Related papers: Discrete Curvatures and Convex Polytopes

We show that for a very general and natural class of curvature functions, the problem of finding a complete strictly convex hypersurface satisfying f({\kappa}) = {\sigma} over (0,1) with a prescribed asymptotic boundary {\Gamma} at infinity…

Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…

A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study whether these metrics have negative Ricci curvatures. Affirmatively, we prove…

Let K be a convex body in $R^d$. A random polytope is the convex hull $[x_1,...,x_n]$ of finitely many points chosen at random in K. $\Bbb E(K,n)$ is the expectation of the volume of a random polytope of n randomly chosen points. I.…

In this paper, we study the fundamental group of the complete steady gradient Ricci soliton with nonnegative sectional curvature. We prove that the fundamental group of such a Ricci soliton is either trivial or infinite. As a corollary, we…

In this paper we prove weak L^{1,p} (and thus C^{\alpha}) compactness for the class of uniformly mean-convex Riemannian n-manifolds with boundary satisfying bounds on curvature quantities, diameter, and (n-1)-volume of the boundary. We…

We study complete non-compact manifolds of positive scalar curvature, with a focus on how curvature decay is constrained by topology at infinity. Our first main result shows that topological linking at infinity forces polynomial decay of…

We determine the combinatorial types of all the 3-dimensional simple convex polytopes in R^3 that can be realized as mean curvature convex (or totally geodesic) Riemannian polyhedra with non-obtuse dihedral angles in Riemannian 3-manifolds…

In this article we study adjoint hypersurfaces of geometric objects obtained by intersecting simple polytopes with few facets in $\mathbb{P}^5$ with the Grassmannian $\mathrm{Gr}(2,4)$. These generalize the positive Grassmannian, which is…

Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete, conformally compact metrics. Related to this issue, an…

Let $\Sigma\subset \mathbb{R}^{2n}$ with $n\geq2$ be any $C^2$ compact convex hypersurface. The stability of closed characteristics has attracted considerable attention in related research fields. A long-standing conjecture states that all…

In this paper, we introduce a new notion of curvature on the edges of a graph that is defined in terms of effective resistances. We call this the Ricci--Foster curvature. We study the Ricci flow resulting from this curvature. We prove the…

Discrete curvatures are quantities associated to the nodes and edges of a graph that reflect the local geometry around them. These curvatures have a rich mathematical theory and they have recently found success as a tool to analyze networks…

We prove new upper and lower bounds on transversal numbers of several classes of simplicial complexes. Specifically, we establish an upper bound on the transversal numbers of pure simplicial complexes in terms of the number of vertices and…

In this paper we study 4d gradient steady Ricci solitons, which are weak $\kappa$-solutions, and admit O(3)-symmetry. Under a weak curvature decay condition, we find precise geometric asymptotics of such solitons, which are similar to those…

We show that if a planar graph $G$ with minimum degree at least $3$ has positive Lin-Lu-Yau Ricci curvature on every edge, then $\Delta(G)\leq 17$, which then implies that $G$ is finite. This is an analogue of a result of DeVos and Mohar…

We explore existence of invariant metrics with positive intermediate Ricci curvature on closed, low-dimensional cohomogeneity one manifolds. For a certain cohomogeneity one $\mathsf{Spin}(4)$-action on $S^3 \times \mathbb{C}\mathrm{P}^2$,…

Given an Einstein structure with positive scalar curvature on a four-dimensional Riemannian manifolds, that is $Ric=\lambda g$ for some positive constant $\lambda$. For convenience, the Ricci curvature is always normalized to $Ric=1$. A…

Let $(M^{n+1},g)$ be a closed Riemannian manifold of dimension $3\le n+1\le 5$. We show that, if the metric $g$ is generic or if the metric $g$ has positive Ricci curvature, then $M$ contains infinitely many geometrically distinct constant…

We show the existence of families of periodic polyhedra in spaces of constant curvature whose fundamental domains can be obtained by attaching prisms and antiprisms to Archimedean solids. These polyhedra have constant discrete curvature and…