Isometric Embedding Research Articles

Generalized Transportation Cost Spaces

The paper is devoted to the geometry of transportation cost spaces and their generalizations introduced by Melleray, Petrov, and Vershik (2008). Transportation cost spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein $1$ spaces. In this work, the existence of metric spaces with the following properties is proved: (1) uniformly discrete infinite metric spaces transportation cost spaces on which do not contain isometric copies of $\ell_1$, this result answers a question raised by Cuth and Johanis (2017); (2) locally finite metric spaces which admit isometric embeddings only into Banach spaces containing isometric copies of $\ell_1$; (3) metric spaces for which the double-point norm is not a norm. In addition, it is proved that the double-point norm spaces corresponding to trees are close to $\ell_\infty^d$ of the corresponding dimension, and that for all finite metric spaces $M$, except a very special class, the infimum of all seminorms for which the embedding of $M$ into the corresponding seminormed space is isometric, is not a seminorm.

Deforming black holes with odd multipolar differential rotation boundary

Motivated by the novel asymptotically global AdS$_4$ solutions with deforming horizon in [JHEP {\bf 1802}, 060 (2018)], we analyze the boundary metric with odd multipolar differential rotation and numerically construct a family of deforming solutions with tripolar differential rotation boundary, including two classes of solutions: solitons and black holes. We find that the maximal values of the rotation parameter $\varepsilon$, below which the stable large black hole solutions could exist, are not a constant for $T> T_{schw}=\sqrt{3}/2\pi\simeq0.2757$. When temperature is much higher than $ T_{schw}$, even though the norm of Killing vector $\partial_{t}$ keeps timelike for some regions of $\varepsilon<2$, solitons and black holes with tripolar differential rotation could be unstable and develop hair due to superradiance. As the temperature $T$ drops toward $T_{schw}$, we find that though there exists the spacelike Killing vector $\partial_{t}$ for some regions of $\varepsilon>2$, solitons and black holes still exist and do not develop hair due to superradiance. Moreover, for $T\leqslant T_{schw}$, the curves of entropy firstly combine into one curve and then separate into two curves again, in the case of each curve there are two solutions at a fixed value of $\varepsilon$. In addition, we study the deformations of horizon for black holes by using an isometric embedding in the hyperbolic three-dimensional space. Furthermore, we also study the quasinormal modes of the solitons and black holes, which have analogous behaviours to that of dipolar rotation and quadrupolar rotation.

The isometric embedding problem for length metric spaces

We prove that every proper [Formula: see text]-dimensional length metric space admits an “approximate isometric embedding” into Lorentzian space [Formula: see text]. By an “approximate isometric embedding” we mean an embedding which preserves the energy functional on a prescribed set of geodesics connecting a dense set of points.

On Lipschitz retraction of finite subsets of normed spaces

If $X$ is a metric space, then its finite subset spaces $X(n)$ form a nested sequence under natural isometric embeddings $X = X(1)\subset X(2) \subset \cdots$. It was previously established, by Kovalev when $X$ is a Hilbert space and, by Ba\v{c}\'{a}k and Kovalev when $X$ is a CAT(0) space, that this sequence admits Lipschitz retractions $X(n)\rightarrow X(n-1)$ for all $n\geq 2$. We prove that when $X$ is a normed space, the above sequence admits Lipschitz retractions $X(n)\rightarrow X$, $X(n)\rightarrow X(2)$, as well as concrete retractions $X(n)\rightarrow X(n-1)$ that are Lipschitz if $n=2,3$ and H\"older-continuous on bounded sets if $n>3$. We also prove that if $X$ is a geodesic metric space, then each $X(n)$ is a $2$-quasiconvex metric space. These results are relevant to certain questions in the aforementioned previous work which asked whether Lipschitz retractions $X(n)\rightarrow X(n-1)$, $n\geq 2$, exist for $X$ in more general classes of Banach spaces.

On isometric embeddings of Wasserstein spaces – the discrete case

The aim of this short paper is to offer a complete characterization of all (not necessarily surjective) isometric embeddings of the Wasserstein space Wp(X), where X is a countable discrete metric space and 0<p<∞ is any parameter value. Roughly speaking, we will prove that any isometric embedding can be described by a special kind of X×(0,1]-indexed family of nonnegative finite measures. Our result implies that a typical non-surjective isometric embedding of Wp(X) splits mass and does not preserve the shape of measures. In order to stress that the lack of surjectivity is what makes things challenging, we will prove alternatively that Wp(X) is isometrically rigid for all 0<p<∞.

