite formability and classifies crystal structures.

Simple crystallizations are edge-coloured graphs representing PL 4-manifolds with the property that the 1-skeleton of the associated triangulation equals the 1-skeleton of a 4-simplex. In the present paper, we prove that any (simply-connected) PL $4$-manifold $M$ admitting a simple crystallization admits a special hand…