New tensors capture intrinsic embedding data of conformal hypersurfaces.
problem Classifying hypersurface invariants in conformal manifolds.
method Constructing curvatures and conformal fundamental forms.
result Finite family of tensors captures extrinsic embedding data.
Study proves higher-order conformal forms don't exist in odd dimensions.
problem Proving non-existence of higher-order conformal forms in odd dimensions.
method Analyzing conformal hypersurface embeddings and differential order invariants.
result General non-existence of higher-order conformal forms in odd dimensions.
New embeddings for manifolds using heat kernels.
problem Constructing canonical conformal embeddings for manifolds.
method Employing heat kernel embedding from Bérard-Besson-Gallot'94 to find canonical conformal embeddings.
result Intrinsic construction of canonical conformal embeddings with dimensions growing exponentially with t. Paper proves embedding theorem for conformally compact manifolds.
problem Embedding conformally compact manifolds into hyperbolic spaces.
method Proves analogous Nash Embedding Theorem for conformally compact manifolds.
result Conformally compact manifolds can be isometrically embedded into hyperbolic spaces.
It is shown that the space of null geodesics of a causally simple Lorentzian manifold is Hausdorff if it admits an open conformal embedding into a globally hyperbolic spacetime. This provides an obstruction to conformal embeddings of causally simple spacetimes into globally hyperbolic ones irrespective of curvature con…
We obtain universal models for several types of locally conformal symplectic manifolds via pullback or reduction. The relation with recent embedding results for locally conformal Kähler manifolds is discussed.
Embedding theorem for tractor bundles applied to conformal geometry.
problem Embedding theorem for tractor bundles in Cartan geometries.
method Extension of Gromov-Zimmer embedding theorem to tractor bundles.
result Rigidity result for conformal actions of special pseudo-unitary groups.
COLoKe adapts Koopman embeddings online, reducing overfitting and improving long-term predictions.
problem Online adaptation of Koopman embeddings to avoid overfitting and maintain long-term predictive accuracy.
method Combines deep feature learning with multistep prediction consistency in a lifted space, using a conformal-style mechanism for selective updates.
result Empirically effective in reducing overfitting and maintaining long-term predictive accuracy.
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.
New tensors help determine if metrics are related to Poincaré-Einstein ones.
problem Determining when metrics on conformally compact manifolds are related to Poincaré-Einstein metrics.
method Developed new tensors and used conformal tractor calculus to analyze metrics.
result The vanishing of these new tensors is a necessary and sufficient condition for a metric to be related to a Poincaré-Einstein metric.
We prove that if two conformal embeddings between Riemann surfaces with finite topology are homotopic, then they are isotopic through conformal embeddings. Furthermore, we show that the space of all conformal embeddings in a given homotopy class deformation retracts into a point, a circle, a torus, or the unit tangent …
The study characterizes geometries of hypersurfaces in warped product and conformal manifolds.
problem Characterizing the geometry of hypersurfaces in warped product and conformal manifolds.
method Using higher fundamental forms and conformal metrics, the study characterizes the geometries of hypersurfaces in warped product and conformal manifolds.
result Higher conformal fundamental forms play a critical role in the characterization of the geometry of hypersurfaces in conformal manifolds.
New operators and curvatures derived from embedded manifolds.
problem Finding obstructions and coupling extrinsic operators.
method Explicit computation of extrinsic Paneitz operator and its applications.
result New extrinsically-coupled fourth and sixth order operators.
The paper simplifies FLRW photon propagators using geometric embeddings.
problem Understanding Friedmann-Lemaître-Robertson-Walker (FLRW) spaces.
method Differential-geometric methods applied to FLRW spaces as submanifolds in \(\mathbb{R}^{n+2}\).
result New and simplified expressions for the photon propagator in four dimensions.
Every nonflat conformal minimal surface is homotopic to a proper one.
problem Proving homotopy of nonflat conformal minimal surfaces to proper ones.
method Analyzing immersions and fluxes of Riemann surfaces into \(\mathbb{R}^n\) and \(\mathbb{C}^n\).
result Every nonflat conformal minimal immersion is homotopic to a proper one.
