scholarly journals Exhaustion of isoperimetric regions in asymptotically hyperbolic manifolds with scalar curvature $R \geq -6$

2018 ◽  
Vol 26 (3) ◽  
pp. 627-658 ◽  
Author(s):  
Dandan Ji ◽  
Yuguang Shi ◽  
Bo Zhu
Keyword(s):  
Scalar Curvature ◽  
Hyperbolic Manifolds ◽  
Asymptotically Hyperbolic Manifolds ◽  
Isoperimetric Regions ◽  
Asymptotically Hyperbolic

scholarly journals Sharp Resolvent Estimates on Non-positively Curved Asymptotically Hyperbolic Manifolds

2020 ◽  
Author(s):  
Yiran Wang
Keyword(s):  
Energy Estimate ◽  
Weighted Sobolev Spaces ◽  
High Energy ◽  
Oscillatory Behavior ◽  
Hyperbolic Manifolds ◽  
Resolvent Estimates ◽  
Positive Sectional Curvature ◽  
Asymptotically Hyperbolic Manifolds ◽  
Asymptotically Hyperbolic ◽  
Positively Curved

Abstract We study the high energy estimate for the resolvent $R(\lambda )$ of the Laplacian on non-trapping asymptotically hyperbolic manifolds (AHMs). In the literature, estimates of $R(\lambda )$ on weighted Sobolev spaces of the order $O(|\lambda |^{N})$ were established for some $N> -1$, $|\lambda |$ large, and $\lambda \in{{\mathbb{C}}}$ in strips where $R(\lambda )$ is holomorphic. In this work, we prove the optimal bound $O(|\lambda |^{-1})$ under the non-positive sectional curvature assumption by taking into account the oscillatory behavior of the Schwartz kernel of the resolvent.

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