arXiv research

Optimizes shapes in uncertain Navier-Stokes flow problems.

Flow turns star-shaped curves into circles.

Paper proves conjecture about star-shaped curves evolving under GAPF, but not always preserves star shape.

Study fourth-order geometric flow of shape operator for co-dimension one immersions.

In 1998 Smoczyk [Smo98] showed that, among others, the blowup limits at singularities are convex for the mean curvature flow starting from a closed star-shaped surface in $\mathbf{R}^3$. We prove in this paper that this is true for the mean curvature flow of star-shaped hypersurfaces in $\mathbf{R}^{n+1}$ in arbitrary …

Inverse curvature flows shape star-shaped hypersurfaces into spheres.

In this paper, we show that the inverse anisotropic mean curvature flow in $\mathbb{R}^{n+1}$, initiating from a star-shaped, strictly $F$-mean convex hypersurface, exists for all time and after rescaling the flow converges exponentially fast to a rescaled Wulff shape in the $C^\infty$ topology. As an application, we p…

Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.

New flow expands hypersurfaces in hyperbolic space, showing round limiting shape for certain powers.

Study anisotropic flow for capillary hypersurfaces, proving new inequalities.

Paper introduces a method to generate stable shapes using Grassmann manifolds.

Proves higher regularity for anisotropic inverse mean curvature flow.

Study examines preservation of curvature-adaptedness during mean curvature flow.

Study on anisotropic curvature flow for noncompact convex hypersurfaces.

Gradient flow expands curves to round shapes.

The paper classifies shapes of translating solitons for a specific flow.

In this paper we complete the study started in [Pi2] of evolution by inverse mean curvature flow of star-shaped hypersurface in non-compact rank one symmetric spaces. We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the quaternionic hyperbolic space. We p…

The paper studies a curve flow preserving anisotropic length for convex curves, leading to a homothetic limit.

In this paper we introduce a Guan-Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan-Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, pro…

Identifies most probable flows for Kunita SDEs in fluid dynamics.

The paper studies the free elastic flow of closed curves and finds their asymptotic shape converges to a circle.

The paper introduces a new flow to converge to a Wulff shape from a smooth convex hypersurface.

Proves a Minkowski inequality for star-shaped hypersurfaces in warped cylinders.

In this paper we consider a star-shaped hypersurface flow by mean curvature. Without any assumption on the convexity, we give a new proof of gradient estimate for a short time. As an application, we also give a lower bound for the blowing up time.

We study the relation between the centro-affine geometry of star-shaped planar curves and the projective geometry of parametrized maps into $\RP^1$. We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution …

Microfluidic devices are utilized to control and direct flow behavior in a wide variety of applications, particularly in medical diagnostics. A particularly popular form of microfluidics -- called inertial microfluidic flow sculpting -- involves placing a sequence of pillars to controllably deform an initial flow field…

Ancient solutions to mean curvature flow have unique shapes.

We study the curvature flow of planar nonconvex lens-shaped domains, considered as special symmetric networks with two triple junctions. We show that the evolving domain becomes convex in finite time; then it shrinks homothetically to a point. Our theorem is the analog of the result of Grayson for curvature flow of clo…

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

The Hodge star mean curvature flow on a 3-dimension Riemannian or pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold, the Schrodingier flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural non-linear dispersive curve flows in geometric analysis. A curve flo…

We derive an upper bound on the waiting time for a variational weak solution to Inverse Mean Curvature Flow in $\mathbb{R}^{n+1}$ to become star-shaped. As a consequence, we demonstrate that any connected surface moving by the flow which is not initially a topological sphere develops a singularity or self-intersection …

Optimal thresholds ensure curves remain embedded in flows.

We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the complex hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to…

Small bubbles sliding on a boundary maintain half-spherical shape.

We consider convex symmetric lens-shaped networks in R^2 that evolve under curve shortening flow. We show that the enclosed convex domain shrinks to a point in finite time. Furthermore, after appropriate rescaling the evolving networks converge to a self-similarly shrinking network, which we prove to be unique in an ap…

In shape analysis, the concept of shape spaces has always been vague, requiring a case-by-case approach for every new type of shape. In this paper, we give a general definition for an abstract space of shapes in a manifold. This notion encompasses every shape space studied so far in the literature, and offers a rigorou…

We consider a compact, star-shaped, mean convex hypersurface $Σ^2\subset \mathbb{R}^3$. We prove that in some cases the flow exists until it shrinks to a point in a spherical manner, which is very typical for convex surfaces as well (see \cite{An1}). We also prove that in the case we have a surface of revolution which …

The study examines evolving star-shaped hypersurfaces in hyperbolic spaces, influenced by ambient geometry.

Proves unknottedness of certain 3D shapes with multiple ends.

In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface $Σ$ is strictly mean convex and star-shaped, then the flow hypersurface $Σ_t$ converges to a large coordinate sphere as $t\rightarrow \infty$ exponentially. We also describe an a…

Study shows how weak inverse anisotropic mean curvature flow behaves at infinity.

The paper studies curvature flows of star-shaped hypersurfaces and proves convergence to spheres.

The paper classifies shapes of translating solitons from isoparametric graphs.

We define a new version of modified mean curvature flow (MMCF) in hyperbolic space $\mathbb{H}^{n+1}$, which interestingly turns out to be the natural negative $L^2$-gradient flow of the energy functional defined by De Silva and Spruck in \cite{DS09}. We show the existence, uniqueness and convergence of the MMCF of com…

How and why stock prices move is a centuries-old question still not answered conclusively. More recently, attention shifted to higher frequencies, where trades are processed piecewise across different timescales. Here we reveal that price impact has a universal non-linear shape for trades aggregated on any intra-day sc…

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

AMF-VI uses adaptive mixtures of flows for robust VI across diverse distributions.

Smooth convergence shown for curve diffusion flows.