On local isometric embeddings of three-dimensional Lie groups

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 determine the least dimensional Euclidean space into which (M, g) can be locally isometrically embedded, even in the case where (M, g) is homogeneous. In this paper, when the space (M, g) is locally isometric to a three-dimensional Lie group equipped with a left-invariant Riemannian metric, we classify all such spaces that can be locally isometrically embedded into the four-dimensional Euclidean space. Two types of algebraic equations, the Gauss equation and the derived Gauss equation, play an essential role in this classification.

On proper holomorphic maps between bounded symmetric domains

We study proper holomorphic maps between bounded symmetric domains D D and Ω \Omega . In particular, when D D and Ω \Omega are of the same rank ≥ 2 \ge 2 such that all irreducible factors of D D are of rank ≥ 2 \ge 2 , we prove that any proper holomorphic map from D D to Ω \Omega is a totally geodesic holomorphic isometric embedding with respect to certain canonical Kähler metrics of D D and Ω \Omega . We also obtain some results regarding holomorphic maps F : D → Ω F:D\to \Omega which map minimal disks of D D properly into rank- 1 1 characteristic symmetric subspaces of Ω \Omega . On the other hand, we obtain new rigidity results regarding semi-product proper holomorphic maps between D D and Ω \Omega under a certain rank condition on D D and Ω \Omega .

Global Embeddings of BTZ and Schwarzschild-ADS Type Black Holes in a Flat Space

We study the problem of construction of global isometric embedding for spherically symmetric black holes with negative cosmological constant in various dimensions. Firstly, we show that there is no such embedding for 4D RN-AdS black hole in 6D flat ambient space, completing the classification which we started earlier. Then we construct an explicit embedding of non-spinning BTZ black hole in 6D flat ambient space. Using this embedding as an anzats, we then construct a global explicit embedding of d-dimensional Schwarzschild-AdS black hole in a flat ( d + 3 ) -dimensional ambient space.

The isometric embedding of the augmented Teichmüller space of a Riemann surface into the augmented Teichmüller space of its covering surface

It is known that every finitely unbranched holomorphic covering $\pi:\widetilde{S}\rightarrow S$ of a compact Riemann surface $S$ with genus $g\geq2$ induces an isometric embedding $\Phi_{\pi} :Teich(S)\rightarrow Teich(\widetilde{S})$. By the mutual relations between Strebel rays in $Teich(S)$ and their embeddings in $Teich(\widetilde{S})$, we show that the augmented Teichmüller space $\widehat{Teich}(S)$ can be isometrically embedded in the augmented Teichmüller space $\widehat{Teich}(\widetilde{S})$.

Conformally isometric embeddings and Hawking temperature

We find necessary and sufficient conditions for existence of a locally isometric embedding of a vacuum space-time into a conformally-flat five-space. We explicitly construct such embeddings for any spherically symmetric Lorentzian metric in 3 + 1 dimensions as a hypersurface in . For the Schwarzschild metric the embedding is global, and extends through the horizon all the way to the r = 0 singularity. We discuss the asymptotic properties of the embedding in the context of Penrose’s theorem on Schwarzschild causality. We finally show that the Hawking temperature of the Schwarzschild metric agrees with the Unruh temperature measured by an observer moving along hyperbolae in .

The semi-global isometric embedding of surfaces with curvature changing signs stably

A semi-global isometric embedding of abstract surfaces with Gaussian curvature changing signs of any finite order is obtained through solving the Darboux equation.

On Turbulence and Geometry: from Nash to Onsager

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.

Kerr black holes with synchronised scalar hair and higher azimuthal harmonic index

Kerr black holes with synchronised scalar hair and azimuthal harmonic index m>1 are constructed and studied. The corresponding domain of existence has a broader frequency range than the fundamental m=1 family; moreover, larger ADM masses, M and angular momenta J are allowed. Amongst other salient features, non-uniqueness of solutions for fixed global quantities is observed: solutions with the same M and J co-exist, for consecutive values of m, and the ones with larger m are always entropically favoured. Our analysis demonstrates, moreover, the qualitative universality of various features observed for m=1 solutions, such as the shape of the domain of existence, the typology of ergo-regions, and the horizon geometry, which is studied through its isometric embedding in Euclidean 3-space.

