Yuri A. Kordyukov, Andrey A. Yakovlev
We present some recent results on the behavior of the spectrum of the differential form Laplacian on a Riemannian foliated manifold when the metric on the ambient manifold is blown up in directions normal to the leaves (in the adiabatic limit).
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In this article we present an intrinsec construction of foliated Brownian motion via stochastic calculus adapted to foliation. The stochastic approach together with a proposed foliated vector calculus provide a natural method to work on harmonic measures. Other results include a decomposition of the Laplacian in terms of the foliated and basic Laplacians, a characterization of totally invariant measures and a differential equation for the density of harmonic measures.
In [4], we gave a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic $1$-forms defined on a compact manifold whose boundary is endowed with a Riemannian flow. In this paper, we extend this result to the case of the first eigenvalue on basic $p$-forms for $p>1$. As in [4], the limiting case allows to characterize the manifold $\mathbb{R} \times B' / Γ$ for some group $Γ$, and where $B'$ denotes the unit closed ball. In particular, we describe the Riemannian product $\mathbb{S}^1\times \mathbb{S}^n$ as the boundary of a manifold.
In this paper, we give a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic $1$-forms defined on a compact manifold whose boundary is endowed with a Riemannian flow. The limiting case gives rise to a particular geometry of the flow and the boundary. Namely, the flow is a local product and the boundary is $η$-umbilical. This allows to characterize the quotient of $\mathbb R\times B'$ by some group $Γ$ as being the limiting manifold. Here $B'$ denotes the unit closed ball. Finally, we deduce several rigidity results describing the product $\mathbb{S}^1\times \mathb…
We use adiabatic limits to study foliated manifolds. The Bott connection naturally shows up as the adiabatic limit of Levi-Civita connections. As an application, we then construct certain natural elliptic operators associated to the foliation and present a direct geometric proof of a vanshing theorem of Connes[Co], which extends the Lichnerowicz vanishing theorem [L] to foliated manifolds with spin leaves, for what we call almost Riemannian foliations. Several new vanishing theorems are also proved by using our method.
In this paper, we discuss some aspects of the averaging method for Poisson connections on foliated manifolds with symmetry generalizing the previous results on the Hannay-Berry connections on fibrations due to \cite{Mn-88,MaMoRa-90} which play an important role in the normal form theory for Hamiltonian systems of adiabatic type.
We discuss asymptotic behavior of the eigenvalue distribution of the differential form Laplacian on a Riemannian foliated manifold when the metric on the ambient manifold is blown up in directions normal to the leaves (in the adiabatic limit). Motivated by analogies with semiclassical spectral asymptotics, we use ideas and notions of noncommutative geometry to suggest a conjectural formula for the eigenvalue distribution in the adiabatic limit, which we call noncommutative Weyl formula. We review known results and discuss the correctness of the noncommutative Weyl formula in each case.
we discuss the decomposition of the zeta-determinant of the square of the Dirac operator into the contributions coming from the different parts of the manifold in the case of an invertible tangential operator.
In this paper we first establish the relation between the zeta-determinant of a Dirac Laplacian with the Dirichlet boundary condition and the APS boundary condition on a cylinder. Using this result and the gluing formula of the zeta-determinant given by Burghelea, Friedlander and Kappeler with some assumptions, we prove the adiabatic decomposition theorem of the zeta-determinant of a Dirac Laplacian. This result was originally proved by J. Park and K. Wojciechowski in [11] but our method is completely different from the one they presented.
The idea of Lichnerowicz or Morse-Novikov cohomology groups of a manifold has been utilized by many researchers to study important properties and invariants of a manifold. Morse-Novikov cohomology is defined using the differential $d_ω=d+ω\wedge$, where $ω$ is a closed $1$-form. We study Morse-Novikov cohomology relative to a foliation on a manifold and its homotopy invariance and then extend it to more general type of forms on a Riemannian foliation. We study the Laplacian and Hodge decompositions for the corresponding differential operators on reduced leafwise Morse-Novikov complexes. In the…
We study spectral asymptotics for the Laplace operator on differential forms on a Riemannian foliated manifold equipped with a bundle-like metric in the case when the metric is blown up in directions normal to the leaves of the foliation. The asymptotical formula for the eigenvalue distribution function is obtained. The relationships with the spectral theory of leafwise Laplacian and with the noncommutative spectral geometry of foliations are discussed.
We study the behaviour of Laplace-type operators H on a complex vector bundle E $\rightarrow$ M in the adiabatic limit of the base space. This space is a fibre bundle M $\rightarrow$ B with compact fibres and the limit corresponds to blowing up directions perpendicular to the fibres by a factor 1/$ε$. Under a gap condition on the fibre-wise eigenvalues we prove existence of effective operators that provide asymptotics to any order in $ε$ for H (with Dirichlet boundary conditions), on an appropriate almost-invariant subspace of L${}^2$(E).
