Schauder estimates on products of cones

不可微分极小乘积锥体无法通过光滑校准来校准

Cones over minimal products cannot be calibrated by smooth calibrations

本文扩展了[Zha26]的关键结果，证明极小乘积结构自动使所有非平凡极小乘积上的锥体无法通过任何全局定义的光滑校准来校准。

Area-minimizing Cones over Products of Grassmannian Manifolds

This paper is the continuation of the previous one \cite{Cui2021}, where we re-proved the area-minimization of cones over Grassmannians of $n$-planes $G(n,m;\mathbb{F})(\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H})$, Cayley plane $\mathbb{O}P^2$ from the point view of Hermitian orthogonal projectors, and gave area-minimizing cones associated to oriented real Grassmannians $\widetilde{G}(n,m;\mathbb{R})$. In this paper, we make a further step on showing that the cones, of dimension no less than $\mathbf{8}$, over minimal products of $G(n,m;\mathbb{F})$ are area-minimizing. Moreover, those cones …

On the stability of minimal cones in warped products

In a seminal paper published in $1968$, J. Simons proved that, for $n\leq 5$, the Euclidean (minimal) cone $CM$, built on a closed, oriented, minimal and non totally geodesic hypersurface $M^n$ of $\mathbb S^{n+1}$ is unstable. In this paper, we extend Simons' analysis to {\em warped} (minimal) cones built over a closed, oriented, minimal hypersurface of a leaf of suitable warped product spaces. Then, we apply our general results to the particular case of the warped product model of the Euclidean sphere, and establish the unstability of $CM$, whenever $2\leq n\leq 14$ and $M^n$ is a closed, or…

锥体和扭曲积流形上的热点

Hot spots on cones and warped product manifolds

研究了扭曲积流形上热方程热点的极值问题及长期行为，证明了Rauch猜想在特定条件下成立，并揭示了谱间隙对长期行为的影响。

Twisted products and $SO(p)\times SO(q)$-invariant special Lagrangian cones

We construct $\sorth{p} \times \sorth{q}$-invariant special Lagrangian (SL) cones in $\C^{p+q}$. These SL cones are natural higher-dimensional analogues of the $\sorth{2}$-invariant SL cones constructed previously by MH and used in our gluing constructions of higher genus SL cones in $\C^{3}$. We study in detail the geometry of these $\sorth{p}\times \sorth{q}$-invariant SL cones, in preparation for their application to our higher dimensional special Legendrian gluing constructions. In particular the symmetries of these cones and their asymptotics near the spherical limit are analysed. All $\s…

Duality on geodesics of Cartan distributions and sub-Riemannian pseudo-product structures

Given a five dimensional space endowed with a Cartan distribution, the abnormal geodesics form another five dimensional space with a cone structure. Then it is shown, if the cone structure is regarded as a control system, then, the space of abnormal geodesics of the cone structure is naturally identified with the original space. In this paper, we provide an exposition on the duality by abnormal geodesics in a wider framework, namely, in terms of quotients of control systems and sub-Riemannian pseudo-product structures. Also we consider the controllability of cone structures and describe the co…

代数锥乘积上的全纯张量

Holomorphic tensors on products of algebraic cones

研究两个代数锥乘积上的全纯张量，证明在特定条件下张量在收缩的Zariski闭包下不变，并应用于Sasaki流形乘积，得到Reeb场流下的不变性。

The automorphism group and the non-self-duality of $p$-cones

In this paper, we determine the automorphism group of the $p$-cones ($p\neq 2$) in dimension greater than two. In particular, we show that the automorphism group of those $p$-cones are the positive scalar multiples of the generalized permutation matrices that fix the main axis of the cone. Next, we take a look at a problem related to the duality theory of the $p$-cones. Under the Euclidean inner product it is well-known that a $p$-cone is self-dual only when $p=2$. However, it was not known whether it is possible to construct an inner product depending on $p$ which makes the $p$-cone self-dual…

Stable and unstable Einstein warped products

In this article, we systematically investigate the stability properties of certain warped product Einstein manifolds. We characterize stability of these metrics in terms of an eigenvalue condition of the Einstein operator on the base manifold. In particular, we prove that all complete manifolds carrying imaginary Killing spinors are strictly stable. Moreover, we show that Ricci-flat and hyperbolic cones over Kähler-Einstein Fano manifolds and over nonnegatively curved Einstein manifolds are stable if the cone has dimension $n\geq10$.

映射锥Thom-Smale复形的瞬子构造

Instanton construction of the mapping cone Thom-Smale complex

本文通过将Morse函数和两个参数变形的映射锥拉普拉斯算子的特征空间，构造了瞬子上链复形，并证明其与拓扑构造的映射锥Thom-Smale复形上链同构。

New G2 holonomy cones and exotic nearly Kaehler structures on the 6-sphere and the product of a pair of 3-spheres

There is a rich theory of so-called (strict) nearly Kaehler manifolds, almost-Hermitian manifolds generalising the famous almost complex structure on the 6-sphere induced by octonionic multiplication. Nearly Kaehler 6-manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional Lie group G2: the metric cone over a Riemannian 6-manifold M has holonomy contained in G2 if and only if M is a nearly Kaehler 6-manifold. A central problem in the field has been the absence of any complete …