Helicity is the sole invariant of volume-preserving transformations.
problem Identifying the sole integral invariant of volume-preserving transformations.
method Proving that any regular integral invariant of volume-preserving transformations is equivalent to the helicity.
result Any functional associated with a well-behaved integral kernel is invariant under volume-preserving diffeomorphisms if and only if it is a function of the helicity.
We study the long time behavior of the volume preserving p-flow in Rn+1 for 1≤p<n−1n+1. By extending Andrews' technique for the flow along the affine normal, we prove that every centrally symmetric solution to the volume preserving p-flow converges sequentially to the unit ball in the $…
Study reveals failure of uniqueness in dynamical invariants for 3D volume-preserving diffeomorphisms.
problem Uniqueness of dynamical invariants for 3D volume-preserving diffeomorphisms.
method Examined failure of uniqueness on integral homology spheres and arbitrary three-manifolds using local and global invariants.
result Failure of uniqueness is severe, with continuous and non-constant invariants appearing in C1-open sets of nonvanishing exact fields of fixed helicity. Algorithm designs neural group actions for symmetric transformations.
problem Designing neural networks for symmetric transformations.
method Develops Neural Group Actions (NGAs) for finite groups, enforcing volume-preserving constraints.
result Demonstrates NGAs for the quaternion group Q8 can learn quantum gate transformations. Study on group cocycles for volume-preserving diffeomorphisms.
problem Understanding group cocycles on volume-preserving diffeomorphisms.
method Constructed two types of group cocycles on the volume-preserving diffeomorphism group.
result One cocycle yields the Euler class of flat sphere bundles for the sphere.
Study proves stability and uniqueness for a specific type of flow.
problem Volume-preserving mean curvature flow stability and uniqueness.
method New gradient flow calibrations for volume preservation, stability estimate in distributional solutions.
result Strong solutions are calibrated and stable under certain conditions.
Sharp 3D Alexandrov inequality applied to volume-preserving flows.
problem Volume-preserving geometric flows in 3D space.
method Sharp quantitative Alexandrov inequality for C2-regular sets. result Established a 3D sharp quantitative version of the Alexandrov inequality.
New invariant 'trunkenness' for 3D volume-preserving vector fields.
problem Characterizing volume-preserving vector fields.
method Constructing and proving properties of trunkenness invariant.
result Trunkenness is independent of helicity and related to knot invariants.
Extends Arnold's linking theory to higher dimensions and submanifolds.
problem Volume-preserving actions in higher dimensions and submanifolds.
method Generalization of V. Arnold's theory to Rk and Rℓ. result Extension of asymptotic linking to higher dimensions and submanifolds.
Volume-preserving neural networks prevent gradient issues.
problem Vanishing and exploding gradients in deep neural networks.
method A new neural network architecture with volume-preserving sublayers.
result Volume-preserving neural networks maintain gradient stability.
Dual pairs constructed for volume preserving diffeomorphisms using symplectic geometry.
problem Understanding the group of volume preserving diffeomorphisms through symplectic geometry.
method Using cotangent bundles of spaces of smooth embeddings, symplectic reduction, and nonlinear Grassmannians of augmented submanifolds.
result Descriptions of coadjoint orbits of the group of volume preserving diffeomorphisms in terms of submanifolds of augmented spaces.
Study shows maps preserving volume on certain complex spaces are unique.
problem Rigidity of volume-preserving maps on Hermitian symmetric spaces.
method Established rigidity for local holomorphic volume-preserving maps.
result Local holomorphic volume-preserving maps are rigid on Hermitian symmetric spaces.
Study a volume preserving flow using symmetric polynomials without curvature pinching assumptions.
problem Volume preserving flow of convex hypersurfaces without curvature pinching constraints.
method Power of the k-th elementary symmetric polynomial in principal curvatures.
result Solution exists for all times and converges to a round sphere in the volume preserving scalar curvature flow case.
Study shows decay of correlations on specific types of flows.
problem Analyzing decay of correlations in specific flow types.
method Asymptotic expansion of correlation function on Abelian covers.
result Established an expansion in inverse powers of time.
New invariants defined for volume-preserving flows on 3-manifolds.
problem Defining invariants for volume-preserving flows.
method Extending wrapping number and trunk to define invariants of links and flows.
result Wrappingness and trunkenness are not functions of helicity.
