The study proves the stability of smooth embeddings of Riemannian metrics into Euclidean space.
problem Stability of smooth embeddings of Riemannian metrics into Euclidean space.
method Local perturbation method to derive a time-dependent local perturbation method.
result Construction of a smooth parametrized family of isometric embeddings for a short time.
This paper classifies Lie groups embeddable into 4D space.
problem Determining the minimum dimensional Euclidean space for Lie group embeddings.
method Used algebraic equations (Gauss and derived Gauss) to classify Lie groups.
result Classified Lie groups embeddable into 4D space.
Proves local isometric embedding of low-differentiability metrics in 3D space.
problem Isometric embedding of metrics of low differentiability in Euclidean 3-space.
method Simplified notation, geodesic and level parameters, solutions of initial value problems for first order non-linear PDEs, classical linear algebraic systems.
result Local isometric embedding exists for metrics of C1 differentiability.
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 point, then local sufficiently smooth isometric embedding exists.
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 intersecting transversely at a point, then local sufficiently smooth isometric embeddings …
Proves isometric embeddings in Euclidean spaces for RCD spaces.
problem Isometric immersions of RCD spaces in Euclidean spaces.
method Analyzes regular isometric immersions and eigenmaps of compact non-collapsed RCD spaces.
result Eigenmaps of compact non-collapsed RCD spaces are locally bi-Lipschitz embeddings to spheres.
Isometric embeddings of Teichmüller spaces are derived from branched coverings.
problem Characterizing isometric embeddings of Teichmüller spaces.
method Holomorphic isometric embeddings induced by branched coverings.
result All isometric embeddings of Teichmüller spaces of dimension at least 2 arise from branched coverings.
We construct smooth metrics on 2-manifold with nonpositive Gauss curvature which cannot be (C^3) locally isometrically embedded in R^3. Moreover, the Gauss curvature of the metric can be made negative except for one point.
Study examines Wang-Yau quasi-local energy in strong fields near apparent horizons.
problem Examining the behavior of Wang-Yau quasi-local energy near apparent horizons in strong fields.
method Analyzing the limit of the Wang-Yau quasi-local energy as a spacelike surface approaches an apparent horizon, considering bounded coordinate functions and spacelike mean curvature.
result The limit of the Wang-Yau quasi-local energy falls into two cases: it blows up or remains finite, depending on whether the horizon can be isometrically embedded into R3. Study of quasilocal mass using isometric embedding in various spacetimes.
problem Understanding quasilocal mass in different spacetimes.
method Application of isometric embedding theory to quasilocal mass.
result Recent progress in quasilocal mass calculations with specific spacetimes.
Proves rigidity for surfaces in Schwarzschild manifold.
problem Rigidity of surfaces in Schwarzschild manifold.
method Variational formula of quasi-local mass.
result Proves rigidity theorem for isometric embeddings.
The paper proves local isometric embeddings for singular metrics near a point.
problem Existence of local isometric embeddings for singular Riemannian metrics.
method Ramified local isometric embeddings using Leray's ramified Cauchy-Kovalevskaya Theorem.
result Existence of local analytic isometric embeddings into Euclidean space.
Paper uses advanced math to embed complex shapes smoothly.
problem Embedding shapes smoothly in high dimensions.
method Convex integration technique with extra iteration.
result Achieved smooth embedding in C1,1−ε class. In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang-Yau quasi-local mass for a surface in spacetime introduced in [7] and [8] is defined by minimizing quasi-local energy associated with admissible isometric embeddings of the …
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 …
This article uses Cartan-Kähler theory to construct local conservation laws from covariantly closed vector valued differential forms, objects that can be given, for example, by harmonic maps between two Riemannian manifolds. We apply the article's main result to construct conservation laws for covariant divergence free…
The paper solves an embedding problem for discs with specific curvature properties.
problem Embedding discs with positive Gauss curvature into Euclidean 3-space.
method Analyzes the free boundary isometric embedding problem with specific curvature conditions.
result Discs with specified curvature can be isometrically embedded into R3 orthogonally to the unit sphere. Defines a new quasi-local mass related to spacetime harmonic functions.
problem Calculating mass for regions in spacetime.
method Quasi-local proof using spacetime harmonic functions and embeddings into Minkowski space.
result Establishes positivity and vanishing conditions for the new quasi-local mass.
