A Reilly Formula and Eigenvalue Estimates for Differential Forms

In this paper, we consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold M isometrically immersed into another Riemannian manifoldM for arbitrary codimension. We first assume the pull back Weitzenböck operator (defined in Section 2) ofM 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 ofM bounded from both sides, we give a lower bound of the first eigenvalue by the Ricci curvature of M and some extrinsic geometry. As a consequence, we prove a weak Ejiri type theorem, that is, if the Ricci curvature bounded from below pointwisely by a function of the norm square of the mean curvature vector, then M is a homology sphere. In the end, we give an example to show that all the eigenvalue estimates and homology sphere theorems are optimal whenM has constant curvature.2010 Mathematics Subject Classification. 58J50; 53C24; 53C40.

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“…On the other hand, B 2 = m α=1 Å α 2 . Hence, the remain proof can be reduced to codimension m = 1 case, which has already done (c. f. [9] ).…”

Section: Now Applying Bochner Formula (23) and By The Assumption W [P]mentioning

confidence: 93%

Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

Chen

Sun

2019

Journal of Differential Equations

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“…As mentionned in [10], the Weingarten map admits a canonical extension to any p-form ϕ on M by the following:…”

Section: Riemannian Flows and Manifolds With Boundarymentioning

confidence: 99%

“…holds, where σ p (M ) is the infimum over M of the lowest p-curvatures σ p . Now, we recall the Reilly formula established in [10]. For this, we denote by J * the restriction of differential forms on N to the boundary M .…”

Section: Riemannian Flows and Manifolds With Boundarymentioning

confidence: 99%

“…In order to finish the proof, it is sufficient to calculate the two sums in the r.h.s. of (10). In fact, the first sum is equal to…”

Section: Lemma 42 If the Equality Is Realized Inmentioning

confidence: 99%

“…1.1]. The main tool in our estimate is the so-called Reilly formula [10] obtained by integrating the Bochner-Weitzenböck formula over N and using the Stokes formula. As a consequence of our estimate, we obtain several rigidity theorems which mainly characterize the flow as a local product and the boundary as an η-umbilical submanifold (see [4,Sec.…”

Section: Introductionmentioning

confidence: 99%

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Eigenvalue Estimate for the Basic Laplacian on Manifolds with Foliated Boundary, Part II

et al. 2018

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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 basic p-forms for p > 1. As in [4], the limiting case allows to characterize the manifold Γ \ R × B ′ for some group Γ, and where B ′ denotes the unit closed ball. In particular, we describe the Riemannian product S 1 × S n as the boundary of a manifold.

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Rigidity results for spin manifolds with foliated boundary

Chami

Ginoux²,

Habib³

et al. 2015

J. Geom.

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In this paper, we consider a compact Riemannian manifold whose boundary is endowed with a Riemannian flow. Under a suitable curvature assumption depending on the O'Neill tensor of the flow, we prove that any solution of the basic Dirac equation is the restriction of a parallel spinor field defined on the whole manifold. As a consequence, we show that the flow is a local product. In particular, in the case where solutions of the basic Dirac equation are given by basic Killing spinors, we characterize the geometry of the manifold and the flow.

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A Reilly Formula and Eigenvalue Estimates for Differential Forms

Cited by 24 publications

References 18 publications

Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

Eigenvalue Estimate for the Basic Laplacian on Manifolds with Foliated Boundary, Part II

Rigidity results for spin manifolds with foliated boundary

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