Colette Anné, Junya Takahashi
The goal of the present paper is to calculate the limit spectrum of the Hodge-de Rham operator under the perturbation of collapsing one part of a manifold obtained by gluing together two manifolds with the same boundary. It appears to take place in the general problem of blowing up conical singularities as introduced in \cite{Maz} and \cite{Row1,Row2}.
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In the present paper, we consider the Hodge-de Rham Laplacian that acts on conformal Killing and projective Killing one-forms of a compact Riemannian manifold.
We analyze the limit of the p-form Laplacian under a collapse, with bounded sectional curvature and bounded diameter, to a smooth limit space. As an application, we characterize when the p-form Laplacian has small positive eigenvalues in a collapsing sequence.
In this paper, we obtain "universal" inequalities for eigenvalues of the weighted Hodge Laplacian on a compact self-shrinker of Euclidean space. These inequalities generalize the Yang-type and Levitin-Parnovski inequalities for eigenvalues of the Laplacian and Laplacian. From the recursion formula of Cheng and Yang \cite{ChengYang07}, the Yang-type inequality for eigenvalues of the weighted Hodge Laplacian are optimal in the sense of the order of eigenvalues.
In this paper, we derive a gradient estimate for the linear combinations of eigenforms of the Hodge Laplacian on a closed manifold. The estimate is given in terms of the dimension, volume, diameter and curvature bound of the manifold. As an application, we obtain directly a sharp estimate for the heat kernel of the Hodge Laplacian.
Eigenvalue estimates for the magnetic Hodge Laplacian on differential forms
本文提出了磁 Hodge Laplacian，将函数上的磁 Laplacian 推广至微分形式，并对比分析了该新算子与已知谱结果的异同。
On the first eigenvalue of the Hodge Laplacian of submanifolds
本文证明在空间形式中闭子流形的第一个p-特征值的严格下界等号成立仅可能在正曲率的球面成立。
In this paper, which is a sequel to math.DG/9902111, we analyze the limit of the p-form Laplacian under a collapse with bounded sectional curvature and bounded diameter to a singular limit space. As applications, we give results about upper and lower bounds on the j-th eigenvalue of the p-form Laplacian, in terms of sectional curvature and diameter.
Combinatorial and Hodge Laplacians: Similarity and Difference
本文通过引入基于离散外微积分的边界诱导图（BIG）拉普拉斯算子，系统比较了组合拉普拉斯算子与霍奇拉普拉斯算子的异同，并实验验证了BIG特征值向霍奇拉普拉斯特征值收敛的条件。
We give some sharp lower bounds of the first eigenvalue for the Hodge Laplacian acting on differential forms on the boundary of a Riemannian manifold. We also give some sharp estimates for the first nonzero Steklov eigenvalue for differential forms.
Small eigenvalues of the Hodge-Laplacian with sectional curvature bounded below
本文在截面曲率有下界的单位体积闭流形上构造度量族，使Hodge-Laplacian作用于p-形式的第k个正特征值趋于零，从而对先前结果施加了曲率约束。
Weighted Hodge Laplacians on Manifolds with Boundary
该研究针对经典霍奇拉普拉斯算子无法适配流形上异质局部特征数据的局限，提出带边界流形的加权霍奇拉普拉斯算子框架，经蛋白质柔性分析实验验证其有效性，可用于分析流形数据的拓扑与几何特征。
We consider a family of compact, oriented and connected Riemannian manifolds shrinking to a metric graph and describe the asymptotic behaviour of the eigenvalues of the Hodge Laplacian. We apply our results to produce manifolds with spectral gaps of arbitrarily large size in the spectrum of the Hodge Laplacian.
