scholarly journals Rigidity and stability of Einstein metrics for quadratic curvature functionals

2015 ◽  
Vol 2015 (700) ◽  
Author(s):  
Matthew J. Gursky ◽  
Jeff A. Viaclovsky
Keyword(s):  
Critical Points ◽  
Moduli Space ◽  
Einstein Metrics ◽  
Riemannian Metrics ◽  
Local Minimizers ◽  
Lagrange Equations ◽  
Critical Metrics ◽  
Local Properties ◽  
Space Of Riemannian Metrics ◽  
Curvature Functionals

AbstractWe investigate rigidity and stability properties of critical points of quadratic curvature functionals on the space of Riemannian metrics. We show it is possible to “gauge” the Euler–Lagrange equations, in a self-adjoint fashion, to become elliptic. Fredholm theory may then be used to describe local properties of the moduli space of critical metrics. We show a number of compact examples are infinitesimally rigid, and consequently, are isolated critical points in the space of unit-volume Riemannian metrics. We then give examples of critical metrics which are strict local minimizers (up to diffeomorphism and scaling). A corollary is a local “reverse Bishop's inequality” for such metrics. In particular, any metric

scholarly journals Critical associated metrics on contact manifolds. II

1986 ◽  
Vol 41 (3) ◽  
pp. 404-410 ◽  
Author(s):  
D. E. Blair ◽  
A. J. Ledger
Keyword(s):  
Scalar Curvature ◽  
Critical Points ◽  
Compact Manifold ◽  
Einstein Metrics ◽  
Contact Structure ◽  
Riemannian Metrics ◽  
Contact Manifold ◽  
Contact Manifolds ◽  
Contact Metric Structure ◽  
Sasakian Metrics

AbstractThe study of the integral of the scalar curvature, ∫MRdVg, as a function on the set of all Riemannian metrics of the same total volume on a compact manifold is now classical, and the critical points are the Einstein metrics. On a compact contact manifold we consider this and ∫M (R − R* − 4n2) dv, with R* the *-scalar curvature, as functions on the set of metrics associated to the contact structure. For these integrals the critical point conditions then become certain commutativity conditions on the Ricci operator and the fundamental collineation of the contact metric structure. In particular, Sasakian metrics, when they exist, are maxima for the second function.

scholarly journals Rigidity of Einstein metrics as critical points of quadratic curvature functionals on closed manifolds

2018 ◽  
Vol 175 ◽  
pp. 237-248 ◽  
Author(s):  
Bingqing Ma ◽  
Guangyue Huang ◽  
Xingxiao Li ◽  
Yu Chen
Keyword(s):  
Critical Points ◽  
Einstein Metrics ◽  
Curvature Functionals

Critical metrics for all quadratic curvature functionals

2021 ◽  
Author(s):  
Miguel Brozos‐Vázquez ◽  
Sandro Caeiro‐Oliveira ◽  
Eduardo García‐Río
Keyword(s):  
Critical Metrics ◽  
Curvature Functionals

scholarly journals Computability Theory and Differential Geometry

2004 ◽  
Vol 10 (4) ◽  
pp. 457-486 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Turing Machine ◽  
Riemannian Metrics ◽  
Computable Function ◽  
Computability Theory ◽  
Settling Time ◽  
Point Of View ◽  
Connected Components ◽  
Local Minima ◽  
Smooth Compact Manifold ◽  
Space Of Riemannian Metrics

Abstract. LetMbe a smooth, compact manifold of dimensionn≥ 5 and sectional curvature ∣K∣ ≤ 1. Let Met(M) = Riem(M)/Diff(M) be the space of Riemannian metrics onMmodulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met(M) such as the diameter. They showed that for every Turing machineTe,eϵ ω, there is a sequence (uniformly effective ine) of homologyn-sphereswhich are also hypersurfaces, such thatis diffeomorphic to the standardn-sphereSn(denoted)iffTehalts on inputk, and in this case the connected sum, so, andis associated with a local minimum of the diameter function on Met(M) whose depth is roughly equal to the settling time ae σe(k)ofTeon inputsy<k.At their request Soare constructed a particular infinite sequence {Ai}ϵωof c.e. sets so that for allithe settling time of the associated Turing machine forAidominates that forAi+1, even when the latter is composed with an arbitrary computable function. From this, Nabutovsky and Weinberger showed that the basins exhibit a “fractal” like behavior with extremely big basins, and very much smaller basins coming off them, and so on. This reveals what Nabutovsky and Weinberger describe in their paper on fractals as “the astonishing richness of the space of Riemannian metrics on a smooth manifold, up to reparametrization.” From the point of view of logic and computability, the Nabutovsky-Weinberger results are especially interesting because: (1) they use c.e. sets to prove structuralcomplexityof the geometry and topology, not merelyundecidabilityresults as in the word problem for groups, Hilbert's Tenth Problem, or most other applications; (2) they usenontrivialinformation about c.e. sets, the Soare sequence {Ai}iϵωabove, not merely Gödel's c.e. noncomputable set K of the 1930's; and (3)withoutusing computability theory there is no known proof that local minima exist even for simple manifolds like the torusT5(see §9.5).

Sign in / Sign up

Export Citation Format

Share Document