Related papers: $L^{2}$ harmonic forms on complete special holonom…

This paper contains some vanishing theorems for $L^2$ harmonic forms on complete Riemannian manifolds with a weighted Poincar\'e inequality and a certain lower bound of the curvature. The results are in the spirit of Li-Wang and Lam, but…

Let $x:M^m\to \bar M$, with $m\geq 3$, be an isometric immersion of a complete noncompact manifold $M$ in a complete simply-connected manifold $\bar M$ with sectional curvature satisfying $-c^2\leq K_{\bar M}\leq 0$, for some constant $c$.…

Let $(X,g)$ be a compact Riemannian stratified space with simple edge singularity. Thus a neighbourhood of the singular stratum is a bundle of truncated cones over a lower dimensional compact smooth manifold. We calculate the various…

In this article, we study the $L^{2}$-harmonic forms on the complete $2n$-dimensional almost K\"{a}her manifold $X$. We observe that the $L^{2}$-harmonic forms can decomposition into Lefschetz powers of primitive forms. Therefore we can…

Let $f:N\rightarrow (M,g)$ be an oriented (or spin), complete, stable, minimal, immersed hypersurface. In this paper we establish various vanishing theorems for the space of $L^2$-harmonic forms and spinors (in the spin case) under suitable…

We give a topological interpretation of the space of $L^2$-harmonic forms on Manifold with flat ends. It is an answer to an old question of J. Dodziuk. We also give a Chern-Gauss-Bonnet formula for the $L^2$-Euler characteristic of some of…

In this article, we first consider the $L^{2}$ \textit{Morse-Novikov cohomology} on a complete Riemannian manifold $M$ equipped with a parallel $1$-form which includes Vaisman manifold. Based on a vanishing theorem of $L^{2}$…

In this paper, we show several vanishing type theorems for $p$-harmonic $\ell$-forms on Riemannian manifolds ($p\geq2$). First of all, we consider complete non-compact immersed submanifolds $M^n$ of ${N}^{n+m}$ with flat normal bundle, we…

In this paper, by using monotonicity formulas for vector bundle-valued $p$-forms satisfying the conservation law, we first obtain general $L^2$ global rigidity theorems for locally conformally flat (LCF) manifolds with constant scalar…

Let $H^p(L^2(M))$ be the space of all $L^2$-harmonic $p$-forms $(2\leq p\leq n-2)$ on complete submanifolds $M$ with flat normal bundle in spheres. In this paper, we first show that $H^p(L^2(M))$ is trivial if the total curvature of $M$ is…

On a compact foliated Riemannian manifold with some transversal curvature conditions, there are no nontrivial basic harmonic forms (M. Min-Oo et al., J. Reine Angew. Math. 415 (1991). In this paper, we extend the above facts to a complete…

We obtain a topological interpretation for the space of $L^2$ harmonic forms for some complete Riemannian manifold : when the geometry at infinity is the geometry of a simply connected nilpotent Lie group, when the geometry at infinity is a…

We introduce Condition W $\,$(1.2) for a smooth differential form $\omega$ on a complete noncompact Riemannian manifold $M$. We prove that $\omega$ is a harmonic form on $M$ if and only if $\omega$ is both closed and co-closed on $M\, ,$…

When $X=\Gamma\backslash \H^n$ is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of $L^2$ harmonic forms vanish. In this paper, we show rigidity…

We prove an $L^2$-$\partial\overline\partial$-Lemma involving smooth square integrable forms on complete K\"ahler manifolds, provided that the unique self-adjoint extension of the Hodge Laplacian on the Hilbert space of $L^2$-forms has a…

Curvature properties of the characteristic connection on an integrable $G_2$ manifold are investigated. We consider integrable $G_2$ manifold of constant type, i.e. the scalar product of the exterior derivative of the $G_2$ form with its…

We study the space of L^2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type. These metrics generalize the geometric structures at infinity of several different well-known classes of metrics, including…

Let $X$ be a singular Hermitian complex space of pure dimension $n$. We use a resolution of singularities to give a smooth representation of the $L^2$-$\overline\partial$-cohomology of $(n,q)$-forms on $X$. The central tool is an…

The harmonicity condition of the curvature 2-form of a pseudo- Riemannian manifold is formulated on the basis of annulment of this form by the de Rham-Lichnerowicz Laplacian. The following theorem is proved: The curvature 2-form of any…

Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew-symmetric. We show that on a compact manifold with holonomy G2 or Spin7 any Killing form has to be parallel. The main tool is a…