In this paper, we consider the problem of building a conformal boundary, embedding a pseudo-Riamnnian manifold as an open subset of a bigger one. We get first results about conformal maximality. We also show that in dimension ≥3, there are rigidity properties for the topological boundary of such a conformal embed…
We develop a new approach to the conformal geometry of embedded hypersurfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner--Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first resu…
We construct isometric and conformally isometric embeddings of some gravitational instantons in R8 and R7. In particular we show that the embedding class of the Einstein--Maxwell instanton due to Burns is equal to 3. For CP2, Eguchi--Hanson and anti-self-dual Taub-NUT we obtain upp…
In this paper, we prove that every confomal minimal immersion of an open Riemann surface into Rn for n≥5 can be approximated uniformly on compacts by conformal minimal embeddings. Furthermore, we show that every open Riemann surface carries a proper conformal minimal embedding into R5. One …
A spacetime can be embedded in an enveloping space with all its extensions.
problem Existence and uniqueness of C0-maximal extensions in globally hyperbolic conformally flat spacetimes.
method Proving conformal embedding into an enveloping space containing all extensions.
result Existence and uniqueness of C0-maximal extensions proven.
Normalizing flows can now estimate densities on unknown manifolds.
problem Normalizing flows struggle with data on unknown low-dimensional manifolds.
method Conformal Embedding Flows, which combine standard flows with trainable conformal embeddings.
result Tractable density estimation on manifold-supported data is possible.
Embeds Lorentzian manifolds in \(\mathbb{R}^{n+2}\) with SO(2,n) compatibility.
problem Embedding Lorentzian manifolds in \(\mathbb{R}^{n+2}\) with specific metric properties.
method Embedding using SO(2,n) compatible metrics.
result Conformal transformations on submanifolds inherited from ambient space.
The standard conformal compactification of Euclidean space is the round sphere. We use conformal geodesics to give an elementary proof that this is the only possible conformal compactification.
For an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained holographically in a construction that uses a singular Yamabe problem and a corresponding minimal hypersurface with boundary. They include an ext…
The invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold: For a given conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem. Using existen…
We extend Garsia's conjecture about surface embeddings.
problem Realizing conformal classes of genus-1 surfaces via embeddings.
method Formulated within the framework of connections on principal bundles, solutions provided in several cases.
result Novel parameterizations of the moduli space of conformal classes of compact surfaces of genus 1.
Generalizes holographic method to higher codimension submanifolds.
problem Extract higher-order local invariants of embeddings.
method Natural generalization of holographic method to higher codimension submanifolds.
result New invariants obstructing the order-by-order construction of unit defining maps.
Improved neural speaker embeddings enhance ASR performance.
problem Few studies have explored neural speaker embeddings for ASR.
method Integrating improved neural speaker embeddings into a conformer-based hybrid HMM ASR system.
result Improved neural embeddings achieve on-par performance with i-vectors.
The conformal Laplacian's algebraic structure is explored in 2D, revealing a central charge.
problem Exploring the algebraic structure of the conformal Laplacian in 2D.
method Using prefactorization algebras and Green functions.
result In 2D, the conformal Laplacian's algebraic structure is revealed through a central charge.
Smoothly approximates embeddings in Lorentzian manifolds.
problem Approximating embeddings in Lorentzian manifolds.
method C^0 approximation of embeddings.
result Approximated embeddings can be made smooth.
The study proves properties of metrics and their conformal classes on specific manifolds.
problem Understanding metrics and their conformal classes on certain manifolds.
method Combining Simons' gap theorem with minimal isometric embeddings and coherent embeddings of standard Einstein metrics.
result The conformal classes of the product metrics and projective spaces realize the sigma invariant uniquely.
Non-linear Hopf manifolds can be embedded into linear ones and admit LCK metrics.
problem Understanding non-linear Hopf manifolds and their properties.
method Holomorphic embeddings and LCK metrics.
result Non-linear Hopf manifolds admit LCK metrics.
