These lecture notes in the De Rham-Hodge theory are designed for a 1-semester undergraduate course (in mathematics, physics, engineering, chemistry or biology). This landmark theory of the 20th Century mathematics gives a rigorous foundation to modern field and gauge theories in physics, engineering and physiology. The only necessary background for comprehensive reading of these notes is Green's theorem from multivariable calculus.
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.
In this paper, we give the Lichnerowicz type formulas for statistical de Rham Hodge operators. Moreover, Kastler-Kalau-Walze type theorems for statistical de Rham Hodge operators on compact manifolds with (respectively without) boundary are proved.
In this paper, we give Lichnerowicz type formulas for the perturbation of the de Rham Hodge operator. We prove the Kastler-Kalau-Walze type theorems for the perturbation of the de Rham Hodge operator on 4-dimensional and 6-dimensional compact manifolds with (resp.without) boundary. Some concrete examples of the perturbation of the de Rham Hodge operator are provided for our main theorems.
In this paper, we get a Kastler-Kalau-Walze type theorem associated to nonminimal de Rham-Hodge operators on compact manifolds with boundary. We give two kinds of operator-theoretic explanations of the gravitational action in the case of four dimensional compact manifolds with flat boundary.
We introduce and study Hodge-de Rham numbers for compact almost complex 4-manifolds, generalizing the Hodge numbers of a complex surface. The main properties of these numbers in the case of complex surfaces are extended to this more general setting, and it is shown that all Hodge-de Rham numbers for compact almost complex 4-manifolds are determined by the cohomology, except for one (the irregularity). Finally, these numbers are shown to prohibit the existence of complex structures on certain manifolds, without reference to the classification of surfaces.
The Hodge-de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a novel way.
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.
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.
This article constructs Von Neumann invariants for constructible complexes and coherent D-modules on compact complex manifolds, generalizing the work of the author on coherent L 2-cohomology. We formulate a conjectural generalization of Dingoyan's L 2-Mixed Hodge structures in terms of Saito's Mixed Hodge Modules and give partial results in this direction. 2020 AMS Classification: 32J27.
The authors study the Hodge theory of the exterior differential operator $d$ acting on $q$-forms on a smoothly bounded domain in $\RR^{N+1}$, and on the half space $\rnp$. The novelty is that the topology used is not an $L^2$ topology but a Sobolev topology. This strikingly alters the problem as compared to the classical setup. It gives rise to a boundary-value problem belonging to a class of problems first introduced by Višik and Eskin, and by Boutet de Monvel.
Given a compact stratified pseudomanifold with a Thom-Mather stratification and a class of riemannian metrics over its regular part, we study the relationships between the $L^{2}$ de Rham and Hodge cohomology and the intersection cohomology of $X$ associated to some perversities. More precisely, to a kind of metric which we call \emph{quasi edge with weights}, we associate two general perversities in the sense of G. Friedman, $p_{g}$ and its dual $q_{g}$. We then show that the absolute $L^{2}$ Hodge cohomology is isomorphic to the maximal $L^{2}$ de Rham cohomology and this is in turn isomorph…
The de Rham-Hodge theory is a landmark of the 20$^\text{th}$ Century's mathematics and has had a great impact on mathematics, physics, computer science, and engineering. This work introduces an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from a filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to close manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropri…
Let $M^n$ be a n-dimensional compact manifold, with $n\geq3$. For any conformal class C of riemannian metrics on M, we set $μ_k^c(M,C)=\inf_{g\in C}μ_{[\frac n2],k}(M,g)\Vol(M,g)^{\frac2n}$, where $μ_{p,k}(M,g)$ is the k-th eigenvalue of the Hodge laplacian acting on coexact p-forms. We prove that $0<μ_k^c(M,C)\leqμ_k^c(S^n,[g_{can}])\leq k^{\frac2n}μ_1^c(S^n,[g_{can}])$.
For any compact manifold of dimension n>=5, we prescribe the volume and any finite part of the spectrum of the Hodge Laplacian acting on diffential forms of degree 1<p<n-1 (exept for p=n/2 if n is even), within a given conformal class. When n<5 and when p=0,1,n-1,n, and p=n/2 if n is even, this simultaneous prescription of the volume, the spectrum and the conformal class is known to be impossible.