We show that every volume preserving codimension one Anosov flow on a closed Riemannian manifold of dimension greater than three admits a global cross section and is therefore topologically conjugate to a suspension of a linear toral automorphism. This proves a conjecture of Verjovsky from the 1970's in the volume pres…
Study shows diffused interface flows to single diffused balls over time.
problem Volume-preserving mean curvature flow in Euclidean space.
method Diffused interface version, exponential convergence proof.
result Exponential convergence to single diffused balls.
Real NVP models improve image modeling with exact computations.
problem Unsupervised learning of probabilistic models.
method Real-valued non-volume preserving transformations.
result Exact log-likelihood computation, sampling, inference, interpretable latent space.
We study the phase field method for the volume preserving mean curvature flow. Given an initial C1 hypersurface we proved the existence of the weak solution for the volume preserving mean curvature flow via the reaction diffusion equation with a nonlocal term. We also show the monotonicity formula and the density up…
Novel weak solutions for volume-preserving mean curvature flow established.
problem Existence and uniqueness of solutions to volume-preserving mean curvature flow.
method Introducing varifold solutions coupled with phase volumes and new calibrations.
result Uniqueness of classical solutions among varifold solutions.
Turing complete flow on 4-sphere preserves volume.
problem Creating a Turing complete flow on a 4-sphere.
method Smooth, conservative flow on the 4-sphere.
result Achieved a Turing complete, volume-preserving flow.
We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the m-th mean curvature plus a volume preserving term, including the case of powers of the mean curvature or of the Gauss curvature. We prove that if the initial hypersurface satisfies a …
For a germ of a smooth map f and a subgroup G_V of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form V in the source or in the target we study the G_V-moduli space of f that parameterizes the G_V-orbits inside the G-orbit of f. We find, for example, that this modu…
The Willmore flow preserves surface volume, leading to convergence to a sphere.
problem Long-term behavior of volume-preserving Willmore flow on surfaces.
method Volume-preserving Willmore flow, blow-up analysis, constrained Lojasiewicz-Simon inequality.
result Smooth solutions exist for spherical surfaces with Willmore energy below 8π and converge to a sphere.
A new flow-based method improves VAE flexibility.
problem Limited flexibility of VAE latent distributions.
method Householder flow for more flexible variational posterior.
result More flexible variational posterior leads to better results.
PPA models data directions as curves for manifold learning.
problem Nonlinear data structure in manifold learning.
method Principal Polynomial Analysis (PPA) models data directions as curves.
result PPA reduces to simple univariate regressions, making it computationally feasible and robust.
New framework explains normalizing flows' power and limitations.
problem Understanding the expressive power and limitations of normalizing flows.
method Theoretical framework for well-conditioned coupling-based normalizing flows and volume-preserving flows.
result RealNVP is distributionally universal, but volume-preserving flows are not.
Estimates for VPMCF show ancient MCF solutions and finite-time behavior.
problem Volume Preserving Mean Curvature Flow (VPMCF) behavior and singularities.
method Nonlocal estimates and blowup analysis.
result Ancient solutions to MCF and finite-time behavior of VPMCF.
We prove that any real-analytic, volume-preserving action of a lattice Γ in a simple Lie group with $\Qrank(Γ)\geq 7$ on a closed 4-manifold of nonzero Euler characteristic factors through a finite group action.
Characterizes 3D steady Euler flows using homologies.
problem Characterizing 3D steady Euler flows.
method Using commuting zero-flux homologies.
result Steady Euler flows cannot be constructed using plugs.
Study on geodesics in the space of volume-preserving immersions.
problem Geodesics in the space of volume-preserving immersions.
method Analysis of Levi-Civita connection and geodesic equations for Sobolev metrics.
result Local well-posedness of geodesic equations in many cases.
Maps between certain Lipschitz manifolds are isometries if they preserve volume.
problem Volume preservation and isometry conditions for Lipschitz manifolds.
method Volume-preserving 1-Lipschitz maps from integral currents onto infinitesimally Euclidean Lipschitz manifolds.
result Volume-preserving maps are isometries under given conditions.
