We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M, g) that are C0-asymptotic to Schwarzschild of mass m > 0. Refining an argument due to H. Bray, we obtain an effective volume comparison theorem in Schwarzschild. We use it to show that isoperimetric regions exist in (M, g) for all sufficiently large volumes, and that they are close to centered coordinate spheres. This implies that the volume-preserving stable constant mean curvature spheres constructed by G. Huisken and S.-T. Yau as well as R. Ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume. This confirms a conjecture of H. Bray. Our results are consistent with the uniqueness results for volumepreserving stable constant mean curvature surfaces in initial data sets obtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian. The additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work.

In this article we show how the classical theory of Reeb and others extends to the case of codimension one singular foliations on closed oriented manifolds, provided that at the singular set sing(F), the foliation is locally defined by Bott-Morse functions which are transversely centers. We prove, in this setting, the equivalent of the local and the complete stability theorems of Reeb. We show that if F has a compact leaf with finite fundamental group, or if a component of sing(F) has codimension ≥ 3 and finite fundamental group, then all leaves of F are compact and diffeomorphic, sing(F) consists of two connected components, and there is a Bott-Morse function f : M → [0, 1] such that f : M \ sing(F) → (0, 1) is a fiber bundle defining F and sing(F) = f.1({0, 1}). This yields a topological description of the type of leaves that appear in these foliations, and also the type of manifolds admitting such foliations. These results unify and generalize well known results for cohomogeneity one isometric actions, and a theorem of Reeb for foliations with Morse singularities of center type.

It is proved that if a compact manifold admits a smooth action by a compact, connected, non-abelian Lie group, then it admits a metric of positive scalar curvature. This result is used to prove that if ^<i>n</i> is an exotic<i>n</i>-sphere which does not bound a spin manifold, then the only possible compact connected transformation groups of ^<i>n</i> are tori of dimension [(<i>n</i>+1)/2].

We consider left-invariant distances d on a Lie group G with the property that there exists a multiplicative one-parameter group of Lie automorphisms (0,∞)→Aut(G), λ↦δλ, so that d(δ_λx,δ_λy)=λd(x,y), for all x,y∈G and all λ>0. First, we show that all such distances are admissible, that is, they induce the manifold topology. Second, we characterize multiplicative one-parameter groups of Lie automorphisms that are dilations for some left-invariant distance in terms of algebraic properties of their infinitesimal generator. Third, we show that an admissible left-invariant distance on a Lie group with at least one nontrivial dilating automorphism is bi-Lipschitz equivalent to one that admits a one-parameter group of dilating automorphisms. Moreover, the infinitesimal generator can be chosen to have spectrum in [1,∞). Fourth, we characterize the automorphisms of a Lie group that are a dilating automorphisms for some admissible distance. Finally, we characterize metric Lie groups admitting a one-parameter group of dilating automorphisms as the only locally compact, isometrically homogeneous metric spaces with metric dilations of all factors. Such metric spaces appear as tangents of doubling metric spaces with unique tangents.

We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disk. The surfaces are constructed by singular perturbation methods and have three boundary components. They are the free boundary analogue of the Costa-Hoffman-Meeks surfaces and the surfaces constructed by Kapouleas by desingularizing coaxial catenoids and planes. It is plausible that the minimal surfaces we constructed here are the same as the ones obtained recently by Ketover using the min-max method.