A Minkowski Inequality for Horowitz–Myers Geon

We investigate the validity and the stability of various Minkowski-like inequalities for C1-perturbations of the ball.Let K⊆Rn be a domain (possibly not convex and not mean-convex) which is C1-close to a ball. We prove the sharp geometric inequality(∫∂K‖II‖1dHn−1)1n−2≥C1(n)Per(K)1n−1, where C1(n) is the constant that yields the equality when K=B1 (and ‖II‖1 is the sum of the absolute values of the eigenvalues of the second fundamental form II of ∂K). Moreover, for any δ>0, if K is sufficiently C1-close to a ball, we show the almost sharp Minkowski inequality(∫∂KH+dHn−1)1n−2≥(C1(n)−δ)Per(K)1n−1. If K is axially symmetric, we prove the Minkowski inequality with the sharp constant (i.e., δ=0).We establish also the sharp quantitative stability (in the family of C1-perturbations of the ball) of the volumetric Minkowski inequality(0.1)(∫∂KH+dHn−1)1n−2≥C2(n)|K|1n, where C2(n) is the constant that yields the equality when K=B1. More precisely, we control the deviation of K from a ball (in a strong norm) with the difference between the left-hand side and the right-hand side of (0.1).Finally, we show, by constructing a counterexample, that the mentioned inequalities are false (even for domains C1-close to the ball) if one replaces H+ with H.

In this paper, we prove a sharp anisotropic Lp Minkowski inequality involving the total Lp anisotropic mean curvature and the anisotropic p-capacity for any bounded domains with smooth boundary in ℝn. As consequences, we obtain an anisotropic Willmore inequality, a sharp anisotropic Minkowski inequality for outward F-minimising sets and a sharp volumetric anisotropic Minkowski inequality. For the proof, we utilize a nonlinear potential theoretic approach which has been recently developed by Agostiniani et al. (2019).

We use norm inequalities for operators, such as Cauchy–Schwarz inequality and Minkowski inequality, to establish sharp norm inequalities for the absolute value of operators. Among other inequalities, it is shown that if A, B and X are operators on a Hilbert space, such that X is self-adjoint, and r ≥ 1, then This inequality generalizes some earlier related results. Other related inequalities are also discussed.

The hyperbolic metric and different hyperbolic type metrics are studied in open sector domains of the complex plane. Several sharp inequalities are proven for them. Our main result describes the behavior of the triangular ratio metric under quasiconformal maps from one sector onto another one.

Unique continuation and extensions of Killing vectors at boundaries for stationary vacuum space-times

The sharp convex mixed Lorentz-Sobolev inequality

Let (M, g) be an (n + 1)-dimensional asymptotically locally hyperbolic (ALH) manifold with a conformal compactification whose conformal infinity is ($\partial$M, [$\gamma$]). We will first observe that Ch(M, g) $\le$ n, where Ch(M, g) is the Cheeger constant of M. We then prove that, if the Ricci curvature of M is bounded from below by --n and its scalar curvature approaches --n(n+1) fast enough at infinity, then Ch(M, g) = n if and only Y($\partial$M, [$\gamma$]) $\ge$ 0, where Y($\partial$M, [$\gamma$]) denotes the Yamabe invariant of the conformal infinity. This gives an answer to a question raised by J. Lee [L].

In this article, we extend Anderson's higher-dimensional Dehn filling construction to a large class of infinite-volume hyperbolic manifolds. This gives an infinite family of topologically distinct asymptotically hyperbolic Einstein manifolds with the same conformal infinity. The construction involves finding a sequence of approximate solutions to the Einstein equations and then perturbing them to exact ones.

On a hyperbolic Poincare manifold, we derive an explicit relationship between the eigenvalues of Weyl-Schouten tensor of a conformal representative of the conformal infinity and the principal curvatures of the level sets of the associated geodesic defining function. This considerably simplifies the arguments and generalizes the results of Galvez, Mira and the second author. In particular, we obtain the equivalence between Christoffel-type problems for hypersurfaces in a hyperbolic Poincare manifold and scalar curvature problems on the conformal infinity.