In this paper, we consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold $M$ isometrically immersed into another Riemannian manifold $\bar M$ for arbitrary codimension. We first assume the pull back Weitzenböck operator (defined in Section 2) of $\bar M$ bounded from below, and obtain an extrinsic lower bound for the first eigenvalue of Hodge-Laplacian. As applications, we obtain some rigidity results and a homology sphere theorem. Second, when the pull back Weitzenböck operator of $\bar M$ bounded from both sides, we give a lower bound of the first eigenvalue by the Ricc…
A vector field induced de Rham-Hodge theory on manifolds
本文提出一种由向量场诱导的德拉姆-霍奇框架，定义了霍奇L²内积、余微分和霍奇拉普拉斯算子，并扩展到有边流形。
An adapted version of the proof (due to A. Weil) of the well-known de Rham Theorem allows us to compare uniformly the spectrum of the Hodge Laplacian acting on differential forms (on a compact Riemannian manifold) to the spectrum of the combinatorial Laplacian acting on cochains associated to an open cover (made of balls of sufficiently small radius). We exhibit then a lower bound for the first positive eigenvalue of the combinatorial Laplacian and deduce a lower bound for the first positive eigenvalue of the Hodge Laplacian.
We study multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on the fact that Hodge Laplacian restricted to the subspace of co-exact two-forms is minus the square of the Beltrami operator, a first-order operator. We prove that for generic metrics the spectrum of the Beltrami operator is simple. Because the Beltrami operator in this setting is …
The goal of the paper is to calculate the limit sectrum of the Hodge-Laplace operator under the perturbation of collapse of one part of a connected sum. This gives some new results concerning the 'conformal spectrum' on differential forms.
We prove $R$-bisectoriality and boundedness of the $H^\infty$-functional calculus in $L^p$ for all $1<p<\infty$ for the Hodge-Dirac operator associated with Witten Laplacians on complete Riemannian manifolds with non-negative Bakry-Emery Ricci curvature on $k$-forms.
We define self-adjoint extensions of the Hodge Laplacian on Lipschitz domains in Riemannian manifolds, corresponding to either the absolute or the relative boundary condition, and examine regularity properties of these operators' domains and form domains. We obtain results valid for general Lipschitz domains, and stronger results for a special class of ``almost convex'' domains, which apply to domains with corners.
Hodge-de Rham and Lichnérowicz Laplacians on double forms and some vanishing theorems
该研究在双形式框架下证明Lichnérowicz拉普拉斯是两类Hodge-de Rham推广拉普拉斯的平均，引入新乘积建立曲率项无指标公式，得到多类消没定理，推广Petersen-Wink结果并揭示曲率算子特征秩对流形结构的影响。
Harmonic vector fields and the Hodge Laplacian operator on Finsler geometry
本文在 Finsler 流形上定义了水平微分、散度与 $p$-调和形式，证明了 Hodge 型定理，提出了调和向量场的新定义，并导出了基于调和 Ricci 标量的 Bochner-Yano 型分类定理。
For each degree p, we construct on any closed manifold a family of Riemannian metrics, with fixed volume such that any positive eigenvalues of the rough and Hodge Laplacians acting on differential p-forms converge to zero. In particular, on the sphere, we can choose these Riemannian metrics as those of non-negative sectional curvature. This is a generalization of the results by Colbois and Maerten in 2010 to the case of higher degree forms.
A Sharp Lower Bound for the Spectrum of the Hodge Laplacian on Kähler Hyperbolic Manifolds and its Applications
本文研究了Kähler超曲面Hodge拉普拉斯算子谱的精确下界，并应用于有界对称域的谱分析。
We prove uniqueness results for a Calderon type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth order potential. The method is based on Carleman estimates for the Hodge Laplacian with relative or absolute boundary conditions, and on the construction of complex geometric optics solutions which reduce the Calderon type problem to a tensor tomography problem for 2-tensors. The arguments in this…
We show that for a closed hyperbolic 3-manifold the size of the first eigenvalue of the Hodge Laplacian acting on coexact 1-forms is comparable to an isoperimetric ratio relating geodesic length and stable commutator length with comparison constants that depend polynomially on the volume and on a lower bound on injectivity radius, refining estimates of Lipnowski and Stern. We use this estimate to show that there exist sequences of closed hyperbolic 3-manifolds with injectivity radius bounded below and volume going to infinity for which the 1-form Laplacian has spectral gap vanishing exponentia…
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.