We provide a variational description of any Liouville (i.e. volume preserving) autonomous vector fields on a smooth manifold. This is obtained via a ``maximal degree'' variational principle; critical sections for this are integral manifolds for the Liouville vector field. We work in coordinates and provide explicit for…
Convex hypersurfaces in hyperbolic space evolve to geodesic spheres.
problem Volume preserving Gauss curvature flow of convex hypersurfaces in hyperbolic space.
method Volume preserving flow with speed given by Gauss curvature power α, using Alexandrov reflection and hyperbolic curvature measures.
result Smooth solution remains convex and converges to a geodesic sphere exponentially.
Classifies stable hypersurfaces in real projective spaces and confirms the isoperimetric conjecture.
problem Volume preserving stability and isoperimetric problem in real projective spaces.
method Classification of stable hypersurfaces and analysis of antipodal invariant hypersurfaces.
result Solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces.
In this paper, we introduce a definition of λ-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that λ-hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete λ-hypersurfaces with …
Study the exponential map on surfaces using fluid dynamics.
problem Exponential map of volume-preserving diffeomorphisms on closed surfaces.
method Fluid dynamical proof of Ebin--Misiołek--Preston theorem and extension of Shnirelman's rigidity result.
result Exponential map is a nonlinear Fredholm mapping of index zero and Fredholm quasiregular.
We prove that a compactly supported homeomorphism of a smooth manifold of dimension greater or equal to 5 can be approximated uniformly by compactly supported diffeomorphisms if and only if it is isotopic to a diffeomorphism. If the given homeomorphism is in addition volume preserving, then it can be approximated unifo…
New method improves Variational Auto-Encoders using convex combination of Inverse Autoregressive Flows.
problem Improving Variational Auto-Encoders (VAEs) for better performance.
method Introducing multiple lower-triangular matrices with ones on the diagonal and combining them using a convex combination to enrich a linear Inverse Autoregressive Flow.
result The proposed method outperforms other volume-preserving flows and is competitive with state-of-the-art linear normalizing flows.
Let (M,g) be a complete 3-dimensional asymptotically flat manifold with everywhere positive scalar curvature. We prove that, given a compact subset K of M, all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid K. This complements the work of G. Huisken and S.-T. Yau and J. …
We consider a connected smooth n-dimensional manifold M endowed with a volume form Ω, and we show that an open subset U of Rn of Lebesgue measure $\Vol (U)$ embeds into M by a smooth volume preserving embedding whenever the volume condition $\Vol (U) \le \Vol (M,Ω)$ is met.
Study on stability of mean curvature flow in hyperbolic space.
problem Stability of volume preserving mean curvature flow in hyperbolic space.
method Analysis of initial conditions and flow behavior in hyperbolic space.
result The flow converges exponentially to an umbilical sphere under certain conditions.
Flow preserves volume on flat torus, converging to stable set.
problem Volume preservation in discrete mean curvature flow on flat torus.
method Discrete mean curvature flow, quantitative Alexandrov estimate, characterization in 2D.
result Flow converges exponentially fast to stable set.
This work replaces pixel-wise reconstruction in VAEs with real NVP transformations.
problem Shortcomings of pixel-wise reconstruction in VAEs.
method Used real-valued non-volume preserving transformations (real NVP) for conditional likelihood computation.
result A simple VAE with real NVP is competitive with complex VAE structures on image modeling tasks.
We study the provenance of singularity formation under mean curvature flow and volume preserving mean curvature flow in an axially symmetric setting. We prove that if the mean curvature is uniformly bounded on any finite time interval, then no singularities can develop during that time under both mean curvature flow an…
Study on λ-hypersurfaces in weighted flow, focusing on volume and radius estimates.
problem Volume and radius estimates of λ-hypersurfaces in weighted flow. method Volume comparison theorem and radius estimates analysis.
result Estimates for intrinsic diameter and extrinsic radius of λ-hypersurfaces. Smoothly conjugate Anosov flows on 3D manifolds are actually smoothly conjugate.
problem Smoothly conjugate 3D Anosov flows are not always smoothly conjugate.
method Proved smooth rigidity for volume preserving Anosov flows on 3-manifolds.
result Smooth conjugacy implies smooth conjugacy for volume preserving Anosov flows.
The paper examines convergence of currents and forms under smooth diffeomorphisms.
problem Analyzing convergence of currents and forms under C0-limits of diffeomorphisms. method Geometric analysis, measure theory, homotopy theory.
result Pushforwards of rectifiable currents converge in the flat norm.