Let (M, g) be an Asymptotically Locally Hyperbolic (ALH) manifold which is the interior of a conformally compact manifold and (∂M, [γ]) its conformal infinity. Suppose that the Ricci tensor of (M, g) dominates that of the hyperbolic space and that its scalar curvature satisfies a certain decay condition at infinity. If the Yamabe invariant of (∂M, [γ]) is non-negative, we prove that there exists a conformal metric on M with non-negative scalar curvature and whose boundary ∂M has either positive or zero constant inner mean curvature. In the spin case, we make use of a previous estimate obtained by X. Zhang and the authors for the Dirac operator of the induced metric on ∂M. As a consequence, we generalize and simplify the proof of the result by Andersson and Dahl in [Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom.16 (1998) 1–27] about the rigidity of the hyperbolic space when the prescribed conformal infinity ∂M is a round sphere. We also provide non-existence results for conformally compact ALH spin metrics when ∂M is conformal to a Riemannian manifold with special holonomy.

With respect to any special boundary defining function, a conformally compact asymptotically hyperbolic metric has an asymptotic expansion near its conformal infinity. If this expansion is even to a certain order and satisfies one extra condition, then it is possible to define its renormalized volume and show that it is independent of choices that preserve this evenness structure. We prove that such expansions are preserved under normalized Ricci flow. We also study the variation of curvature functionals in this setting, and as one application, obtain the variation formula $$ \frac{{\rm d}}{{\rm d}t} {\rm RenV}\big(M^n, g(t)\big) = -\mathop{\vphantom{T}}^R \! \! \! \int_{M^n} (S(g(t))+n(n-1)) {\rm d}V_{g(t)},$$ where $S(g(t))$ is the scalar curvature for the evolving metric $g(t)$, and $\mathop{\vphantom{T}}^R \! \! \! \int (\cdot) {\rm d}V_g$ is Riesz renormalization. This extends our earlier work to a broader class of metrics.

This chapter analyzes the ambient and Poincaré metrics for locally conformally flat manifolds and for conformal classes containing an Einstein metric. The obstruction tensor vanishes for even dimensional conformal structures of these types. It shows that for these special conformal classes, there is a way to uniquely specify the formally undetermined term at order n/2 in an invariant way and thereby obtain a unique ambient metric up to terms vanishing to infinite order and up to diffeomorphism, just like in odd dimensions. It derives a formula of Skenderis and Solodukhin [SS] for the ambient or Poincaré metric in the locally conformally flat case which is in normal form relative to an arbitrary metric in the conformal class, and proves an elated unique continuation result for hyperbolic metrics in terms of data at conformal infinity. The case n = 2 is special for all of these considerations. The chapter also derives the form of the GJMS operators for an Einstein metric.

The Cauchy slicings for globally hyperbolic spacetimes and their relation with the causal boundary are surveyed and revisited, starting at the seminal conformal boundary constructions by R. Penrose. Our study covers: (1) adaptive possibilities and techniques for their Cauchy slicings, (2) global hyperbolicity of sliced spacetimes, (3) critical review on the conformal and causal boundaries for a globally hyperbolic spacetime, and (4) procedures to compute the causal boundary of a Cauchy temporal splitting by using isocausal comparison with a static product. New simple counterexamples on $\mathbb{R}^2$ illustrate a variety of possibilities related to these splittings, such as the logical independence (for normalized sliced spacetimes) between the completeness of the slices and global hyperbolicity, the necessity of uniform bounds on the slicings in order to ensure global hyperbolicity, or the insufficience of these bounds for the computation of the causal boundary. A refinement of one of these examples shows that the space of all the (normalized, conformal classes of) globally hyperbolic metrics on a smooth product manifold $\mathbb{R}\times S$ is not convex, even though it is path connected by means of piecewise convex combinations.

We consider a (2 + 1)-dimensional holographic CFT on a static spacetime with globally timelike Killing vector. Taking the spatial geometry to be closed but otherwise general we expect a non-trivial vacuum energy at zero temperature due to the Casimir effect. We assume a thermal state has an AdS/CFT dual description as a static smooth solution to gravity with a negative cosmological constant, which ends only on the conformal boundary or horizons. A bulk geometric argument then provides an upper bound on the ratio of CFT free energy to temperature. Considering the zero temperature limit of this bound implies the vacuum energy of the CFT is non-positive. Furthermore the vacuum energy must be negative unless the boundary metric is locally conformal to a product of time with a constant curvature space. We emphasise the argument does not require the zero temperature bulk geometry to be smooth, but only that singularities are ‘good’ so are hidden by horizons at finite temperature.

We construct families of asymptotically locally hyperbolic Riemannian metrics with constant scalar curvature (i.e., time symmetric vacuum general relativistic initial data sets with negative cosmological constant), with prescribed topology of apparent horizons and of the conformal boundary at infinity, and with controlled mass. In particular we obtain new classes of solutions with negative mass.