A Cheng-type Eigenvalue-Comparison Theorem for the Hodge Laplacian
本文提出了Hodge拉普拉斯算子在特定几何条件下的特征值上界定理，扩展了Cheng的经典结果，并应用于1形式的连接拉普拉斯算子的特征值估计。
Eigenvalue Estimates of the Hodge Laplacian Under Lower Ricci Curvature Bound
本文在下界里奇曲率限制条件下，研究了闭合黎曼流形上霍奇拉普拉斯算子的特征值估计，并建立了连接拉普拉斯算子的特征值界及微分形式的全局皮卡尔不等式。
Hodge-Laplacian Eigenvalues on Surfaces with Boundary
本文利用微分形式将Rohleder关于Neumann与Dirichlet特征值不等式的变分方法推广至带边界Riemann流形，通过分解Hodge-Laplacian绝对边界问题的谱，得到了更一般的特征值不等式定理。
By introducing a weight function to the Laplace operator, Bakry and Émery defined the "drift Laplacian" to study diffusion processes. Our first main result is that, given a Bakry-Émery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new relationship between Dirichlet eigenvalues of domains in $\R^n$ and Neumann eigenvalues of domains in $\R^{n+1}$ and a new maximum principle. Using our main result and maximum principle, we are able to gen…
On the Schwartz estimate for Hodge Laplacians on semisimple Lie groups
本文将对称空间上的Schwartz估计推广到半单李群，证明了Hodge Laplacian与Dirac算子的相应估计，并给出李代数上同调的Kuga引理版本，以利用Fourier分析研究向量丛截面的热问题。
Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning
本文提出欧拉表示下的持久 Hodge 拉普拉斯算子，用于流形拓扑学习，并开发相应学习算法，在蛋白质-配体结合亲和力预测中验证了其有效性。
On empirical Hodge Laplacians under the manifold hypothesis
研究欧几里得空间闭子流形上独立同分布观测值，将图拉普拉斯算子高阶推广为霍奇拉普拉斯算子近似拉普拉斯 - 贝尔特拉米算子，给出相关狄利克雷形式高概率误差界，改进现有误差界并为拓扑数据分析提供见解。
Carrollian $\mathbb{R}^\times$-bundles III: The Hodge Star and Hodge--de Rham Laplacians
利用主丛工具在Carrollian $\mathbb{R}^\times$-丛上构造Hodge星算子和Hodge--de Rham拉普拉斯算子，填补Carrollian几何与伪黎曼几何之间的空白，并应用于Schwarzschild黑洞视界及Carrollian电磁学。
We develop Hodge theory for a Riemannian manifold $(M,g)$ with a background closed 3-form, H. Precisely, we prove that if the metric connections with torsion $\pm H$ have holonomy groups $G_\pm$, then the $d^H$-Laplacian preserves the irreducible representations of the Lie algebras of the holonomy groups on the space of forms.
