arXiv research

The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced. A backward Bäcklund transform of a generalized harmonic inverse mean curvature surface is defined. A Darboux transform of a generalized harmonic inverse mean curvature surface is constructed by a backward Bäck…

The paper proves smoothness of mean curvature flow for generic initial data in 3D and 4D.

Study of mean curvature flows with conical singularities using mathematical techniques.

Study constant mean curvature surfaces with integrable boundary conditions.

Study proves a new flow method for mean curvature with volume change analysis.

Formula extends mean curvature to convex sets.

In this paper we prove that stable, compact without boundary, oriented, nonzero constant mean curvature surfaces in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds are the slices, provided its mean curvature satisfies some positive lower bound. More generally, we prove that stable, compact without boundary…

Constructs approximate mean curvature flows for general varifolds.

Symmetry groups help define solitons in curved spaces.

Proves existence of shrinkers via mean curvature flow.

Smooth convergence to equidistant surfaces in Fuchsian manifolds.

Proves uniqueness of geometric flow in various Riemannian manifolds.

Study shows how to preserve Lagrangian condition in mean curvature flow on Kim-McCann metrics.

Paper relaxes convexity assumptions in mean curvature flow results.

In this paper, we prove a general halfspace theorem for constant mean curvature surfaces. Under certain hypotheses, we prove that, in an ambient space M^3, any constant mean curvature H_0 surface on one side of a constant mean curvature H_0 surface Σ_0 is an equidistant surface to Σ_0. The main hypotheses of the theore…

Localized deformation of scalar curvature and mean curvature on manifolds.

Stability of capillary hypersurfaces with higher order mean curvature.

The paper explores rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space.

Paper discusses solving generalized Hessian inequalities with various operators.

Study classifies ruled surfaces from mean curvature flow solutions.

In this paper, we discuss uniqueness and backward uniqueness for mean curvature flow of non-compact manifolds. We use an energy argument to prove two uniqueness theorems for mean curvature flow with possibly unbounded curvatures. These generalize the results by Chen and Yin. Using similar method, we also obtain a uniqu…

Study proves existence and uniqueness of ancient flows from cones.

The paper constructs surfaces with prescribed mean curvature in a specific space.

The paper examines properties of spherical Finsler metrics, proving their semi-C-reducibility and conditions for vanishing mean stretch curvature.

This paper studies mean curvature flows near cylindrical singularities.

Entropy defined for submanifolds; applies to mean curvature flow limits of surfaces.

In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time $T$ of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based …

We introduce the notion of Kähler manifolds that are almost Einstein and we define a generalized mean curvature vector field along submanifolds in them. We prove that Lagrangian submanifolds remain Lagrangian, when deformed in direction of the generalized mean curvature vector field. For a Kähler manifold that is almos…

Calculates S-curvature and mean Berwald curvature in homogeneous Finsler spaces.

Study proves mean curvature flow in GRW spacetimes with perpendicular boundary condition.

Geodesic boundaries of surfaces with constant mean curvature are studied.

The paper proves a theorem about mean curvature in Euclidean and hyperbolic spaces.

The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regular…

Study finds unique self-expanders for mean curvature flow.

Compact capillary hypersurfaces with specific curvatures are proven.

Mean curvature flow evolves isometrically immersed base manifolds $M$ in the direction of their mean curvatures in an ambient manifold $\bar{M}$. If the base manifold $M$ is compact, the short time existence and uniqueness of the mean curvature flow are well-known. For complete isometrically immersed submanifolds of ar…

In this note we generalize an extension theorem in [5] and [9] of the mean curvature flow to the H^{k} mean curvature flow under some extra conditions. The main difficult problem in proving the extension theorem is to find a suitable version of Michael-Simon inequality for the H^{k} mean curvature flow, and to do a sui…

Study of prescribing scalar curvature and mean curvature on compact manifolds with boundary.

The paper defines a frequency for mean curvature flow and proves its monotonicity.

In this paper, we first investigate the integral curvature condition to extend the mean curvature flow of submanifolds in a Riemannian manifold with codimension $d\geq1$, which generalizes the extension theorem for the mean curvature flow of hypersurfaces due to Le-Šešum \cite{LS} and the authors \cite{XYZ1,XYZ2}. Usin…

In this paper, we obtain a basic Chen's inequality for a C-totally real submanifold in a generalized $(κ,μ)$-contact space forms involving intrinsic invariants, namely the scalar curvature and the sectional curvatures of the submanifold on left hand side and the main extrinsic invariant, namely the squared mean curvatu…

Generic level sets in mean curvature flow are BV solutions.

Mean curvature flow shows singularities on smooth surfaces.

We study inverse mean curvature flows of starshaped, mean convex hypersurfaces in warped product manifolds with a positive warping factor $\varphi(r)$. If $\varphi'(r)>0$ and $\varphi''(r)\geq 0$, we show that these flows exist for all times, remain starshaped and mean convex. Plus the positivity of $\varphi''(r)$ and …

Proves rigidity of boundaries with constant mean curvature in warped product manifolds.

The study examines constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.

Existence of double bubbles with high constant mean curvatures in Riemannian manifolds.

New proof shows symmetry for certain curved surfaces in higher dimensions.