arXiv research

Huisken studied asymptotic behavior of a mean curvature flow in a Euclidean space when it develops a singularity of type I, and proved that its rescaled flow converges to a self-shrinker in the Euclidean space. In this paper, we generalize this result for a Ricci-mean curvature flow moving along a Ricci flow constructe…

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

We give two new proofs of Perelman's theorem that shrinking breathers of Ricci flow on closed manifolds are gradient Ricci solitons, using the fact that the singularity models of type I solutions are shrinking gradient Ricci solitons and the fact that non-collapsed type I ancient solutions have rescaled limits being sh…

We show that a rescale limit at any degenerate singularity of Ricci flow in dimension 3 is a steady gradient soliton. In particular, we give a geometric description of type I and type II singularities.

Study steady gradient Ricci solitons with cylindrical tangent flows at infinity.

We prove gradient estimates for hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1},$ expanding by negative powers of a certain class of homogeneous curvature functions. We obtain optimal gradient estimates for hypersurfaces evolving by certain powers $p>1$ of $F^{-1}$ and smooth convergence of the properly rescale…

Riemannian gradient descent escapes some spurious critical points on low-rank matrix manifold.

In this short notes, we discuss monotonicity formulas under various rescaled versions of Ricci flow. The main result is Theorem \ref{theo rescaled}.

The $H^1(ds)$-gradient flow shrinks circles with radius $r_0$ to a point.

Study of Bach flow on specific nilmanifolds, converging to a soliton.

Study proves existence and uniqueness of ancient flows from cones.

Rescaling expansiveness proven for k*-expansive vector fields.

In this paper we study the Teichmüller harmonic map flow as introduced by Rupflin and Topping [15]. It evolves pairs of maps and metrics $(u,g)$ into branched minimal immersions, or equivalently into weakly conformal harmonic maps, where $u$ maps from a fixed closed surface $M$ with metric $g$ to a general target manif…

Functional central limit theorem for kernel gradient flow and infinitesimal gradient boosting

Study shows uniform decay rate for singular mean curvature flows.

Study of null mean curvature flow on de Sitter lightcone, related to 2d-Ricci flow.

Proves convergence of mean curvature flow on cylinders with unique continuation.

Study shows unique tangent flow for Lagrangian surfaces with bounded mean curvature.

Quantitative estimate for curvature in mean curvature flow.

Study of tori of revolution under Willmore flow converges to Clifford Torus.

We show, for mean curvature flows in Euclidean space, that if one of the tangent flows at a given space-time point consists of a closed, multiplicity-one, smoothly embedded self-similar shrinker, then it is the unique tangent flow at that point. That is the limit of the parabolic rescalings does not depend on the chose…

In this thesis, we consider the knot energy "integral Menger curvature" which is the triple integral over the inverse of the classic circumradius of three distinct points on the given knot to the power $p\in [2,\infty)$. We prove the existence of the first variation for a subset of a certain fractional Sobolev space if…

The study examines gradient Ricci solitons with nonnegative curvature, proving properties of their blow-downs.

Geometric flows study nearly parallel G2-structures on 3-Sasakian 7-manifolds.

Sharp convergence rate for curvature stability in planar free elastic flow.

Study mean curvature flow of high codimension submanifolds in complex projective space.

We consider in this work a system of two stochastic differential equations named the perturbed compositional gradient flow. By introducing a separation of fast and slow scales of the two equations, we show that the limit of the slow motion is given by an averaged ordinary differential equation. We then demonstrate that…

Estimates the rate of convergence of mean curvature flow solutions.

Study high codimension mean curvature flow in Riemannian manifolds, proving limiting flow in Euclidean space.

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…

Method determines latent dimensionality in international trade flows.

The study examines 4D steady gradient Ricci solitons with nonnegative curvature away from a compact set.

Paper proves stability of quermassintegral inequalities using inverse curvature flow.

Paper constructs flows converging to cones and foliations.

We consider some elementary aspects of the geometry of the space of probability measures endowed with Wasserstein distance. In such a setting, we discuss the various terms entering Perelman's shrinker entropy, and characterize two new monotonic functionals for the volume-normalized Ricci flow. One is obtained by a resc…

The paper extends Ricci flow theory with Type-I scalar curvature bounds, proving entropy convergence and characterizing singular sets.

J.J.L. Vel$\acute{a}$zquez in 1994 used the degree theory to show that there is a perturbation of Simons' cone, starting from which the mean curvature flow develops a type $\mathrm{II}$ singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled…

We prove that the Ricci flow g(t) starting at any metric on the euclidean space that is invariant by a transitive nilpotent Lie group N, can be obtained by solving an ODE for a curve of nilpotent Lie brackets. By using that this ODE is the negative gradient flow of a homogeneous polynomial, we obtain that g(t) is type-…

We consider a one-parameter family of strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ moving with speed $- K^αν$, where $ν$ denotes the outward-pointing unit normal vector and $α\geq \frac{1}{n+2}$. For $α> \frac{1}{n+2}$, we show that the flow converges to a round sphere after rescaling. In the affine invariant ca…

We consider contracting and expanding curvature flows in $\Ss$. When the flow hypersurfaces are strictly convex we establish a relation between the contracting hypersurfaces and the expanding hypersurfaces which is given by the Gauß map. The contracting hypersurfaces shrink to a point $x_0$ while the expanding hypersur…

We consider contracting flows in $(n+1)$-dimensional hyperbolic space and expanding flows in $(n+1)$-dimensional de Sitter space. When the flow hypersurfaces are strictly convex we relate the contracting hypersurfaces and the expanding hypersurfaces by the Gauss map. The contracting hypersurfaces shrink to a point $x_0…

MonoFlow rethinks GANs using Wasserstein gradient flows.

Study Chen's flow of curves in two settings: closed circles and lines, identifying geometric conditions for global behavior.

Sigmoid-type networks avoid vanishing gradients with regularization and rescaling.

Study on stability of cylindrical singularities in MCF of finite codimensions.

We study the curve diffusion flow for closed curves immersed in the Minkowski plane $\mathcal{M}$, which is equivalent to the Euclidean plane endowed with a closed, symmetric, convex curve called an indicatrix that scales the length of a vector in $\mathcal{M}$ depending on its length. The indiactrix $\partial\mathcal{…

We prove that the leaves of the rescaled curvature flow considered in arXiv:math/0403485 [math.DG] converge to the graph of a constant function.

We show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor $ψ$ given a strictly convex initial hypersurface in Euclidean space suitably pinched. We also prove the convergence of rescaled surfaces to a smooth limit manifold whic…