Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1609.03043

The kinematic formula in the 3D-Heisenberg group

Yen-Chang Huang
View original source ↗Read source document ↗Export metadata ↓

Abstract

By studying the group of rigid motions, $PSH(1)$, in the 3D-Heisenberg group $H_1$, we define the density and the measure for the sets of horizontal lines. We show that the volume of a convex domain $D\subset H_1$ is equal to the integral of length of chord over all horizontal lines intersecting $D$. As the classical result in integral geometry, we also define the kinematic density for $PSH(1)$ and show the probability of randomly throwing a vector $v$ interesting the convex domain $D\subset D_0$ under the condition that $v$ is contained in $D_0$. Both results show the relationship connecting the geometric probability and the natural geometric quantity in Cheng-Hwang-Malchiodi-Yang's work approached by the variational method.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yen-Chang Huang. 2016-09-10. The kinematic formula in the 3D-Heisenberg group. https://doi.org/10.1515/agms-2016-0020

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the topology of manifolds with nonnegative Ricci curvature and linear volume growth

Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature and linear volume growth. First, we show that the fundamental group of such a manifold contains a subgroup $\mathbb{Z}^k$ of finite index, where $0\le k\le n-1$. Second, we prove that if the Ricci curvature is positive everywhere, then the fundamental group is finite. The proofs are based on an analysis of the equivariant asymptotic geometry of successive covering spaces and a plane/halfplane rigidity result for RCD spaces.

math.DG↗
arXiv2026

K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers

Recently, Sun-Zhang have developed an algebraic theory for Kähler-Ricci shrinkers showing that they admit the structure of a polarized Fano fibration $(π: X \to Y, ξ)$. In particular, they conjecture that existence of a Kähler-Ricci shrinker metric is equivalent to a notion of K-stability. We prove one direction of this conjecture, namely that existence of a Kähler-Ricci shrinker metric $g$ implies K-polystability of $(π: X \to Y, ξ)$, in the case that the Ricci curvature of $g$ decays at infinity. As an application, we give a non-existence result: if $M$ is the blowup of a six-dimensional quadric along a two-dimensional subquadric, then the total space $X$ of the cube root of $K_M$ is a polarized Fano fibration not admitting a Kähler-Ricci shrinker.

math.DG↗
arXiv2026

Observações sobre funções potenciais de variedades quase-Einstein não compactas

Neste artigo, estudamos o conjunto de funções potenciais em variedades quase Einstein não compactas. Mostramos que o espaço de todas as funções potenciais positivas em uma variedade tridimensional não compacta quase-Einstein tem dimensão no máximo dois, e que a igualdade vale se e somente se a variedade for isométrica a um produto $B\times\mathbb{R}$, onde $B$ é uma superfície $λ$-Einstein ou um dos exemplos obtidos por L. Berard Bergery e descritos no livro de Besse. Além disso, provamos que qualquer variedade quase-Einstein assintoticamente plana $n$-dimensional com $λ=0$ é necessariamente Ricci-plana.

math.DG↗