Universal bounds on the size of a black hole

For 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.

Estimates for eigenvalues of ℒr operator on self-shrinkers

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.

Applications of the ‘Ham Sandwich Theorem’ to Eigenvalues of the Laplacian

We 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.