arXiv research

Study on group cocycles for volume-preserving diffeomorphisms.

Study proves stability and uniqueness for a specific type of flow.

Sharp 3D Alexandrov inequality applied to volume-preserving flows.

Extends Arnold's linking theory to higher dimensions and submanifolds.

Dual pairs constructed for volume preserving diffeomorphisms using symplectic geometry.

Study shows decay of correlations on specific types of flows.

New invariants defined for volume-preserving flows on 3-manifolds.

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.

We study the phase field method for the volume preserving mean curvature flow. Given an initial $C^1$ 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…

We study the long time behavior of the volume preserving $p$-flow in $\mathbb{R}^{n+1}$ for $1\leq p<\frac{n+1}{n-1}$. 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.

Novel weak solutions for volume-preserving mean curvature flow established.

We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional $\mathcal I$ defined on exact divergence-free vector fields of class $C^1$ on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that $\mat…

Turing complete flow on 4-sphere preserves volume.

In this paper, we establish the rigidity result for local holomorphic volume preserving maps from an irreducible Hermitian manifold of compact type into its Cartesian products.

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…

We propose a novel approach to addressing the vanishing (or exploding) gradient problem in deep neural networks. We construct a new architecture for deep neural networks where all layers (except the output layer) of the network are a combination of rotation, permutation, diagonal, and activation sublayers which are all…

The Willmore flow preserves surface volume, leading to convergence to a sphere.

New framework explains normalizing flows' power and limitations.

Estimates for VPMCF show ancient MCF solutions and finite-time behavior.

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.

Maps between certain Lipschitz manifolds are isometries if they preserve volume.

We study a volume preserving curvature flow of convex hypersurfaces, driven by a power of the $k$-th elementary symmetric polynomial in the principal curvatures. Unlike most of the previous works on related problems, we do not require assumptions on the curvature pinching of the initial datum. We prove that the solutio…

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.

We construct a new invariant-the trunkenness-for volume-perserving vector fields on S^3 up to volume-preserving diffeomorphism. We prove that the trunkenness is independent from the helicity and that it is the limit of a knot invariant (called the trunk) computed on long pieces of orbits.

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 …

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…

Study the exponential map on surfaces using fluid dynamics.

We consider a connected smooth $n$-dimensional manifold $M$ endowed with a volume form $Ω$, and we show that an open subset $U$ of $R^n$ 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.

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. …

Study on stability of mean curvature flow in hyperbolic space.

Flow preserves volume on flat torus, converging to stable set.

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…

Smoothly conjugate Anosov flows on 3D manifolds are actually smoothly conjugate.

We consider curvature flows in hyperbolic space with a monotone, symmetric, homogeneous of degree 1 curvature function F. Furthermore we assume F to be either concave and inverse concave or convex. For compact initial hypersurfaces, which are strictly convex by horospheres, we show the long time existence of mixed volu…

We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the volume-preserving diffeomorphism group of a three-dimensional manifold and study its properties. Despite the fact that the space $\mathcal{D}_μ(M^3)$ is infinite-dimensional, we succeed in defining the signature of the bi-invariant quadr…

Let G be a connected semisimple Lie group without compact factors whose real rank is at least 2, and let Γ\subset G be an irreducible lattice. We provide a C^\infty classification for volume-preserving Cartan actions of Γand G. Also, if G has real rank at least 3, we provide a C^\infty classification for volume-preserv…

New model improves MCMC efficiency and multi-modal distribution exploration.

Metrics are isometric for certain Anosov magnetic systems.

Study shows smooth convergence of round surfaces in flat space-time models.

The paper studies how convex hypersurfaces in hyperbolic space evolve under a specific curvature flow.

We study the convergence of an axially symmetric hypersurface evolving by volume preserving mean curvature flow. Assuming the surface is not pinching off along the axis at any time during the flow, and without any additional conditions, as for example on the curvature, we prove that it converges to a hemisphere, when t…

We prove: "If $M$ is a compact hypersurface of the hyperbolic space, convex by horospheres and evolving by the volume preserving mean curvature flow, then it flows for all time, convexity by horospheres is preserved and the flow converges, exponentially, to a geodesic sphere". In addition, we show that the same conclus…

We determine the Riemannian manifolds for which the group of exact volume preserving diffeomorphisms is a totally geodesic subgroup of the group of volume preserving diffeomorphisms, considering right invariant $L^2$-metrics. The same is done for the subgroup of Hamiltonian diffeomorphisms as a subgroup of the group of…

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a $3$-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan's homological characterization of geodesible flows in the volume-preserving case. As an application, we…