scholarly journals Bounds for the first eigenvalue of the horizontal Laplacian in positively curved sub-Riemannian manifolds

2012 ◽  
Vol 164 (1) ◽  
pp. 155-177 ◽  
Author(s):  
Robert K. Hladky
Keyword(s):  
Riemannian Manifolds ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Positively Curved

scholarly journals Lichnerowicz-Obata Estimate, Almost Parallel p-form and Almost Product Manifolds

2021 ◽  
Author(s):  
Masayuki Aino
Keyword(s):  
Riemannian Manifolds ◽  
First Eigenvalue ◽  
Type Estimate ◽  
The First Eigenvalue ◽  
Hausdorff Approximation

AbstractWe show a Lichnerowicz-Obata type estimate for the first eigenvalue of the Laplacian of n-dimensional closed Riemannian manifolds with an almost parallel p-form ($$2\le p \le n/2$$ 2 ≤ p ≤ n / 2 ) in $$L^2$$ L 2 -sense, and give a Gromov-Hausdorff approximation to a product $$S^{n-p}\times X$$ S n - p × X under some pinching conditions when $$2\le p<n/2$$ 2 ≤ p < n / 2 .

scholarly journals A survey on extensions of Riemannian manifolds and Bartnik mass estimates

2021 ◽  
pp. 1-30
Author(s):  
Armando Cabrera Pacheco ◽  
Carla Cederbaum
Keyword(s):  
General Relativity ◽  
Riemannian Manifolds ◽  
Gaussian Curvature ◽  
First Eigenvalue ◽  
Content Type ◽  
Asymptotically Flat ◽  
Adm Mass ◽  
Local Mass ◽  
Novel Technique ◽  
The First Eigenvalue

Mantoulidis and Schoen developed a novel technique to handcraft asymptotically flat extensions of Riemannian manifolds ( Σ ≅ S 2 , g ) (\Sigma \cong \mathbb {S}^2,g) , with g g satisfying λ 1 ≔ λ 1 ( − Δ g + K ( g ) ) > 0 \lambda _1 ≔\lambda _1(-\Delta _g + K(g))>0 , where λ 1 \lambda _1 is the first eigenvalue of the operator − Δ g + K ( g ) -\Delta _g+K(g) and K ( g ) K(g) is the Gaussian curvature of g g , with control on the ADM mass of the extension. Remarkably, this procedure allowed them to compute the Bartnik mass in this so-called minimal case; the Bartnik mass is a notion of quasi-local mass in General Relativity which is very challenging to compute. In this survey, we describe the Mantoulidis–Schoen construction, its impact and influence in subsequent research related to Bartnik mass estimates when the minimality assumption is dropped, and its adaptation to other settings of interest in General Relativity.

scholarly journals Gromov’s convergence theorem and its application

1985 ◽  
Vol 100 ◽  
pp. 11-48 ◽  
Author(s):  
Atsushi Katsuda
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Convergence Theorem ◽  
Riemannian Geometry ◽  
Main Tool ◽  
First Eigenvalue ◽  
Sphere Theorem ◽  
Positive Ricci Curvature ◽  
The First Eigenvalue ◽  
Differentiable Sphere Theorem

One of the basic questions of Riemannian geometry is that “If two Riemannian manifolds are similar with respect to the Riemannian invariants, for example, the curvature, the volume, the first eigenvalue of the Laplacian, then are they topologically similar?”. Initiated by H. Rauch, many works are developed to the above question. Recently M. Gromov showed a remarkable theorem ([7] 8.25, 8.28), which may be useful not only for the above question but also beyond the above. But it seems to the author that his proof is heuristic and it contains some gaps (for these, see § 1), so we give a detailed proof of 8.25 in [7]. This is the first purpose of this paper. Second purpose is to prove a differentiable sphere theorem for manifolds of positive Ricci curvature, using the above theorem as a main tool.

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