universal inequalities
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2020 ◽  
Vol 80 (10) ◽  
Author(s):  
Run-Qiu Yang ◽  
H. Lü

AbstractFor static black holes in Einstein gravity, if matter fields satisfy a few general conditions, we conjecture that three characteristic parameters about the spatial size of black holes, namely the outermost photon sphere area $$A_{\mathrm {ph,out}}$$ A ph , out , the corresponding shadow area $$A_{\mathrm {sh,out}}$$ A sh , out and the horizon area $$A_{{\mathcal {H}}}$$ A H satisfy a series of universal inequalities $$9A_{{\mathcal {H}}}/4\le A_{\mathrm {ph,out}}\le A_{\mathrm {sh,out}}/3\le 36\pi M^2$$ 9 A H / 4 ≤ A ph , out ≤ A sh , out / 3 ≤ 36 π M 2 , where M is the ADM mass. We present a complete proof in the spherically symmetric case and some pieces of evidence to support it in general static cases. We also discuss the properties of the photon spheres in general static spacetimes and show that, similar to horizon, photon spheres are also conformal invariant structures of the spacetimes.


2018 ◽  
Vol 227 (2) ◽  
pp. 843-859 ◽  
Author(s):  
Gabriele Balletti ◽  
Akihiro Higashitani

2017 ◽  
Vol 28 (13) ◽  
pp. 1750097
Author(s):  
Guangyue Huang ◽  
Xuerong Qi ◽  
Hongjuan Li

Let [Formula: see text] be an [Formula: see text]-dimensional compact self-shrinker in [Formula: see text] with smooth boundary [Formula: see text]. In this paper, we study eigenvalues of the operator [Formula: see text] on [Formula: see text], where [Formula: see text] is defined by [Formula: see text] with [Formula: see text] denoting a positive definite (0,2)-tensor field on [Formula: see text]. We obtain “universal” inequalities for eigenvalues of the operator [Formula: see text]. These inequalities generalize the result of Cheng and Peng in [Estimates for eigenvalues of [Formula: see text] operator on self-shrinkers, Commun. Contemp. Math. 15(6) (2013), Article ID:1350011, 23 pp.]. Furthermore, we also consider the case that equalities occur.


2017 ◽  
Vol 37 (5) ◽  
pp. 1536-1544
Author(s):  
Feng DU ◽  
Chuanxi WU ◽  
Guanghan LI ◽  
Changyu XIA

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Kei Funano

AbstractWe apply Gromov’s ham sandwich method to get: (1) domain monotonicity (up to a multiplicative constant factor); (2) reverse domain monotonicity (up to a multiplicative constant factor); and (3) universal inequalities for Neumann eigenvalues of the Laplacian on bounded convex domains in Euclidean space.


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