We extend the study of the de Rham operator with ideal boundary conditions from the case of isolated conic singularities, as analyzed by Cheeger, to the case of arbitrary stratified pseudomanifolds. We introduce a class of ideal boundary operators and the notion of mezzoperversity, which intermediates between the standard lower and upper middle perversities in intersection theory, as interpreted in this de Rham setting, and show that the de Rham operator with these boundary conditions is Fredholm and has compact resolvent. We also prove an isomorphism between the resulting Hodge and L2 de Rham…
Recent years have witnessed a trend that advanced mathematical tools, such as algebraic topology, differential geometry, graph theory, and partial differential equations, have been developed for describing biological macromolecules. These tools have considerably strengthened our ability to understand the molecular mechanism of macromolecular function, dynamics and transport from their structures. However, currently, there is no unified mathematical theory to analyze, describe and characterize biological macromolecular geometry, topology, flexibility and natural mode at a variety of scales. We …
We look at several problems in even dimensional conformal geometry based around the de Rham complex. A leading and motivating problem is to find a conformally invariant replacement for the usual de Rham harmonics. An obviously related problem is to find, for each order of differential form bundle, a ``gauge'' operator which completes the exterior derivative to a system which is both elliptically coercive and conformally invariant. Treating these issues involves constructing a family of new operators which, on the one hand, generalise Branson's celebrated Q-curvature and, on the other hand, com…
A measured solenoid is a compact laminated space endowed with a transversal measure. The De Rham $L^2$-cohomology of the solenoid is defined by using differential forms which are smooth in the leafwise directions and $L^2$ in the transversal direction. We develop the theory of harmonic forms for Riemannian measured solenoids, and prove that this computes the De Rham $L^2$-cohomology of the solenoid. This implies in particular a Poincare duality result.
We compute the de Rham cohomology of the weak stable foliation of the geodesic flow of a connected orientable closed hyperbolic surface with various coefficients. For most of the coefficients, we also give certain "Hodge decompositions" of the corresponding de Rham complexes, which are not obtained by the usual Hodge theory of foliations. These results are based on unitary representation theory of $\mathrm{PSL}(2,\mathbb{R})$. As an application we obtain an answer to a problem considered by Haefliger and Li around 1980.
We are interested in the spectrum of the Hodge-de Rham operator on a cyclic covering $X$ over a compact manifold $M$ of dimension $n+1$. Let $Σ$ be a hypersurface in $M$ which does not disconnect $M$ and such that $M-Σ$ is a fundamental domain of the covering. If the cohomology group $H^{n/2 (Σ)$ is trivial, we can construct for each $N \in \N$ a metric $g=g_N$ on $M$, such that the Hodge-de Rham operator on the covering $(X,g)$ has at least $N$ gaps in its (essential) spectrum. If $H^{n/2}(Σ) \ne 0$, the same statement holds true for the Hodge-de Rham operators on $p$-forms provided $p \notin…
Let M be a hypercomplex Hermitian manifold, (M,I) the same manifold considered as a complex Hermitian with a complex structure I induced by the quaternions. The standard linear-algebraic construction produces a canonical nowhere degenerate (2,0)-form on (M,I). It is well known that M is hyperkaehler if and only if the form Ωis closed with respect to the de Rham differential. The M is called HKT (hyperkaehler with torsion) if this form is closed with respect to the Dolbeault differential (this condition is weaker. Conjecturally, all compact hypercomplex manifolds admit an HKT-metrics. We exploi…
Hamiltonian-Type Spectral Functionals for Time-Dependent Perturbed Hodge--de Rham Operators
School of Data Science and Artificial Intelligence Dongbei University of Finance and Economics（东北财经大学数据科学与人工智能学院）；School of Mathematics and Statistics Northeast Normal University（东北师范大学数学与统计学院）
We show that the kernel of the Rumin Laplacian agrees with that of the Hodge-de Rham Laplacian on compact Sasakian manifolds. As a corollary, we obtain another proof of primitiveness of harmonic forms. Moreover, the space of harmonic forms coincides with the sub-Riemann limit of Hodge-de Rham Laplacian when its limit converges. Finally, we express the analytic torsion function associated with the Rumin complex in terms of the Reeb vector field.
