On stability and scalar curvature rigidity of quaternion-Kähler manifolds

A rigidity result for weakly asymptotically hyperbolic manifolds with lower bounds on Ricci curvature is proved without assuming that the manifolds are spin.The argument makes use of a quasilocal mass characterization of Euclidean balls from [9] [14] and eigenfunction compactification ideas from [12].

An Einstein manifold is called scalar curvature rigid if there are no compactly supported volume-preserving deformations of the metric which increase the scalar curvature. We give various characterizations of scalar curvature rigidity for open Einstein manifolds as well as for closed Einstein manifolds. As an application, we construct mass-decreasing deformations of the Riemannian Schwarzschild metric and the Taub–Bolt metric.

In this paper we extend the local scalar curvature rigidity result in Brendle and Marques (J Differ Geom 88:379–394, 2011) to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper (Corvino, Commun Math Phys 214:137–189, 2000). We obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces. We also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains, where the lapse functions are positive, on vacuum static spaces with positive scalar curvature, and show such domains are maximal, which generalizes the work in Hang and Wang (Commun Anal Geom 14:91–106, 2006).

In this paper, we prove a scalar curvature rigidity result for geodesic balls in $S^n$. This result contrasts sharply with the counterexamples to Min-Oo’s conjecture constructed in "Deformations of the hemisphere that increase scalar curvature," Invent. Math. 185 (2011), 175–197.

We prove a rigidity result for non-negative scalar curvature perturbations of the Euclidean metric on $$\mathbb {R}^n$$ , which may be regarded as a weak version of the rigidity statement of the positive mass theorem. We prove our result by analyzing long time solutions of Ricci DeTurck flow. As a byproduct in doing so, we extend known $$L^p$$ bounds and decay rates for Ricci DeTurck flow and prove regularity of the flow at the initial data.

Motivated by Brendle-Marques-Neves' counterexample to the Min-Oo's conjecture, we prove a volume constrained scalar curvature rigidity theorem which applies to the hemisphere.

This paper presents a scalar curvature rigidity result of real hyperbolic product manifolds in analogy to M. Min–Oo’s result in [14]. In order to prove this, we consider Dirac bundles obtained from the spinor bundle, and we derive Killing equations trivializing these Dirac bundles. Moreover, an integrated Bochner–Weitzenbock formula is shown which allows the usage of the non–compact Bochner technique.

Marginally Outer Trapped Surfaces and Scalar Curvature Rigidity

Scalar curvature rigidity of almost Hermitian manifolds which are asymptotic to [formula omitted

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.

For closed connected Riemannian spin manifolds, an upper estimate of the smallest eigenvalue of the Dirac operator in terms of the hyperspherical radius is proved. When combined with known lower Dirac eigenvalue estimates, this has a number of geometric consequences. Some are known and include Llarull’s scalar curvature rigidity of the standard metric on the sphere, Geroch’s conjecture on the impossibility of positive scalar curvature on tori, and a mean curvature estimate for spin fill-ins with nonnegative scalar curvature due to Gromov, including its rigidity statement recently proved by Cecchini, Hirsch and Zeidler. New applications provide a comparison of the hyperspherical radius with the Yamabe constant and improved estimates of the hyperspherical radius for Kähler manifolds, Kähler–Einstein manifolds, quaternionic Kähler manifolds, and manifolds with a harmonic 1-form of constant length.

The new positive energy conjecture was first formulated by Horowitz and Myers in the late 1990s to probe for a possible extended, nonsupersymmetric AdS/CFT correspondence. We consider a version formulated for complete, asymptotically Poincaré-Einstein Riemannian metrics (M, g) with bounded scalar curvature R ≥ −n(n − 1) and with no (inner) boundary except possibly a finite union of compact, totally geodesic hypersurfaces (horizons). This version then asserts that any such (M, g) must have mass not less than a certain bound which is realized as the mass m0 of a metric g0 induced on a time-symmetric slice of a spacetime called an AdS soliton. This conjecture remains unproved, having so far resisted standard techniques. Little is known other than that the conjecture is true for metrics which are sufficiently small perturbations of g0. We pose another test for the conjecture. We assume its validity and attempt to prove as a corollary the corresponding scalar curvature rigidity statement, which is that g0 is the unique asymptotically Poincaré-Einstein metric with mass m0 obeying R ≥ −n(n − 1). Were a second such metric g1 not isometric to g0 to exist, it then may well admit perturbations of lower mass, contradicting the assumed validity of the conjecture. We find enough rigidity to show that the minimum mass metric must be static Einstein, so the problem is reduced to that of static uniqueness. When n = 3 the manifold must be isometric to a time-symmetric slice of an AdS soliton spacetime, or must have a non-compact horizon. En route we study the mass aspect, obtaining and generalizing known results: (i) we relate the mass aspect of static metrics to the holographic energy density, (ii) we obtain the conformal invariance of the mass aspect when the bulk dimension is odd, and (iii) we show the vanishing of the mass aspect for negative Einstein manifolds with Einstein conformal boundary.

In this paper we study the hyperbolic version of this result using Witten's method and show that a spin manifold which is (strongly) asymptotically hyperbolic cannot have scalar curvature R≥ = −n(n−1) everywhere unless it is isometric to hyperbolic space

Rigidity results for asymptotically locally hyperbolic manifolds with lower bounds on scalar curvature are proved using spinor methods related to the Witten proof of the positive mass theorem. The argument is based on a study of the Dirac operator defined with respect to the Killing connection. The existence of asymptotic Killing spinors is related to the spin structure on the end. The expression for the mass is calculated and proven to vanish for conformally compact Einstein manifolds with conformal boundary a spherical space form, giving rigidity. In the 4-dimensional case, the signature of the manifold is related to the spin structure on the end and explicit formulas for the relevant invariants are given.

We characterize the standard S 3 as the closed Ricci-positive 3-manifold with scalar curvature at least 6 having isoperimetric surfaces of largest area: 4π. As a corollary we answer in the affirmative an interesting special case of a conjecture of M. Min-Oo's on the scalar curvature rigidity of the upper hemisphere.