Polar actions on Damek-Ricci spaces

Damek-Ricci空间上的微分共形刚性

Infinitesimal Conformal Rigidity on Damek-Ricci Spaces

本文研究了Damek-Ricci空间上的共形向量场，证明其必为Killing向量场，采用解析方法给出了构造性证明。

Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces

In this article we consider solvable hypersurfaces of the form $N \exp(\R H)$ with induced metrics in the symmetric space $M = SL(3,\C)/SU(3)$, where $H$ a suitable unit length vector in the subgroup $A$ of the Iwasawa decomposition $SL(3,\C) = NAK$. Since $M$ is rank $2$, $A$ is $2$-dimensional and we can parametrize these hypersurfaces via an angle $α\in [0,π/2]$ determining the direction of $H$. We show that one of the hypersurfaces (corresponding to $α= 0$) is minimally embedded and isometric to the non-symmetric $7$-dimensional Damek-Ricci space. We also provide an explicit formula for th…

Damek-Ricci空间中的球状等参超曲面

Sphere-like isoparametric hypersurfaces in Damek-Ricci spaces

本文在Damek-Ricci空间中通过过度膨胀超球面构造新的球状等参超曲面族，连接测地球面与已知等参超曲面，并研究其齐次性和焦簇的全测地条件。

Damek-Ricci空间上某些有理等参函数的分类

Classification of Certain Rational Isoparametric Functions on Damek-Ricci Spaces

本文分类了Damek-Ricci空间上以标准坐标多项式除以t形式的有理等参函数，涵盖已知例子并产生新例，确定焦点簇与平均曲率，并处理了C H^k中R H^k管状邻域的例外情形。

关于超几何型调和流形的体积熵的一个注记

A note on the volume entropy of harmonic manifolds of hypergeometric type

本文研究超几何型调和流形在 Ricci 曲率归一化条件下的体积熵不等式，证明上界仅由截面曲率为 -1 的实双曲空间达到，并指出 Damek-Ricci 空间达到下界的例子仅有四种。

Codimension one Ricci soliton subgroups of solvable Iwasawa groups

Recently, Jablonski proved that, to a large extent, a simply connected solvable Lie group endowed with a left-invariant Ricci soliton metric can be isometrically embedded into the solvable Iwasawa group of a non-compact symmetric space. Motivated by this result, we classify codimension one subgroups of the solvable Iwasawa groups of irreducible symmetric spaces of non-compact type whose induced metrics are Ricci solitons. We also obtain the classifications of codimension one Ricci soliton subgroups of Damek-Ricci spaces and generalized Heisenberg groups.

Explicit p-harmonic functions on rank-one Lie groups of Iwasawa type

We construct explicit proper p-harmonic functions on rank-one Lie groups of Iwasawa type. This class of Lie groups includes many classical Riemannian manifolds such as the rank-one symmetric spaces of non-compact type, Damek-Ricci spaces, rank-one Einstein solvmanifolds and Carnot spaces.

On homogeneous manifolds whose isotropy actions are polar

We show that simply connected Riemannian homogeneous spaces of compact semisimple Lie groups with polar isotropy actions are symmetric, generalizing results of Fabio Podesta and the third named author. Without assuming compactness, we give a classification of Riemannian homogeneous spaces of semisimple Lie groups whose linear isotropy representations are polar. We show for various such spaces that they do not have polar isotropy actions. Moreover, we prove that Heisenberg groups and non-symmetric Damek-Ricci spaces have non-polar isotropy actions.

非负 Ricci 曲率且万有覆盖在无穷远处几乎 $k$-极的流形的基本群的有限生成性

Finite generation of fundamental groups for manifolds with nonnegative Ricci curvature whose universal cover is almost $k$-polar at infinity

本文证明非负 Ricci 曲率完备开流形在万有覆盖满足几乎 $k$-极条件时基本群有限生成且几乎阿贝尔，并在几乎最大体积增长时形变收缩到平坦流形。

Space of Ricci flows (II)

Based on the compactness of the moduli of non-collapsed Calabi-Yau spaces with mild singularities, we set up a structure theory for polarized Kähler Ricci flows with proper geometric bounds. Our theory is a generalization of the structure theory of non-collapsed Kähler Einstein manifolds. As applications, we prove the Hamilton-Tian conjecture and the partial-$C^0$-conjecture of Tian.

RCD 空间上凸函数 Hessian 与 Ricci 张量的精细表示

Fine representation of Hessian of convex functions and Ricci tensor on RCD spaces

本文在 RCD 空间上为分布 Ricci 张量建立极分解，并推广至凸函数 Hessian 与截面曲率算子，解决了若干开放问题。

On the Hodge Metric of the Universal Deformation Space of Calabi-Yau Threefolds

In this paper, we represent the Hodge metric in terms of the Weil-Petersson metric and its Ricci curvature on the moduli spaces of polarized Calabi-Yau threefolds.