Infinite Geodesics of Sub-Finsler Distances in Heisenberg Groups

Abstract We consider Heisenberg groups equipped with a sub-Finsler metric. Using methods of optimal control theory we prove that in this geometric setting the infinite geodesics are horizontal lines under the assumption that the sub-Finsler metric is defined by a strictly convex norm. This answers a question posed in [8] and has applications in the characterization of isometric embeddings into Heisenberg groups.

Isometric embedding and Darboux integrability

Given a smooth 2-dimensional Riemannian or pseudo-Riemannian manifold $$(M, \varvec{g})$$ and an ambient 3-dimensional Riemannian or pseudo-Riemannian manifold $$(N, \varvec{h})$$ , one can ask under what circumstances does the exterior differential system $$\mathcal {I}$$ for an isometric embedding $$M\hookrightarrow N$$ have particularly nice solvability properties. In this paper we give a classification of all 2-dimensional metrics $$\varvec{g}$$ whose isometric embedding system into flat Riemannian or pseudo-Riemannian 3-manifolds $$(N, \varvec{h})$$ is Darboux integrable. As an illustration of the motivation behind the classification, we examine in detail one of the classified metrics, $$\varvec{g}_0$$ , showing how to use its Darboux integrability in order to construct all its embeddings in finite terms of arbitrary functions. Additionally, the geometric Cauchy problem for the embedding of $$\varvec{g}_0$$ is shown to be reducible to a system of two first-order ODEs for two unknown functions—or equivalently, to a single second-order scalar ODE. For a large class of initial data, this reduction permits explicit solvability of the geometric Cauchy problem for $$\varvec{g}_0$$ up to quadrature. The results described for $$\varvec{g}_0$$ also hold for any classified metric whose embedding system is hyperbolic.

Remarks on holomorphic isometric embeddings between bounded symmetric domains

In this article, we study holomorphic isometric embeddings between bounded symmetric domains. In particular, we show the total geodesy of any holomorphic isometric embedding between reducible bounded symmetric domains with the same rank.

EMBEDDING OF METRIC GRAPHS ON HYPERBOLIC SURFACES

An embedding of a metric graph $(G,d)$ on a closed hyperbolic surface is essential if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_{e}(G)$ of $(G,d)$ is the lowest genus of a surface on which such an embedding is possible. We establish a formula to compute $g_{e}(G)$ and show that, for every integer $g\geq g_{e}(G)$ , there is an embedding of $(G,d)$ (possibly after a rescaling of $d$ ) on a surface of genus $g$ . Next, we study minimal embeddings where each complementary region has Euler characteristic $-1$ . The maximum essential genus $g_{e}^{\max }(G)$ of $(G,d)$ is the largest genus of a surface on which the graph is minimally embedded. We describe a method for an essential embedding of $(G,d)$ , where $g_{e}(G)$ and $g_{e}^{\max }(G)$ are realised.

Sobolev spaces of maps and the Dirichlet problem for harmonic maps

We give a self-contained treatment of the existence of a regular solution to the Dirichlet problem for harmonic maps into a geodesic ball on which the squared distance function from the origin is strictly convex. No curvature assumptions on the target are required. In this route we introduce a new deformation result which permits to glue a suitable Euclidean end to the geodesic ball without violating the convexity property of the distance function from the fixed origin. We also take the occasion to analyze the relationships between different notions of Sobolev maps when the target manifold is covered by a single normal coordinate chart. In particular, we provide full details on the equivalence between the notions of traced Sobolev classes of bounded maps defined intrinsically and in terms of Euclidean isometric embeddings.

Isometric embedding with nonnegative Gauss curvature under the graph setting

We study the regularity of the isometric embedding $ X: $ $ (B(O, r), g) $ $ \rightarrow $ $ (\mathbb{R}^3, g_{can}) $ of a 2-ball with nonnegatively curved $ C^4 $ metric into $ \mathbb{R}^3 $. Under the assumption that $ X $ can be expressed in the graph form, we show $ X \in C^{2,1} $ near $ P $, which is optimal by Iaia's example.

Limits of quasi-local angular momentum on an isolated gravitating system

I shall discuss the Chen-Wang-Yau quasilocal angular momentum, which is defined based on the theory of optimal isometric embedding and quasilocal mass of Wang-Yau, and the limits of which at spatial and null infinity of an isolated gravitating system. This is based on joint work with Po-Ning Chen, Jordan Keller, Ye-Kai Wang, and Shing-Tung Yau.