Study limits of quasi-local angular momentum at infinity of gravitating systems.
problem Understanding limits of quasi-local angular momentum at infinity of gravitating systems.
method Based on optimal isometric embedding and quasilocal mass theory, the study defines and analyzes the limits of quasi-local angular momentum at spatial and null infinity.
result Limits of quasi-local angular momentum are discussed at spatial and null infinity of an isolated gravitating system.
First explicit isometric immersion of a flat Klein bottle in 3D space.
problem Finding an isometric embedding of a Klein bottle in 3D.
method Piecewise-linear map from a Klein bottle to Euclidean 3-space.
result Explicit numerical data for a flat Klein bottle isometrically immersed in 3D.
New examples of embeddings defy Anosov representation limits.
problem Examples of robust quasi-isometric embeddings not approximated by Anosov representations.
method Exhibited non-locally rigid, Zariski dense embeddings in SLm(K). result Higher rank Anosov representation theorems fail for m≥30. We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in …
This text is the extended version of a talk given at 6th Meeting of Integrable Systems and Quantum Filed Theory at Peyresq hold from June 10 2006 to June 17, 2006 at Peyresq, France. The goal of this lecture is to give a brief introduction to Cartan-Kähler's theory. As examples to the application of this theory, we cho…
Two families of curvature inhomogeneous manifolds with constant Ricci eigenvalues are constructed.
problem Constructing curvature inhomogeneous Riemannian manifolds with constant Ricci eigenvalues.
method Derived from Einstein warped products and Riemannian Schwarzschild--Tangherlini manifold.
result Admits local isometric immersions and isometric embeddings of minimum codimension.
New framework for manifold convolutions using toric embeddings.
problem Computational intractability of manifold convolutions.
method Isometric embeddings into tori for global manifold convolutions.
result Global definition of manifold convolutions on finite approximations.
Researchers compute quasi-local mass of Kerr black hole horizon.
problem Computing the quasi-local mass of Kerr black hole horizons.
method Isometric embedding of Kerr black hole horizon in hyperbolic space.
result Quasi-local mass varies with Kerr black hole spin parameter.
LOCA learns standardized data coordinates from measurements.
problem Learning invariant data coordinates from non-linearly deformed manifolds.
method LOCA, a LOcal Conformal Autoencoder, learns an isometric embedding.
result LOCA preserves geometric information while learning invariant coordinates.
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 impor…
New energy measure for isolated systems in general relativity.
problem Quantifying energy in isolated systems in general relativity.
method Optimal isometric embedding and conformal Killing fields.
result Finite quasi-local energies for asymptotically flat spacetimes.
The paper classifies metrics allowing isometric embedding into flat 3-manifolds with Darboux integrability.
problem Isometric embedding of 2-manifolds into flat 3-manifolds with specific solvability properties.
method Classification of metrics allowing Darboux integrable isometric embedding into flat 3-manifolds.
result Explicit construction of isometric embeddings for a specific metric and reduction of Cauchy problem to ODEs.
In \cite{ly, ly2}, Liu and the second author propose a definition of the quasi-local mass and prove its positivity. This is demonstrated through an inequality which in turn can be interpreted as a total mean curvature comparison theorem for isometric embeddings of a surface of positive Gaussian curvature. The Riemannia…
We reduce the question of local nonsolvability of the Darboux equation, and hence of the isometric embedding problem for surfaces, to the local nonsolvability of a simple linear equation whose type is explicitly determined by the Gaussian curvature.
