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We prove a rigidity theorem that shows that, under many circumstances, quasi-isometric embeddings of equal rank, higher rank symmetric spaces are close to isometric embeddings. We also produce some surprising examples of quasi-isometric embeddings of higher rank symmetric spaces. In particular, we produce embeddings of $SL(n,mathbb R)$ into $Sp(2(n-1),mathbb R)$ when no isometric embeddings exist. A key ingredient in our proofs of rigidity results is a direct generalization of the Mostow-Morse Lemma in higher rank. Typically this lemma is replaced by the quasi-flat theorem which says that maximal quasi-flat is within bounded distance of a finite union of flats. We improve this by showing that the quasi-flat is in fact flat off of a subset of codimension $2$.
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In this paper, we prove that certain spaces are not quasi-isometric to Cayley graphs of finitely generated groups. In particular, we answer a question of Woess and prove a conjecture of Diestel and Leader by showing that certain homogeneous graphs are not quasi-isometric to a Cayley graph of a finitely generated group. This paper is the first in a sequence of papers proving results announced in [EFW0]. In particular, this paper contains many steps in the proofs of quasi-isometric rigidity of lattices in Sol and of the quasi-isometry classification of lamplighter groups. The proofs of those results are completed in [EFW1]. The method used here is based on the idea of coarse differentiation introduced in [EFW0].
The definition of quasi-local mass for a bounded space-like region in space-time is essential in several major unsettled problems in general relativity. The quasi-local mass is expected to be a type of flux integral on the boundary two-surface and should be independent of whichever space-like region it bounds. An important idea which is related to the Hamiltonian formulation of general relativity is to consider a reference surface in a flat ambient space with the same first fundamental form and derive the quasi-local mass from the difference of the extrinsic geometries. This approach has been taken by Brown-York and Liu-Yau (see also related works) to define such notions using the isometric embedding theorem into the Euclidean three-space. However, there exist surfaces in the Minkowski space whose quasilocal mass is strictly positive. It appears that the momentum information needs to be accounted for to reconcile the difference. In order to fully capture this information, we use isometric embeddings into the Minkowski space as references. In this article, we first prove an existence and uniqueness theorem for such isometric embeddings. We then solve the boundary value problem for Jangs equation as a procedure to recognize such a surface in the Minkowski space. In doing so, we discover new expression of quasi-local mass. The new mass is positive when the ambient space-time satisfies the dominant energy condition and vanishes on surfaces in the Minkowski space. It also has the nice asymptotic behavior at spatial and null infinity. Some of these results were announced in [29].
J. Nash proved that the geometry of any Riemannian manifold M imposes no restrictions to be embedded isometrically into a (fixed) ball B_{mathbb{R}^{N}}(1) of the Euclidean space R^N. However, the geometry of M appears, to some extent, imposing restrictions on the mean curvature vector of the embedding.
We give a new proof for the local existence of a smooth isometric embedding of a smooth $3$-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into $6$-dimensional Euclidean space. Our proof avoids the sophisticated arguments via microlocal analysis used in earlier proofs. In Part 1, we introduce a new type of system of partial differential equations, which is not one of the standard types (elliptic, hyperbolic, parabolic) but satisfies a property called strong symmetric positivity. Such a PDE system is a generalization of and has properties similar to a system of ordinary differential equations with a regular singular point. A local existence theorem is then established by using a novel local-to-global-to-local approach. In Part 2, we apply this theorem to prove the local existence result for isometric embeddings.
We consider a general notion of snowflake of a metric space by composing the distance by a nontrivial concave function. We prove that a snowflake of a metric space $X$ isometrically embeds into some finite-dimensional normed space if and only if $X$ is finite. In the case of power functions we give a uniform bound on the cardinality of $X$ depending only on the power exponent and the dimension of the vector space.
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