arXiv research

Turing complete flow on 4-sphere preserves volume.

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

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

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

Study shows decay of correlations on specific types of flows.

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

Study shows diffused interface flows to single diffused balls over time.

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

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 $…

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…

New framework explains normalizing flows' power and limitations.

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…

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

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

Study on stability of mean curvature flow in hyperbolic space.

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…

In this paper, we investigate the volume-prserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space. We prove that the tubeness is preserved along the flow under certain conditions.

Convex hypersurfaces in hyperbolic space evolve to geodesic spheres.

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 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 …

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

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 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…

Flow preserves quermassintegrals, converging to a geodesic sphere.

Study proves a new formula for capillary hypersurfaces and shows a flow converging to a special shape.

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 …

In this article, we continue the work in \cite{GL} and study a normalized hypersurface flow in the more general ambient setting of warped product spaces. This flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. As an application, the i…

The paper studies the stability of volume and area preserving mean curvature flows in Schwarzschild and asymptotic Schwarzschild spaces.

Surface diffusion and mean curvature flows converge to stable critical sets in flat tori.

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

This paper concerns the evolution of a closed hypersurface of dimension $n(\geq 2)$ in the Euclidean space ${\mathbb{R}}^{n+1}$ under a mixed volume preserving flow. The speed equals a power $β(\geq 1)$ of homogeneous, either convex or concave, curvature functions of degree one plus a mixed volume preserving term, incl…

New theorem shows shapes close to balls, flow converges to balls in 2D and 3D.

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

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 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…

We investigate the formation of singularities for surfaces evolving by volume preserving mean curvature flow. For axially symmetric flows - surfaces of revolution - in $\mathbb{R}^3$ with Neumann boundary conditions, we prove that the first developing singularity is of Type I. The result is obtained without any additio…

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

The paper studies curvature measures and volume-preserving flows on convex bodies.

Study shows how curved surfaces evolve smoothly to spherical shapes.

In this paper, we study the complete bounded $λ$-hypersurfaces in weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded $λ$-hypersurfaces with $|A|\leqα$ and get some applications of the volume comparison theorem. Secondly, we consider the relation amo…

In this paper we consider a class of weighted-volume preserving curvature flows acting on hypersurfaces that are trapped within two parallel hyperplanes and satisfy an orthogonal boundary condition. In the author's thesis the stability of cylinders under the flows was considered; it was found that they are stable provi…

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 …

Let N be a (n+1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface. We consider curvature flows in N with different curvature functions F (including the mean curvature, the gauss curvature and the second elementary symmetric polynomial) and a volume preserving term. Under suitable a…

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

Non-vanishing steady Euler flows and Beltrami fields found in high dimensions.

Bi-contact surgery operations can be applied to Anosov flows.

We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed convex hypersurface converges to a round sphere. The proof is based on the mono…

First we investigate the evolutions of the radius function and its gradient along the volume-preserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space under certain condition for the radius function. Next, we prove that the …