Using our Theorem 1.1, minimalist

= 2 dvol g for all ,

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Let G be a second-countable, locally compact group. In this article we study amenable G-actions on Kirchberg algebras that admit an approximately central embedding of a canonical quasi-free action on the Cuntz algebra O_∞. If G is discrete, this coincides with the class of amenable and outer G-actions on Kirchberg algebras. We show that the resulting G-C*-dynamical systems are classified by equivariant Kasparov theory, up to cocycle conjugacy. This is the first classification theory of its kind applicable to actions of arbitrary locally compact groups. Among various applications, our main result solves a conjecture of Izumi for actions of discrete amenable torsion-free groups, and recovers the main results of recent work by Izumi–Matui for actions of poly-Z groups.

More generally, we recall that a minimal immersion M 3 !(N 4 , g) is stable iffor all f C 0 (M \M ).Theorem 1.3.Let (N 4 , g) be a closed Riemannian manifold.There exists C = C(N, g) such that, if M 3 !N 4 is a 2-sided, stable minimal immersion, thenWe have recently generalized Theorem 1.3 to hold in (non-compact) ambient (N 4 , g) with bounded sectional curvature in a joint work with Stryker [18, Corollary 2.5], which resolves [22, Conjecture 2.13].Theorems 1.2 and 1.3 are the 4-dimensional analogue of the well-known curvature estimate of Schoen [46] for minimal surfaces in three dimensions.Note that, by the work of Schoen-Simon-Yau [49], such an estimate was previously known to hold where C depended on an upper bound for volume of M 3 in small balls.Remark 1.4.There have been several interesting developments since the first version of this paper was posted.The authors have discovered [16] a new proof of Theorem 1.1 that can be localized to obtain (interior) volume estimates (in the spirit of Pogorelov [45]; cf.[20]).This new proof is related to the study of uniformly positive scalar curvature, while the current paper is related to the study of non-negative scalar curvature.Subsequently, Catino-Mastrolia-Roncoroni have discovered [11] a completely different proof of Theorem 1.1, related to the study of Bakry-mery-Ricci curvature.Interestingly, the dimension restriction n+1=4 enters each proof in a different way.( 1 ) A slight modification of the proof of Theorem 1.1 yields a structure theorem for finite-index minimal hypersurfaces in R 4 , analogous to the well-known results of Gulliver, Fischer-Colbrie, and Osserman [27], [30], [43].Recall that a complete, 2-sided, immersedwhere( 1 ) Added in proof: Theorem 1.1 has recently been generalized to the cases M 4 !R 5 by the authors along with Minter and Stryker [17], as well as M 5 !R 6 by Mazet [36].Nktp4jcrcuP09pwXQggfOTSZkVxS7gfwjwEzfN3Ci1c= Nktp4jcrcuP09pwXQggfOTSZkVxS7gfwjwEzfN3Ci1c= stable minimal hypersurfaces in R 4 3 Theorem 1.5.A complete, 2-sided, minimal immersion M 3 !R 4 has finite Morse index if and only if it has finite total curvature M |A M | 3 <.We remark that Tysk [59] proved the same statement for a complete, 2-sided, minimal immersion M n !R n+1 (for 3n6) under the assumption that M has Euclidean volume growth.Theorem 1.5 has strong consequences on the structure of M near infinity.We recall the following definition.Definition 1.6.([47, 2]) Suppose n3 and let M n !R n+1 be a complete minimal immersion.An end E of M is regular at infinity if it is the graph of a function w over a hyperplane with the asymptoticsfor some constants a, b, and c j , where x 1 , ..., x n are the coordinates in .mersion with finite total curvature, then each end of M is regular at infinity.Moreover, by [34], a 2-sided minimal immersion with finite Morse index has finitely many ends.Combined with Theorem 1.5, this yields the following result.Corollary 1.7.Suppose M 3 !R 4 is a complete, 2-sided, minimal immersion with finite Morse index.Then M has finitely many ends, each of which is regular at infinity.In particular, M has cubic volume growth, i.e. sup

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