On an asymptotically conic manifold $(M,g)$, we analyze the asymptotics of the integral kernel of the resolvent $R_q(k):=(Δ_q+k^2)^{-1}$ of the Hodge Laplacian $Δ_q$ on $q$-forms as the spectral parameter $k$ approaches zero, assuming that 0 is not a resonance. The first application we give is an $L^p$ Sobolev estimate for $d+δ$ and $Δ_q$. Then we obtain a complete characterization of the range of $p>1$ for which the Riesz transform for $q$-forms $T_q=(d+δ)Δ_q^{-1/2}$ is bounded on $L^p$. Finally, we obtain an asymptotic formula for the analytic torsion of a family of smooth compact Riemannian…
The Heat Kernel of the Hodge Laplacian on an Invariant Subspace of 1-forms on the Hyperbolic Space $\mathbb{H}^{n}$
本文构造了双曲空间上1-形式不变子空间Hodge Laplacian热核的显式公式，利用扭曲积分解和Millson型递推关系，得到了尖锐一致估计与长时间渐近行为，将标量热核界推广至1-形式。
Let $g$ and $\tilde{g}$ be Riemannian metrics on a noncompact manifold $M$, which are conformally equivalent. We show that under a very mild \emph{first order} control on the conformal factor, the wave operators corresponding to the Hodge-Laplacians $Δ_g$ and $Δ_{\tilde{g}}$ acting on differential forms exist and are complete. We apply this result to Riemannian manifolds with a bounded geometry and more specifically, to warped product Riemannian manifolds with a bounded geometry. Finally, we combine our results with some explicit calculations by Antoci to determine the absolutely continuous sp…
We examine the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on $p$-forms by computing the heat invariants associated to the $p$-spectrum. We show that the heat invariants of the $0$-spectrum together with those of the $1$-spectrum for the corresponding Hodge Laplacians are sufficient to distinguish orbifolds with singularities from manifolds as long as the singular sets have codimension $\le 3.$ This is enough to distinguish orbifolds from manifolds for dimension $\le 3.$
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。
On any compact manifold of dimension greater than 3, we exhibit a metric whose first positive eigenvalue for the Laplacian acting on p-form is of multiplicity 2. As a corollary, we prescribe the volume and any finite part of the spectrum of the Hodge Laplacian with multiplicity 1 or 2.
Hodge Laplacian on $1$-forms of homogeneous $3$-spheres
本文研究了SU(2)和SO(3)上的左不变度量下Hodge-Laplacian的谱，首次明确计算了Berger 3球面的特征值并提出并证明了齐性度量下第一个特征值的显式公式，展示了1形式谱决定度量的全等性。
On spectral simplicity of the Hodge Laplacian and Curl Operator along paths of metrics
本文证明在闭定向3-流形上，旋度算子的特征空间在一般度量下是一维的，即使沿C^k度量族也成立，并分析了Hodge Laplacian非简单谱的两种来源。
Eigenvalue estimates on weighted manifolds
该研究推导了加权黎曼流形上微分形式的Hodge拉普拉斯算子的特征值估计，统一拓展了已有结果并给出几何应用，建立了(f)-极小超曲面Jacobi算子特征值与Hodge拉普拉斯谱的关联不等式。
Hodge Laplacian and geometry of Kuranishi family of Fano manifolds
本文利用 Bochner-Kodaira 公式估计 Fano 流形上 Hodge Laplacian 的特征值，并据此分析其形变 Kuranishi 族，证明 Kähler 形式的保持性、给出 Ricci 势公式且重新解释 Donaldson-Fujiki 图景。
Lower bounds for the eigenvalues of the Hodge Laplacian on certain non-convex domains
研究非凸域上Hodge拉普拉斯算子最小正特征值的几何下界，推导凸域多孔情况下的高阶精确特征值下界，并改进Cheeger不等式。
On the Laplacian coflow of invariant $G_2$-structures and its solitons
本研究利用霍奇拉普拉斯算子的不可约$G_2$-分解等公式研究余闭$G_2$-结构的拉普拉斯余流，刻画相关对称与孤立子条件，给出紧收缩孤立子不存在的简化证明，并基于括号流视角证明几乎阿贝尔李群上该类余流解的长时间存在性。
This paper constructs a Hodge theory of noncompact topologically tame manifolds $M$. The main result is an isomorphism between the de Rham cohomology with compact supports of $M$ and the kernel of the Hodge--Witten--Bismut Laplacian $\lap_μ$ associated to a measure $dμ$ which has sufficiently rapid growth at infinity on $M$. This follows from the construction of a space of forms associated to $\lap_μ$ which satisfy an ``extension by zero'' property. The ``extension by zero'' property is proved for manifolds with cylindrical ends possessing gaussian growth measures.
