Equifocal submanifolds are an extension of the notion of isoparametric submanifolds in Euclidean spaces to symmetric spaces and consequently they share many of the properties well-known for their isoparametric relatives. An important step in understanding isoparametric submanifolds was Thorbergsson's proof of their homogeneity in codimension at least two which in particular solved the classification problem in this case. In this paper we prove the analogous result for equifocal submanifolds using the generalization of Thorbergsson's result to infinite dimensions due to Heintze and Liu.

A sharp affine L_p Sobolev inequality for functions on Euclidean n-space is established. This new inequality is significantly stronger than (and directly implies) the classical sharp L_p Sobolev inequality of Aubin and Talenti, even though it uses only the vector space structure and standard Lebesgue measure on ℝⁿ. For the new inequality, no inner product, norm, or conformal structure is needed; the inequality is invariant under all affine transformations of ℝⁿ.

We prove that an isometric action of a compact Lie group on a compact symmetric space is variationally complete if and only if it is hyperpolar.

We study the behavior of the Cheeger isoperimetric constant on infinite families of graphs and Riemann surfaces, and its relationship to the first eigenvalue λ₁ of the Laplacian. We adapt probabilistic arguments of Bollobás to the setting of Riemann surfaces, and then show that Cheeger constants of the modular surfaces are uniformly bounded from above away from the maximum value. We extend this result to the class of Ramanujan surfaces, defined below.

In this paper, we study the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary. Using a general version of Positive Mass Theorem of Schoen-Yau and Witten, we prove the following theorem: For any compact manifold with boundary and nonnegative scalar curvature, if it is spin and its boundary can be isometrically embedded into Euclidean space as a strictly convex hypersurface, then the integral of mean curvature of the boundary of the manifold cannot be greater than the integral of mean curvature of the embedded image as a hypersurface in Euclidean space. Moreover, equality holds if and only if the manifold is isometric with a domain in the Euclidean space. Conversely, under the assumption that the theorem is true, then one can prove the ADM mass of an asymptotically flat manifold is nonnegative, which is part of the Positive Mass Theorem.

We show that there is a C^∞ open and dense set of positively curved metrics on S² whose geodesic flow has positive topological entropy, and thus exhibits chaotic behavior. The geodesic flow for each of these metrics possesses a horseshoe and it follows that these metrics have an exponential growth rate of hyperbolic closed geodesics. The positive curvature hypothesis is required to ensure the existence of a global surface of section for the geodesic flow. Our proof uses a new and general topological criterion for a surface diffeomorphism to exhibit chaotic behavior.

Very shortly after this manuscript was completed, the authors learned about remarkable recent work by Hofer, Wysocki, and Zehnder [14, 15] on three dimensional Reeb flows. In the special case of geodesic flows on S², they show that if the geodesic flow has no parabolic closed geodesics (this holds for an open and C^∞ dense set of Riemannian metrics on S²), then it possesses either a global surface of section or a heteroclinic orbit. It then immediately follows from the proof of our main theorem that there is a C^∞ open and dense set of Riemannian metrics on S² whose geodesic flow has positive topological entropy.

This concludes a program to show that every orientable compact surface has a C^∞ open and dense set of Riemannian metrics whose geodesic flow has positive topological entropy.

In this paper, we continued our investigation of complete manifolds whose spectrum of the Laplacian has an optimal positive lower bound. In particular, we proved a splitting type theorem for n-dimensional manifolds that have a finite volume end. This can be viewed as a study of the equality case of a theorem of Cheng.

We show that any formal embedding sending a real-analytic strongly pseudoconvex hypersurface M ⊂ ℂ^N, N > 1, into another such hypersurface M^′ ⊂ ℂ^N+1 is convergent.