Let X be a smooth compact manifold with boundary. For smooth foliations on the boundary of X admitting a `resolution' in terms of a fibration, we construct a pseudodifferential calculus generalizing the fibred cusp calculus of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a simple description of the Fredholm operators inside the calculus. When the leaves of the fibration `resolving' the foliation are compact, we also obtain an index formula for Fredholm perturbations of Dirac-type operators. Along the way, we obtain a formula for the adiabatic limit of the eta inva…
Given a compact boundaryless Riemannian manifold $Y$ on which a compact Lie group $G$ acts, there is always a metric on $Y$ such that the action is by isometries. Assuming $Y$ is equipped with such a metric, recall that the $G$-invariant Laplacian is the restriction of the ordinary Laplacian to the space of functions which are constant along the orbits of $G$. In this paper, the author analyzes the wave trace of the $G$-invariant Laplacian and shows that the singularities of this wave trace occur at the lengths of certain geodesic arcs--those that are orthogonal to the orbits and whose endpoin…
The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution
研究具有全测地纤维和可积水平分布的黎曼浸没的水平拉普拉斯算子的谱性质，证明其酉等价于基流形上某个无限秩平坦向量丛的扭曲拉普拉斯算子，并应用于缩放第一非零特征值的渐近行为。
We consider the $SL(2,R)$ action on moduli spaces of quadratic differentials. If $μ$ is an $SL(2,R)$-invariant probability measure, crucial information about the associated representation on $L^2(μ)$ (and in particular, fine asymptotics for decay of correlations of the diagonal action, the Teichmüller flow) is encoded in the part of the spectrum of the corresponding foliated hyperbolic Laplacian that lies in $(0,1/4)$ (which controls the contribution of the complementary series). Here we prove that the essential spectrum of an invariant algebraic measure is contained in $[1/4,\infty)$, i.e., f…
Under suitable conditions, we show that the Euler characteristic of a foliated Riemannian manifold can be computed only from curvature invariants which are transverse to the leaves. Our proof uses the hypoelliptic sub-Laplacian on forms recently introduced by two of the authors.
The gluing formula of the zeta-determinant of a Laplacian given by Burghelea, Friedlander and Kappeler contains an unknown constant. In this paper we compute this constant to complete the formula under the assumption of the product structure near boundary. As applications of this result,we prove the adiabatic decomposition theorems of the zeta-determinant of a Laplacian with respect to the Dirichlet and Neumann boundary conditions and of the analytic torsion with respect to the absolute and relative boundary conditions.
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。
Adiabatic Limit and Analytic Torsion of Vector Bundles
本文研究了向量丛在绝热参数下的指标与解析扭量，并证明其与底流形的度量一致。
Motivated by our conjecture of an earlier work predicting the degeneration at the second page of the Frölicher spectral sequence of any compact complex manifold supporting an SKT metric $ω$ (i.e. such that $\partial\bar\partialω=0$), we prove degeneration at $E_2$ whenever the manifold admits a Hermitian metric whose torsion operator $τ$ and its adjoint vanish on $Δ''$-harmonic forms of positive degrees up to $\mbox{dim}_\C X$. Besides the pseudo-differential Laplacian inducing a Hodge theory for $E_2$ that we constructed in earlier work and Demailly's Bochner-Kodaira-Nakano formula for Hermit…
We study the nodal sets of non-degenerate eigenfunctions of the Laplacian on fibre bundles $π{:}\, M\to B$ in the adiabatic limit. This limit consists in considering a family $G_\varepsilon$ of Riemannian metrics, that are close to Riemannian submersions, for which the ratio of the diameter of the fibres to that of the base is given by $\varepsilon \ll 1$. We assume $M$ to be compact and allow for fibres $F$ with boundary, under the condition that the ground state eigenvalue of the Dirichlet-Laplacian on $F_x$ is independent of the base point. We prove for $\mathrm{dim} B \leq 3$ that the noda…
Given a complex manifold $X$ and a smooth positive function $η$ thereon, we perturb the standard differential operator $d=\partial + \bar\partial$ acting on differential forms to a first-order differential operator $D_η$ whose principal part is $η\partial + \bar\partial$. The role of the zero-th order part is to force the integrability property $D_η^2=0$ that leads to a cohomology isomorphic to the de Rham cohomology of $X$, while the components of types $(0,\,1)$ and $(1,\,0)$ of $D_η$ induce cohomologies isomorphic to the Dolbeault and conjugate-Dolbeault cohomologies. We compute Bochner-Kod…
A Twisted Adiabatic Limit Approach to Vanishing Theorems for Complex Line Bundles
本文通过将扭曲绝热极限构造推广到复线丛上的联络，在非 Kähler 流形和非全纯线丛的一般情形下证明了复线丛 D''-上同调群的消失定理，并建立了扭曲 Laplacian 的比较公式。
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