We prove using an integral criterion the existence and completeness of the wave operators $W_{\pm}(Δ_h^{(k)}, Δ_g^{(k)}, I_{g,h}^{(k)})$ corresponding to the Hodge Laplacians $Δ_ν^{(k)}$ acting on differential $k$-forms, for $ν\in\{g,h\}$, induced by two quasi-isometric Riemannian metrics $g$ and $h$ on a complete open smooth manifold $M$. In particular, this result provides a criterion for the absolutely continuous spectra $σ_{\mathrm{ac}}(Δ_g^{(k)}) = σ_{\mathrm{ac}}(Δ_h^{(k)})$ of $Δ_ν^{(k)}$ to coincide. The proof is based on gradient estimates obtained by probabilistic Bismut-type formula…
A weighted Reilly type integral formula for differential forms and its applications
本文推导了带权Reilly型积分公式，并应用于微分形式的谱估计及边界曲率性质研究。
In the present paper we show properties of a little-known Laplacian operator acting on symmetric tensors. This operator is an analogue of the well known Hodge-de Rham Laplacian which acts on exterior differential forms. Moreover, this operator admits the Weitzenböck decomposition and we study it using the analytical method, due to Bochner, of proving vanishing theorems for the null space of a Laplace operator admitting a Weitzenböck decomposition and further of estimating its lowest eigenvalue.
Resolving the viscosity operator ambiguity on Riemannian manifolds via a kinematic selection principle
本文研究了在黎曼流形上Navier-Stokes方程的不同粘性算子选择问题，通过运动构造原理唯一选择变形拉普拉斯算子，并证明了在特定边界条件下，变形拉普拉斯算子在超曲面上具有良好的 coercivity 性质，从而解决了分析上的障碍。
Do the Hodge spectra distinguish orbifolds from manifolds? Part 2
本文研究了紧致黎曼轨形奇异集与Hodge Laplacian的p-谱的关系，针对单独考虑的各p-谱给出了正反两方面的谱反演结果，并给出了保证奇异集体积可确定的余维数条件及反例。
We review the recent development of Hodge theory for almost complex manifolds. This includes the determination of whether the Hodge numbers defined by $\bar\partial$-Laplacian are almost complex, almost Kähler, or birational invariants in dimension four.
In this paper we study the heat equation (of Hodge-Laplacian) deformation of $(p, p)$-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a $(p, p)$-form solution is preserved under such an invariant condition we prove the sharp differential Harnack (in the sense of Li-Yau-Hamilton) estimates for the positive solutions of the Hodge-Laplacian heat equation. We also prove a nonlinear version coupled with the Kähler-Ricci flow and some interpolating matrix differential Harnack type estimates for both the Kähler-Ricci flow and the Ricci flow.
We develop a Fredholm Theory for the Hodge Laplacian in weighted spaces on ALG$^*$ manifolds in dimension four. We then give several applications of this theory. First, we show the existence of harmonic functions with prescribed asymptotics at infinity. A corollary of this is a non-existence result for ALG$^*$ manifolds with non-negative Ricci curvature having group $Γ= \{e\}$ at infinity. Next, we prove a Hodge decomposition for the first de Rham cohomology group of an ALG$^*$ manifold. A corollary of this is vanishing of the first betti number for any ALG$^*$ manifold with non-negative Ricci…
On any compact manifold of dimension greater than 6, we prescribe the volume and any finite part of the spectrum Hodge Laplacian acting on $p$-form for $1\leq p<\frac n2$. In particular, we prescribe the multiplicity of the first eigenvalues.
Short survey about small eigenvalues of the Hodge Laplacian under bounded curvature collapsing.
The Dirac-Dolbeault Operator Approach to the Hodge Conjecture
该研究利用Dirac-Dolbeault算子相关的Green函数表示与Nash-Moser广义反函数定理，在特定假设下证明复射影流形满足条件的复子流形存在性，进而证明复射影流形的Hodge猜想。
We study symplectic Laplacians on compact symplectic manifolds with boundary. These Laplacians are associated with symplectic cohomologies of differential forms and can be of fourth-order. We introduce several natural boundary conditions on differential forms and use them to establish Hodge theory by proving various form decomposition and also isomorphisms between the symplectic cohomologies and the spaces of harmonic fields. These novel boundary conditions can be applied in certain cases to study relative symplectic cohomologies and Lefschetz maps between relative de Rham cohomologies. As an …
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