Eigenforms of the Laplacian for Riemannian V-submersions

Lefschetz decompositions for eigenforms on a Kähler manifold

We show that the eigenspaces of the Laplacian $Δ_k$ on $k$-forms on a compact Kähler manifold carry Hodge and Lefschetz decompositions. Among other consequences, we show that the positive part of the spectrum of $Δ_k$ lies in the spectrum of $Δ_{k+1}$.

本征形式与特殊和乐

Eigenforms and special holonomy

本研究针对特殊和乐完备非紧里奇平坦流形上霍奇拉普拉斯算子的$L^2$本征形式，证明了其存在与非存在结果，聚焦引力瞬子并阐述了相关物理意义。

$η$-Ricci-Yamabe Soliton on Riemannian Submersions from Riemannian manifolds

In this research article, we establish the geometrical bearing on Riemannian submersions in terms of $η$-Ricci-Yamabe Soliton with the potential field and giving the classification of any fiber of Riemannian submersion is an $η$-Ricci-Yamabe soliton, $η$-Ricci soliton and $η$-Yamabe soliton. We also discuss the various conditions for which the target manifold of Riemannian submersion is an $η$-Ricci-Yamabe soliton, $η$-Ricci soliton, $η$-Yamabe soliton and quasi-Yamabe soliton. In a particular case when the potential filed $V$ of the $η$-Ricci-Yamabe soliton is of gradient type, we derive a La…

关于具有全测地纤维的拉普拉斯算子与黎曼淹没的注记

Remark on Laplacians and Riemannian Submersions with Totally Geodesic Fibers

本文研究具有全测地纤维的黎曼淹没的典型变分中拉普拉斯算子的第一正特征值，在纤维为Einstein且Ricci曲率满足条件时得到特征值界，并应用于Yamabe泛函临界点稳定性及四元数Kähler流形的twistor纤维化。

A Lichnerowicz estimate for the spectral gap of the sub-Laplacian

For a second order operator on a compact manifold satisfying the strong Hörmander condition, we give a bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian of a Riemannian manifold. We consider a wide class of such operators which includes horizontal lifts of the Laplacian on Riemannian submersions with minimal leaves.

On the fundamental tone of the $p$-Laplacian on Riemannian manifolds and applications

We present a general lower bound for the fundamental tone for the $p$-Laplacian on Riemannian manifolds carrying a special kind of function. We then apply our result to the cases of negatively curved simply connected manifolds, a class of warped product manifolds and for a class of Riemannian submersions.

黎曼浸没的基本谱的Dirichlet--Neumann对偶性：超对称视角

Dirichlet--Neumann duality for the Basic Spectrum of Riemannian Submersions: A Supersymmetric Perspective

研究纤维具有基本平均曲率的黎曼浸没的谱几何，通过限制拉普拉斯-贝尔特拉米算子于基本函数空间，将谱问题简化为基流形上的加权拉普拉斯问题，并利用超对称量子力学建立基本Dirichlet与Neumann谱在变换S↦1/S下的对偶性。

极化群上次拉普拉斯算子的等价性

Equivalence of sub-Laplacian on Polarized groups

本研究刻画了次黎曼李群间与次拉普拉斯算子可交换的光滑映射，证明其为次黎曼淹没映射，还明确了Carnot群中次拉普拉斯算子可决定次黎曼结构，澄清了相关前期分析。

Nondegeneracy of the eigenvalues of the Hodge Laplacian for generic metrics on 3-manifolds

In this paper we analyze the eigenvalues and eigenfunctions of the Hodge Laplacian for generic metrics on a closed 3-manifold $M$. In particular, we show that the nonzero eigenvalues are simple and the zero set of the eigenforms of degree 1 or 2 consists of isolated points for a residual set of $C^r$ metrics on $M$, for any integer $r\geq2$. The proof of this result hinges on a detailed study of the Beltrami (or rotational) operator on co-exact 1-forms.

Curvature Properties of 3-$(α,δ)$-Sasaki Manifolds

We investigate curvature properties of 3-$(α,δ)$-Sasaki manifolds, a special class of almost 3-contact metric manifolds generalizing 3-Sasaki manifolds (corresponding to $α= δ= 1$) that admit a canonical metric connection with skew torsion and define a Riemannian submersion over a quaternionic Kähler manifold with vanishing, positive or negative scalar curvature, according to $δ= 0$, $αδ> 0$ or $αδ< 0$. We shall investigate both the Riemannian curvature and the curvature of the canonical connection, with particular focus on their curvature operators, regarded as symmetric endomorphisms of the …

Witten deformation for noncompact manifolds with bounded geometry

Motivated by the Landau-Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function $f$ near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of Thom-Smale complex of $f$ as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large $T$. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomol…

磁测地线、Hodge Laplacian特征值与等周不等式

Magnetic geodesics, Hodge Laplacian eigenvalues, and isoperimetric inequalities

本文利用与共存1-形式微分相关的磁测地流及其在Mañé临界能量水平的行为，给出了Hodge Laplacian最小特征值的Cheeger型不等式的新证明，改进了常数和体积依赖性，并探讨了双曲流形上Mañé临界值与特征值的关系。

关于纤维流形上拉普拉斯算子特征值的研究

On the eigenvalues of the Laplacian on fibred manifolds

本文研究了纤维流形上拉普拉斯算子特征值的比较定理，推广了李钦罗维奇和法伯-克拉恩不等式，并证明了在里奇曲率下界条件下陈定理的对应结果。

Eigenvalues of the Laplacian acting on $p$-forms and metric conformal deformations

Let $(M,g)$ be a compact connected orientable Riemannian manifold of dimension $n\ge4$ and let $λ_{k,p} (g)$ be the $k$-th positive eigenvalue of the Laplacian $Δ_{g,p}=dd^*+d^*d$ acting on differential forms of degree $p$ on $M$. We prove that the metric $g$ can be conformally deformed to a metric $g'$, having the same volume as $g$, with arbitrarily large $λ_{1,p} (g')$ for all $p\in[2,n-2]$. Note that for the other values of $p$, that is $p=0, 1, n-1$ and $n$, one can deduce from the literature that, $\forall k >0$, the $k$-th eigenvalue $λ_{k,p}$ is uniformly bounded on any conformal class…

黎曼叶状结构的绝热极限与谱序列

Adiabatic limits and spectral sequences for Riemannian foliations

本文研究一般黎曼叶状结构在绝热极限下 Laplacian 的小特征值与特征形式渐近，用可微谱序列刻画其数量并给出 Hodge 理论解释，推广了已有结果。

测地随机游动提升的不变性原理

Invariance principle for Lifts of Geodesic Random Walks

本文研究 Riemannian 淹没下测地随机游动的提升，在速度分布适当条件下证明其收敛于水平 Brown 运动，并据此给出水平 Laplace 算子与 Laplace-Beltrami 算子几何恒等式的概率证明。