arXiv research

Study shows convergence for mean curvature flow on almost minimal totally real submanifolds.

Ancient flows converge fast with finite curvature and convexity.

New formulas compare total mean curvatures of nested hypersurfaces.

A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3-space with constant curvature -1 has two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute cur…

Formula extends mean curvature to convex sets.

Total curvatures of certain hypersurfaces are continuous.

We review recent results on classifying complete constant mean curvature 1 (CMC 1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature" -- one is the total absolute curvature, which is the integral over the surface of the absolute value of the Gaussian curvature, a…

The study classifies flows of finite curvature in 3D space.

Study establishes Pólya-Szegő inequalities on submanifolds with small total mean curvature.

We survey our recent results on classifying complete constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature,…

We derive total mean curvature integration formulae of a three co-dimensional foliation $\mathcal{F}^{n}$ on a screen integrable half-lightlike submanifold, $M^{n+1}$ in a semi-Riemannian manifold $\overline{M}^{n+3}$. We give generalized differential equations relating to mean curvatures of a totally umbilical half-li…

Using Green's theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field and obtain the following well-known theorem as an immediate consequence: the total mean curvature of a closed smooth surface in the Euclidean 3-space is sta…

Proves Gromov's conjecture on total mean curvature using surgery and positive mass theorems.

Sharp inequality proved in 3D hyperbolic spaces using flow methods.

Sharp Minkowski inequality for convex surfaces in curved spaces.

Study soap bubbles with almost constant higher-order mean curvature, proving unique limits under certain conditions.

Study minimizes Willmore energy with constraints on surface properties.

Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.

Ancient curve shortening flows have entropy and curvature bounds equivalent.

The paper studies singularities in mean curvature flow and bounds on minimal surface total curvature.

The flow of a torus by inverse mean curvature keeps total curvature bounded until singularity.

Study bounds total mean curvature of fill-ins with scalar curvature constraints.

Totally umbilic surfaces in hyperbolic 3-manifolds are constructed and characterized.

We consider surfaces with parallel mean curvature vector field and finite total curvature in product spaces of type $\mathbb{M}^n(c)\times\mathbb{R}$, where $\mathbb{M}^n(c)$ is a space form, and characterize certain of these surfaces. When $n=2$, our results are similar to those obtained in \cite{bds} for surfaces wit…

The paper finds conditions for certain hypersurfaces to be totally umbilical.

Estimates Bartnik mass for metrics with nonnegative Gauss curvature.

We consider the integrals of $r$-mean curvatures $S_r$ of a complete hypersurface $M$ in space forms $\mathcal{Q}_c^{n+1}$ which generalize volume $(r=0)$, total mean curvature $(r=1)$, total scalar curvature $(r=2)$ and total curvature $(r=n)$. Among other results we prove that a complete properly immersed hypersurfac…

In this work, complete constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space with total absolute curvature at most 4 pi are classified. This classification suggests that the Cohn-Vossen inequality can be sharpened for surfaces with odd numbers of ends, and a proof of this is given.

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…

Study totally umbilic submanifolds using planar pseudo-geodesics.

Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. This paper continues our recent study and presents new integral formulae and their applications for codimension-one foliated Randers spaces. The goal is a gener…

Upper bound for total mean curvature of spin fill-ins is proven.

New inequality shows all special submanifolds in light cone are totally umbilical spheres.

The paper derives inequalities for mean curvatures of hypersurfaces in Riemannian manifolds.

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

The paper proves bounds on mean curvature for CMC foliations with Ricci curvature constraints.

We prove that a totally umbilical biharmonic surface in any $3$-dimensional Riemannian manifold has constant mean curvature. We use this to show that a totally umbilical surface in Thurston's 3-dimensional geometries is proper biharmonic if and only if it is a part of $S^2(1/\sqrt{2})$ in $S^3$. We also give complete c…

Study variational properties of curves in half-plane with area constraints.

The authors showed in a preceding paper that in a connected locally harmonic manifold, the volume of a tube of small radius about a regularly parameterized simple arc depends only on the length of the arc and the radius. In this paper, we show that this property characterizes harmonic manifolds even if it is assumed on…

We show that under certain curvature conditions of the ambient space an entire Killing graph of constant mean curvature lying inside a slab must be a totally geodesic slice.

We investigate the integral conditions to extend the mean curvature flow in a Riemannian manifold. We prove that the mean curvature flow solution with finite total mean curvature on a finite time interval $[0,T)$ can be extended over time $T$. Moreover, we show that the condition is optimal in some sense.

Under some dimension restrictions, we prove that totally umbilical hypersurfaces of Spin$^c$ manifolds carrying a parallel, real or imaginary Killing spinor are of constant mean curvature. This extends to the Spin$^c$ case the result of O. Kowalski stating that, every totally umbilical hypersurface of an Einstein manif…

Ancient mean curvature flows start from unstable minimal hypersurfaces.

In Kaehler manifolds are investigated conformally flat totally real submanifolds, which are semiparallel or have semiparallel mean curvature vector.

We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consis…

We find a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in $\mathbb R^n$. As an application, we derive a sharp geometric inequality for mean convex, star-shaped hypersurfaces which relates the volume enclosed by a hypersurface to a weighted total mean curvature of the hypers…

Let $F:Σ^n \times [0,T)\to \R^{n+m}$ be a family of compact immersed submanifolds moving by their mean curvature vectors. We show the Gauss maps $γ:(Σ^n, g_t)\to G(n, m)$ form a harmonic heat flow with respect to the time-dependent induced metric $g_t$. This provides a more systematic approach to investigating higher c…

In this paper we prove that the H^k (k is odd and larger than 2) mean curvature flow of a closed convex hypersurface can be extended over the maximal time provided that the total L^p integral of the mean curvature is finite for some p