arXiv research

Nash's theorem proved with Günther's trick

An complete exposition of Matthias Gunther's elementary proof of Nash's isometric embedding theorem.

The paper proves isometric embeddings for smooth manifolds.

We obtain global extensions of the celebrated Nash-Kuiper theorem for $C^{1,θ}$ isometric immersions of compact manifolds with optimal Hölder exponent. In particular for the Weyl problem of isometrically embedding a convex compact surface in 3-space, we show that the Nash-Kuiper non-rigidity prevails upto exponent $θ<1…

Paper proves embedding theorem for conformally compact manifolds.

This note is about a little extension of Nash's embedding theorem in the case of complete manifolds.

We use the Nash embedding theorem to construct generators for the space of algebraic covariant derivative curvature tensors.

The Nash-Kuiper Theorem states that the collection of $C^1$-isometric embeddings from a Riemannian manifold $M^n$ into $\mathbb{E}^N$ is $C^0$-dense within the collection of all smooth 1-Lipschitz embeddings provided that $n < N$. This result is now known to be a consequence of Gromov's more general $h$-principle. Ther…

We explore the practicability of Nash's Embedding Theorem in vision and imaging sciences. In particular, we investigate the relevance of a result of Burago and Zalgaller regarding the existence of isometric embeddings of polyhedral surfaces in $\mathbb{R}^3$ and we show that their proof does not extended directly to hi…

The h-cobordism theorem is a noted theorem in differential and PL topology. A generalization of the h-cobordism theorem for possibly non simply connected manifolds is the so called s-cobordism theorem. In this paper, we prove semialgebraic and Nash versions of these theorems. That is, starting with semialgebraic or Nas…

We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash $C^1$ Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also sh…

Lying at the intersection of Ado's theorem and the Nash embedding theorem, we consider the problem of finding faithful representations of Lie groups which are simultaneously isometric embeddings. Such special maps are found for a certain class of solvable Lie groups which includes all Einstein and Ricci soliton solvman…

Study of metrics on spheres and their complex structure properties.

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.

Study on embedding properties of Riemannian manifolds with specific geometric constraints.

Given a multifunction from $X$ to the $k-$fold symmetric product $Sym_k(X)$, we use the Dold-Thom Theorem to establish a homological selection Theorem. This is used to establish existence of Nash equilibria. Cost functions in problems concerning the existence of Nash Equilibria are traditionally multilinear in the mixe…

Kuranishi's proof of complex deformation theory revisited

This article is a short nontechnical survey of recent progresses in fluid dynamics and differential geometry, relating a conjecture of Lars Onsager to the work of Nash on isometric embeddings.

Extends Nash-Kuiper theorem to higher Hölder exponents.

The paper explores how Finsler manifolds differ from Riemannian ones in functional inequalities.

What is the suitable Laplace operator on vector fields for the Navier-Stokes equation on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden's Laplacian. A probabilistic repr…

In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper says that any short embedding in codimension one can be uniformly approximated by $C^1$ isometric embeddings. This statement clearly cannot be true for $C^2$ embeddings in general, due to the classi…

Paper proves Allard's theorem in Alexandrov spaces.

First I will explain my motivation to introduce the $δ$-invariants for Riemannian manifolds. I will also recall the notions of ideal immersions and best ways of living. Then I will present a few of the many applications of $δ$-invariants to several areas in mathematics. Finally, I will present two optimal inequalities …

Ancient Ricci flows with bounded Nash entropy have uniform Sobolev inequalities.

Let (N,g) be a Riemannian manifold. For a compact, connected and oriented submanifold M of N. we define the space of volume preserving embeddings Emb_μ(M,N) as the set of smooth embeddings f:M \rightarrow N such that f*μ^{f}=μ, where μ^{f} (resp. μ) is the Riemannian volume form on f(M) (resp. M) induced by the ambient…

New Zoll families of minimal spheres found in spheres and projective spaces.

Constructs harmonic spinors and 1-forms on 3-manifold connected sums and torus sums.

In this article, we study local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic m…

The study examines Nash equilibria in utility maximization games with multiplicative performance criteria.

Second part of proving linearization theorem for sl2(C).

The paper constructs metrics on spheres with families of minimal hypersurfaces.

Study metric perturbations to make degenerate harmonic forms non-degenerate.

Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.

Crochet models of a hyperbolic plane is a popular educational tool as they help to visualize complicated objets in hyperbolic geometry. We present another way how to make crochet models when we view them as a part of a triangulated hyperbolic plane. We also provide a model of a cylinder in a hyperbolic space. This appr…

The paper studies gradient estimates for solutions of a nonlinear elliptic equation on Riemannian manifolds.

Although the Nash theorem solves the isometric embedding problem, matters are inherently more involved if one is further seeking an embedding that is well-behaved from the standpoint of submanifold geometry. More generally, consider a Lipschitz map $F:M^m\to\mathbb R^n$, where $M^m$ is a Hadamard manifold whose curvatu…

New insights on quantifying space deformation.

The study finds invariants of smooth metrics on surfaces through embeddings into spheres.

Sharp gradient estimates for a weighted p-Laplacian equation on metric measure spaces.

The paper establishes gradient estimates and Liouville theorems for Φ-Laplacian equations on Riemannian manifolds.

Researchers solve a complex equation to embed graphs with negative curvature.

We consider a symmetric multi-players zero-sum game with two strategic variables. There are $n$ players, $n\geq 3$. Each player is denoted by $i$. Two strategic variables are $t_i$ and $s_i$, $i\in \{1, \dots, n\}$. They are related by invertible functions. Using the minimax theorem by \cite{sion} we will show that Nas…

Proves the Hodge conjecture for complex projective manifolds.

Deep fictitious play converges to Nash equilibrium in stochastic differential games.

Existence of smooth valuations on subspaces is shown for certain conditions.

Paper proves volume growth estimate for steady gradient Ricci solitons.

This paper is about the rigidity of compact group actions in the Poisson context. The main resut is that Hamiltonian actions of compact semisimple type are rigid. We prove it via a Nash-Moser normal form theorem for closed subgroups of SCI-type. This Nash-Moser normal form has other applications to stability results th…