arXiv research

Study on steady solitons' curvature behavior near infinity.

The paper studies the curvature behavior near the boundary of certain domains.

Study behavior of curvatures near singular points of frontals.

In this paper, we study the positive cross curvature flow on locally homogeneous 3-manifolds. We describe the long time behavior of these flows. We combine this with earlier results concerning the asymptotic behavior of the negative cross curvature flow to describe the two sided behavior of maximal solutions of the cro…

In this paper, we introduce a monotonicity formula for the mean curvature flow which is related to self-expanders. Then we use the monotonicity to study the asymptotic behavior of Type III mean curvature flow on noncompact hypersurfaces.

The article explores Helfrich flow with spontaneous curvature, finding singularities and convergence behaviors.

Study of lightcone framed surfaces in Lorentz-Minkowski 3-space, focusing on curvature behavior.

In this paper we investigate the behavior of three-dimensional homogeneous solutions of the cross curvature flow using Riemannian groupoids. The Riemannian groupoid technique, introduced by John Lott, allows us to investigate the long term behavior of collapsing solutions of the flow, producing soliton solutions in the…

The article studies curvature operator behavior in 3D under Ricci flow.

Study of Moncrief lines' behavior in curved space-times.

Study finds solitons on curved spaces with varying behavior.

Bach flow studied on special product manifolds, revealing interesting behavior.

Study asymptotic behavior of Weingarten surfaces at infinity.

In this paper, we study the existence, uniqueness and asymptotic behavior of rotationally symmetric translating solitons of the mean curvature flow in Minkowski space. We also study the asymptotic behavior and the strict convexity of general solitons of such flows.

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

Study on harmonic functions in spaces with collapsing behaviors.

Anisotropic curvature flow of networks shows unique solutions and behavior under finite time.

In the 3-dimensional Lorentz-Minkowski space we prove that the sign of the Gaussian curvature of any timelike minimal surface is determined by the degeneracy and the orientations of the two null curves that generate the surface. Moreover, we also investigate the behavior of the Gaussian curvature near singular points o…

The paper proves properties of surfaces with finite curvature in 3D space.

Study finite curvature solutions on surfaces with nonnegative Gauss curvature.

Study on metric properties near boundary of tube domains.

We show precompactness results for solutions to parabolic fourth order geometric evolution equations. As part of the proof we obtain smoothing estimates for these flows in the presence of a curvature bound, an improvement on prior results which also require a Sobolev constant bound. As consequences of these results we …

We define cuspidal curvature $κ_c$ (resp. normalized cuspidal curvature $μ_c$) along cuspidal edges (resp. at swallowtail singularity) in Riemannian $3$-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the product $κ_Π$ called the product curva…

Improved curvature behavior on hyperbolic surfaces.

Geometric tools study two- and threepeakons in Camassa-Holm equation.

Study rotational surfaces with prescribed Gauss curvature in 3D space.

Study the long-time behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.

Curves in 3-sphere with constant curvature and torsion are described.

In this paper we consider the geometric behavior near infinity of some Einstein manifolds $(X^n, g)$ with Weyl curvature belonging to a certain $L^p$ space. Namely, we show that if $(X^n, g)$, $n \geq 7$, admits an essential set and has its Weyl curvature in $L^p$ for some $1<p<\frac{n-1}{2}$, then $(X^n, g)$ must be a…

Flow of curved surfaces converges to a specific shape over time.

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

We prove that the scalar curvature of a homogeneous Ricci flow solution blows up at a forward or backward finite-time singularity.

Study on Einstein solitons with bounds and asymptotic behavior.

Study on Kähler manifolds connects curvature decay with growth of holomorphic functions.

Constructs flows with prescribed tangent and asymptotic behavior.

Graph Ricci flow reveals hidden hierarchies in stock market correlations.

In this paper, we study the convexity, interior gradient estimate, Liouville type theorem and asymptotic behavior at infinity of translating solutions to mean curvature flow as well as the nonlinear flow by powers of the mean curvature.

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

We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the exi…

The study classifies certain types of incomplete surfaces with low curvature.

In this paper is studied the behavior of principal curvature lines near a curve of umbilic points of a smooth surface.

Survey on geodesics on tetrahedra in curved spaces.

We consider the evolution by mean curvature flow of a closed submanifold of the complex projective space. We show that, if the submanifold has small codimension and satisfies a suitable pinching condition on the second fundamental form, then the evolution has two possible behaviors: either the submanifold shrinks to a …

Study curvature lines of a vector field on surfaces.

Focal loss reduces model curvature for better calibration.

In this paper, we analyze the asymptotic behavior of $κ$-noncollapsed and positively curved steady Ricci solitons and prove that any $n$-dimensional $κ$-noncollapsed steady Kähler-Ricci soliton with non-negative sectional curvature must be flat.

We study asymptotic behavior of nonparametric hypersurfaces moving by $α$ powers of Gauss curvature $α> 1/n$. Our work generalizes the results of V. Oliker [Oli91] for $α= 1$.

The paper studies a flow of Legendre curves, generalizing the inverse curvature flow of regular curves.