arXiv research

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…

The paper studies constant mean curvature hypersurfaces in Finsler manifolds.

New definition of stable $(r+1)$-th capillary hypersurfaces proposed.

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 derive a formula for the first variation of horizontal perimeter measure for $C^2$ hypersurfaces of completely general sub-Riemannian manifolds, allowing for the existence of characteristic points. For $C^2$ hypersurfaces in vertically rigid sub-Riemannian manifolds we also produce a second variation formula for var…

In this paper, we propose a new volume-preserving flow and show that it performs similarly to the linear general normalizing flow. The idea is to enrich a linear Inverse Autoregressive Flow by introducing multiple lower-triangular matrices with ones on the diagonal and combining them using a convex combination. In the …

Efficient algorithm for clustering and classification using MBO scheme.

Alternative proof of weak solutions to mean curvature flow using minimizing movements.

Study on group cocycles for volume-preserving diffeomorphisms.

The study finds instability conditions for specific surfaces in sub-Riemannian 3-space forms.

We use variational arguments to introduce a notion of mean curvature for surfaces in the Heisenberg group H^1 endowed with its Carnot-Carathéodory distance. By analyzing the first variation of area, we characterize C^2 stationary surfaces for the area as those with mean curvature zero (or constant if a volume-preservin…

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

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

The main result is the identification of the orthogonal complement of the subalgebra of conformal vector field inside the algebra of all vector fields of a compact flat 2-manifold. As a fundamental tool, the complete Hodge decomposition for manifold with boundary is used. The identification allows the derivation of gov…

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

Volume-preserving neural networks prevent gradient issues.

Variational auto-encoders (VAE) are scalable and powerful generative models. However, the choice of the variational posterior determines tractability and flexibility of the VAE. Commonly, latent variables are modeled using the normal distribution with a diagonal covariance matrix. This results in computational efficien…

Dual pairs constructed for volume preserving diffeomorphisms using symplectic geometry.

Study a volume preserving flow using symmetric polynomials without curvature pinching assumptions.

Study shows decay of correlations on specific types of flows.

Since $n$-dimensional $λ$-hypersurfaces in the Euclidean space $\mathbb {R}^{n+1}$ are critical points of the weighted area functional for the weighted volume-preserving variations, in this paper, we study the rigidity properties of complete $λ$-hypersurfaces. We give a gap theorem of complete $λ$-hypersurfaces with po…

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.

Variational autoencoders (VAEs), that are built upon deep neural networks have emerged as popular generative models in computer vision. Most of the work towards improving variational autoencoders has focused mainly on making the approximations to the posterior flexible and accurate, leading to tremendous progress. Howe…

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…

Novel discretization of Euler equations for incompressible fluids.

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

Let $M$ be smooth $n$-dimensional manifold, fibered over a $k$-dimensional submanifold $B$ as $π:M \to B$, and $\vartheta \in Λ^k (M)$; one can consider the functional on sections $φ$ of the bundle $π$ defined by $\int_D φ^* (\vartheta)$, with $D$ a domain in $B$. We show that for $k = n-2$ the variational principle ba…

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.

SGNs use Hamiltonian mechanics for invertible deep generative modeling.

Characterizes 3D steady Euler flows using homologies.

Minimal Gaussian surface area is achieved by cones over a regular simplex for $m>3$ sets partitioning $R^n$.

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

The paper examines stable CMC hypersurfaces with boundaries on parallel hyperplanes.

Study on surfaces in Heisenberg group with constant mean curvature.

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.