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The isometric immersion of two-dimensional Riemannian manifolds or surfaces with negative Gauss curvature into the three-dimensional Euclidean space is studied in this paper. The global weak solutions to the Gauss-Codazzi equations with…

A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as isometric immersions into $\R^3$. This problem can be formulated as…

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…

The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is…

In this paper, we study the smooth isometric immersion of a complete, simply connected surface with a negative Gauss curvature into the three-dimensional Euclidean space. A fundamental and longstanding problem is to find a sufficient…

We survey recent developments on the analysis of Gauss--Codazzi--Ricci equations, the first-order PDE system arising from the classical problem of isometric immersions in differential geometry, especially in the regime of low Sobolev…

We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a…

This paper is devoted to investigating the isometric immersion problem of Riemannian manifolds in a high codimension. It has recently been demonstrated that any short immersion from an $n$-dimensional smooth compact manifold into…

We study the problem of isometrically embedding a two-dimensional Riemannian manifold into Euclidean three-space. It is shown that if Gaussian curvature vanishes to finite order and its zero set consists of two smooth curves tangent at a…

We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding…

We investigate isometric immersions $f\colon M^n\to\R^{n+2}$, $n\geq 3$, of Riemannian manifolds into Euclidean space with codimension two that admit isometric deformations that preserve the metric of the Gauss map. In precise terms, the…

An important theorem about biharmonic submanifolds proved independently by Chen-Ishikawa [CI] and Jiang [Ji] states that an isometric immersion of a surface into 3-dimensional Euclidean space is biharmonic if and only if it is harmonic…

We use a new method to give conditions for the existence of a local isometric immersion of a Riemannian $n$-manifold $M$ in $\mathbb{R}^{n+k}$, for a given $n$ and $k$. These equate to the (local) existence of a $k$-tuple of scalar fields…

We provide integral curvature bounds for compact Riemannian manifolds that allow isometric immersions into a Euclidean space with low codimension in terms of the Betti numbers.

We study the Darboux equation, a fundamental PDE arising in the theory of isometric immersions of two-dimensional Riemannian manifolds into $\mathbb{R}^3$, in the low-regularity regime. We introduce a notion of weak solution for $u\in…

The celebrated Nash Embedding Theorem asserts that every closed Riemannian manifold can be isometrically embedded into a sufficiently high-dimensional Euclidean space. In this paper, we prove an analogous result in the conformally compact…

We study curvature invariants of a sub-Riemannian manifold (i.e., a manifold with a Riemannian metric on a non-holonomic distribution) related to mutual curvature of several pairwise orthogonal subspaces of the distribution, and prove…

A fundamental result in global analysis and nonlinear elasticity asserts that given a solution $\mathfrak{S}$ to the Gauss--Codazzi--Ricci equations over a simply-connected closed manifold $(\mathcal{M}^n,g)$, one may find an isometric…

We study the old problem of isometrically embedding a 2-dimensional Riemannian manifold into Euclidean 3-space. It is shown that if the Gaussian curvature vanishes to finite order and its zero set consists of two Lipschitz curves…

We develop a compactness result near the boundary for families of locally convex immersions. We also develop a mod 2 degree theory for immersion of constant (and prescribed) Gaussian curvature with prescribed boundary. These are then used…