We find necessary and sufficient conditions for existence of a locally isometric embedding of a vacuum space-time into a conformally-flat 5-space. We explicitly construct such embeddings for any spherically symmetric Lorentzian metric in 3+1 dimensions as a hypersurface in R4,1. For the Schwarzschild metric the…
Study properties of hypersurfaces in spacetimes with conformal transformations.
problem Properties of embedded hypersurfaces in spacetimes with a preferred spatial direction.
method Analysis of hypersurfaces with conformal transformations, scalar curvature conditions, and Riemannian manifold properties.
result Hypersurfaces are either Einstein or have vanishing twist, and under certain conditions, they are isomorphic to the 3-sphere.
Renormalized volume invariant for knots in 3-sphere computed.
problem Computing renormalized volume for knot embeddings in 3-sphere.
method Renormalizing volume associated to singular Yamabe metric.
result Renormalized volume is a global conformal invariant for knots in 3-sphere.
The study finds invariants of smooth metrics on surfaces through embeddings into spheres.
problem Understanding invariants of smooth metrics on surfaces through embeddings into spheres.
method Defining the Willmore functional over Nash isometric embeddings and analyzing its infimum.
result The study identifies unique conformal classes of metrics with specific invariants and lower bounds.
The paper proves properties of minimal isometric embeddings and conformal deformations of Riemannian surfaces.
problem Minimal isometric embeddings and conformal deformations of Riemannian surfaces.
method Analyzes minimal isometric embeddings and conformal deformations of Riemannian surfaces.
result Minimal isometric embeddings and conformal deformations of Riemannian surfaces have specific properties.
A conformal structure on a manifold Mn induces natural second order conformally invariant operators, called Möbius and Laplace structures, acting on specific weight bundles of M, provided that n≥3. By extending the notions of Möbius and Laplace structures to the case of surfaces and curves, we develop here th…
DANCE improves prediction set efficiency for deep learning models.
problem Inefficient, overly conservative prediction sets for pre-trained models.
method DANCE combines adaptive kernel regression and nearest-neighbor approach.
result DANCE produces more efficient and robust prediction sets.
We study the non-embddability property for a class of real hypersurfaces, called real hypersurfaces of involution type, into the sphere in the low codimensional case, by making use of property of a naturally related Gauss curvature. We also study rigidity problems for conformal maps between a class of Kähler manifolds …
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.
In this work we present new fundamental tools for studying the variations of the Willmore functional of immersed surfaces into Rm. This approach gives for instance a new proof of the existence of a Willmore minimizing embedding of an arbitrary closed surface in arbitrary codimension. We explain how the same approach…
Study of metrics on spheres and their complex structure properties.
problem Identifying metrics on spheres and their complex structure properties.
method Identify metrics via Nash isometric embeddings, use isotopic extension theorem, and analyze extrinsic quantities.
result No sphere of dimensions 6 or higher can be diffeomorphic to a complex manifold.
Identifies metrics on manifolds and their embeddings into spheres, characterizing constant curvature metrics.
problem Characterizing metrics on manifolds and their embeddings into spheres.
method Identifies metrics on manifolds and their embeddings into spheres, characterizes metrics of constant scalar curvature, and uses Yamabe metrics and almost Hermitian structures.
result Characterizes metrics of constant scalar curvature by properties of extrinsic quantities of their embeddings.
In this paper we prove that every bordered Riemann surface M admits a complete proper null holomorphic embedding into a ball of the complex Euclidean 3-space C3. The real part of such an embedding is a complete conformal minimal immersion M→R3 with bounded image. For any such M we also co…
Harmonic maps intersect all minimal surfaces with bounded curvature.
problem Intersection of harmonic maps with minimal surfaces.
method Nonconstant conformal harmonic maps intersecting bounded curvature minimal surfaces.
result Harmonic maps intersect every nonflat properly embedded minimal surface of bounded curvature.
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…
In this paper we prove that a complete, embedded minimal surface M in R3 with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface M with boundary punctured in a finite number of interior points and that M can be represented in terms of meromorphic …