Related papers: Counterexamples for Local Isometric Embedding

In this paper we define Fermi-type coordinates in a 2-dimensional Lorentz manifold, and use this coordinate system to provide a local characterization of constant Gaussian curvature metrics for such manifolds, following a classical result…

A proof of the isometric embedding of a given two-metric in E^3 of class C^1. The method uses the theory of first order partial differential equations. The curvature of the metric plays no role in the proof.

We give new examples of closed smooth 4-manifolds which support singular metrics of nonpositive curvature, but no smooth ones, thereby answering affirmatively a question of Gromov. The obstruction comes from patterns of incompressible…

Nontrivial isometric embeddings for flat metrics (i.e., those which are not just planes in the ambient space) can serve as useful tools in the description of gravity in the embedding gravity approach. Such embeddings can additionally be…

For appropriately values of $H$, we obtain an area estimate for a complete non-compact $H$-surface of finite topology and finite area, embedded in a three-manifold of negative curvature. Moreover, in the case of equality and under…

Due to Janet-Cartan's theorem, any analytic Riemannian manifolds can be locally isometrically embedded into a sufficiently high dimensional Euclidean space. However, for an individual Riemannian manifold (M,g), it is in general hard to…

In this article, we give the integrability conditions for the existence of an isometric immersion from an orientable simply connected surface having prescribed Gauss map and positive extrinsic curvature into some unimodular Lie groups. In…

This is a significantly improved version with new applications. We show that there are many cohomogeneity one manifolds which do not admit an analytic invariant metric with non-negative sectional curvature, although they do have a smooth…

We provide examples of non-locally compact geodesic Ptolemy metric spaces which are not uniquely geodesic. On the other hand, we show that locally compact, geodesic Ptolemy metric spaces are uniquely geodesic. Moreover, we prove that a…

We show that a closed non-orientable $3$-manifold admits a positive scalar curvature metric if and only if its orientation double cover does; however, for each $4\le n\le 7$, there exist infinitely many smooth non-orientable $n$-manifolds…

We show the contractibility of spaces of invariant Riemannian metrics of positive scalar curvature on compact connected manifolds of dimension at least two, with and without boundary and equipped with compact Lie group actions. On manifolds…

In this manuscript we consider non-degenerate surfaces $\Sigma^2$ immersed in a 3-dimensional homogeneous space $\mathbb{L}^3(\kappa,\tau)$ endowed with two different metrics, the one induced by the Riemannian metric of…

Our aim in this paper is to study local rigidity for metrics defined on a compact manifold $M$ with boundary satisfying constant scalar curvature on $M$ and constant mean curvature on $\partial M$. We present some geometrical hypotheses…

We construct examples of smooth 4-dimensional manifolds M supporting a locally CAT(0)-metric, whose universal cover X satisfy Hruska's isolated flats condition, and contain 2-dimensional flats F with the property that the boundary at…

We construct self-dual(SD) but not locally conformally flat(LCF) metrics on families of non-simply connected 4-manifolds with small signature. We construct various sequences with bounded or unbounded Betti numbers and Euler characteristic.…

The isometric immersion of two-dimensional Riemannian manifold with negative Gauss curvature into the three-dimensional Euclidean space is considered through the Gauss-Codazzi equations for the first and second fundamental forms. The large…

A classic result of Shi and Tam states that a 2-sphere of positive Gauss and mean curvature bounding a compact 3-manifold with nonnegative scalar curvature, must have total mean curvature not greater than that of the isometric embedding…

We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold $(N^{n+1}, \bar g)$ through regularity study of a degenerate fully nonlinear curvature equation in general…

Given a smooth 2-dimensional Riemannian or pseudo-Riemannian manifold $(M, \boldsymbol{g})$ and an ambient 3-dimensional Riemannian or pseudo-Riemannian manifold $(N, \boldsymbol{h})$, one can ask under what circumstances does the exterior…

We review notions of mass of asymptotically locally Anti-de Sitter three-dimensional spacetimes, and apply them to some known solutions. For two-dimensional general relativistic initial data sets the mass is not invariant under asymptotic…