3 Harmonic Forms on Manifolds withNon-Negative Bakry-Émery-Ricci Curvature (original) (raw)

Harmonic measures on covers of compact surfaces of nonpositive curvature

Transactions of the American Mathematical Society, 1993

Let M M be the universal cover of a compact nonflat surface N N of nonpositive curvature. We show that on the average the Brownian motion on M M behaves similarly to the Brownian motion on negatively curved manifolds. We use this to prove that harmonic measures on the sphere at infinity have positive Hausdorff dimension and if the geodesic flow on N N is ergodic then the harmonic and geodesic measure classes at infinity are singular unless the curvature is constant.

L 2-Harmonic 1-Forms on Submanifolds with Finite Total Curvature

Journal of Geometric Analysis, 2014

Let x : M m →M, m ≥ 3, be an isometric immersion of a complete noncompact manifold M in a complete simply connected manifoldM with sectional curvature satisfying −k 2 ≤ KM ≤ 0, for some constant k. Assume that the immersion has finite total curvature in the sense that the traceless second fundamental form has finite L m-norm. If KM ≡ 0, assume further that the first eigenvalue of the Laplacian of M is bounded from below by a suitable constant. We prove that the space of the L 2 harmonic 1-forms on M has finite dimension. Moreover, there exists a constant Λ > 0, explicitly computed, such that if the total curvature is bounded from above by Λ then there are no nontrivial L 2-harmonic 1-forms on M.

Vanishing theorems for L^{2}$$ L 2 harmonic forms on complete Riemannian manifolds

Geometriae Dedicata, 2016

This paper contains some vanishing theorems for L 2 harmonic forms on complete Riemannian manifolds with a weighted Poincaré inequality and a certain lower bound of the curvature. The results are in the spirit of Li-Wang and Lam, but without assumptions of sign and growth rate of the weight function, so they can be applied to complete stable hypersurfaces.

Vanishing properties of ppp-harmonic ell\ellell-forms on Riemannian manifolds

arXiv: Differential Geometry, 2017

In this paper, we show several vanishing type theorems for ppp-harmonic ell\ellell-forms on Riemannian manifolds ($p\geq2$). First of all, we consider complete non-compact immersed submanifolds MnM^nMn of Nn+m{N}^{n+m}Nn+m with flat normal bundle, we prove that any ppp-harmonic ell\ellell-forms on MMM is trivial if NNN has pure curvature tensor and MMM satisfies some geometric condition. Then, we obtain a vanishing theorem on Riemannian manifolds with weighted Poincar\'{e} inequality. Final, we investigate complete simply connected, locally conformally flat Riemannian manifolds MMM and point out that there is no nontrivial ppp-harmonic ell\ellell-form on MMM provided that operatornameRic\operatorname{Ric}operatornameRic has suitable bound.

Harmonic 1-Forms on Compact f-Manifolds

Mediterranean Journal of Mathematics, 2007

We consider a Riemannian manifold with a compatible f-structure which admits a parallelizable kernel. With some additional integrability conditions it is called S-manifold. This class of manifolds is a natural generalization of the Sasakian manifolds. We study properties of harmonic 1-forms on such a manifold and deduce some topological properties.

Vanishing theorems for square-integrable harmonic forms

Proceedings Mathematical Sciences, 1981

There are two ways of proving vanishing theorems for cohomology of compact Riem&rmian manifolds. One is based on Bochner's lemma, a variant of maximum principle, the other is an application of Weitzenb6ck identity and integration by parts. Both are discussed in great detail in . In this paper we show that the latter method yields vanishing theorems for L2-1~rmonic forms on complete manifolds. The extra ingredient required in the proof is due to Andreotti arid Vesentini [1]; (see also , theorem 26).

Harmonic and holomorphic 1-forms on compact balanced Hermitian manifolds

Differential Geometry and Its Applications, 2001

On compact balanced Hermitian manifolds we obtain obstructions to the existence of harmonic 1-forms, 9-harmonic (1,0)-forms and holomorphic (1,0)-forms in terms of the Ricci tensors with respect to the Riemannian curvature and the Hermitian curvature. Vanishing of the first Dolbeault cohomology groups of the twistor space of a compact irreducible hyper Kahler manifold is shown. A necessary and sufficient condition the (1,0)-part of a harmonic 1-form to be holomorphic and vice versa, a real 1-form with a holomorphic (1,0)-part to be harmonic are found.