This text is the extended version of a talk given at the conference Geometry, Topology, QFT and Cosmology hold from May 28 to May 30, 2008 at the Observatoire de Paris. Using exterior differential systems, I provide a positive answer to the generalized isometric embedding problem of vector bundles, and show how conserv…
Study holomorphic isometric embeddings between symmetric domains, focusing on total geodesy.
problem Holomorphic isometric embeddings between bounded symmetric domains.
method Analyzing total geodesy in reducible bounded symmetric domains with the same rank.
result Total geodesy of holomorphic isometric embeddings in reducible domains with the same rank.
Study quasi-isometric embeddings in symmetric spaces and Lie groups.
problem Understanding embeddings between symmetric spaces and Lie groups.
method Decompose embeddings into irreducible components and analyze examples.
result Rigidity results extended to semisimple Lie groups, including counterexamples.
Embeds spherically symmetric space-times into 5D flat space, matching Hawking temperature.
problem Finding conditions for embedding spherically symmetric space-times into flat 5D space.
method Explicitly constructs embeddings for any spherically symmetric Lorentzian metric in 3+1 dimensions.
result Hawking temperature matches Unruh temperature for Schwarzschild metric.
We prove a priori bounds for the trace of the second fundamental form of a C4 isometric embedding into Rn+1 of a metric g of non-negative sectional curvature on Sn, in terms of the scalar curvature, and the diameter of g. These estimates give a bound on the extrinsic geometry in terms of intrinsic quanti…
The study proves a localized ellipsoid characterization for convex bodies and applies it to Finsler surfaces.
problem Characterizing convex bodies and their sections by planes.
method Localized ellipsoid characterization applied to Finsler surfaces in normed spaces.
result In certain cases, the intrinsic metric of a Finsler surface imposes restrictions on its extrinsic geometry.
We study isometric actions of tree automorphism groups on the infinite-dimensional hyperbolic spaces. On the one hand, we exhibit a general one-parameter family of such representations and analyse the corresponding equivariant embeddings of the trees, showing that they are convex-cocompact and asymptotically isometric.…
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 C1 isometric embeddings. This statement clearly cannot be true for C2 embeddings in general, due to the classi…
Harmonic maps prove quasi-isometric embeddings are close to unique.
problem Understanding quasi-isometric embeddings between pinched Hadamard manifolds.
method Proving quasi-isometric maps are close to harmonic maps.
result Quasi-isometric maps are within bounded distance from a unique harmonic map.
Long spacelike embeddings can be approximated by isometric ones.
problem Approximating long embeddings to isometric embeddings in Lorentzian spaces.
method Proving approximation by constructing C1 isometric embeddings. result Long spacelike embeddings can be C0-approximated by C1 isometric embeddings. Study holomorphic isometric embeddings of a Grassmannian into quadrics.
problem Holomorphic isometric embeddings of complex Grassmannian into quadrics.
method Generalization of do Carmo--Wallach theory to study moduli space.
result Moduli space of embeddings up to equivalence discussed.
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 paper extends isometric embedding results to null cones and spheres.
problem Isometric embedding of metrics on spheres in null cones.
method Extending Li-Wang's result to compact manifolds, specializing to 2D, developing existence and uniqueness theorems, and proving foliations.
result Existence and uniqueness of isometric embeddings in null cones.
The study examines critical Hölder spaces for isometric embeddings of polar caps.
problem Characterizing critical Hölder spaces for isometric embeddings of polar caps.
method Analyzing the Levi-Civita connection and constructing specific embeddings.
result For $α > rac{1}{2}$, the Levi-Civita connection is induced by the Euclidean connection; for $α < rac{1}{2}$, such embeddings exist.
A Ricci surface is a Riemannian 2-manifold (M,g) whose Gaussian curvature K satisfies KΔK+g(dK,dK)+4K3=0. Every minimal surface isometrically embedded in R3 is a Ricci surface of non-positive curvature. At the end of the 19th century Ricci-Curbastro has proved that conversely, every point x of a Ri…
This paper proposes grid cells encode position via a conformal isometric embedding of 2D physical space.
problem Hexagonal grid firing patterns in grid cells.
method Learning a distance-preserving position embedding in neural space using a recurrent neural network.
result The conformal isometric embedding of 2D physical space into neural space explains hexagonal grid firing patterns.