arXiv research

We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after finite time. The same holds if the initial data is only weakly convex or non-smoot…

Study on anisotropic curvature flow for noncompact convex hypersurfaces.

The paper studies a flow of convex hypersurfaces expanding by their support and curvature functions.

Flow of convex hypersurfaces in hyperbolic space converges to geodesic spheres.

The paper extends a Harnack inequality to noncompact evolving hypersurfaces.

The paper studies curvature flows in hyperbolic space and proves convergence to spheres under certain conditions.

Anisotropic expanding flow of convex hypersurfaces converges to a soliton under certain conditions.

We prove $ε$-closeness of hypersurfaces to a sphere in Euclidean space under the assumption that the traceless second fundamental form is $δ$-small compared to the mean curvature. We give the explicit dependence of $δ$ on $ε$ within the class of uniformly convex hypersurfaces with bounded volume.

Paper finds unique solutions for curved surfaces with specific gradient.

New control on diameter and curvature for evolving surfaces.

Unified flow solves $L^p$ Christoffel-Minkowski problem for $p>1$.

The paper proves a unique solution to a shrinking flow equation converging to a soliton.

We prove that convex hypersurfaces in ${\mathbb R}^{n+1}$ contracting under the flow by any power $α>\frac{1}{n+2}$ of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the ini…

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

This paper solves the Christoffel problem in hyperbolic space and its equivalent on spheres.

Constructs uniformly positive scalar curvature metrics on open manifolds

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

Study shows linear growth of index for free boundary minimal hypersurfaces.

The paper studies foliation flows with logarithmic speeds and finds convergence to translating solutions.

The study shows that close hypersurfaces have uniformly bounded inequalities.

Classifies ancient solutions to curvature flows, finding two main types.

Compactness of singular minimal hypersurfaces with bounded volumes and eigenvalues.

The study calculates the average number of tangent k-dimensional subspaces to multiple convex hypersurfaces in random position.

Study of Brown--York mass for four-dimensional asymptotically flat manifolds.

The paper studies hypersurfaces in spheres using mean curvature flow with surgery.

Derives a simpler equation for CR-curvature vanishing on certain complex hypersurfaces.

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(κ)=σ\in(0, 1)$ with a prescribed asymptotic boundary $Γ$ at infinity has at least one smooth solution with uniformly bounded hyperbol…

Proves convexity of certain hypersurfaces with negative λ.

Flow of spacelike hypersurfaces converges to flat slice in asymptotically flat spacetimes.

New global section found for geodesic flows on convex hypersurfaces.

The study proves the existence of $k$-convex hypersurfaces for specific curvature equations.

Local rigidity proved for convex hypersurfaces in spaces of constant curvature.

Groups with specific subgroup actions have proper actions on uniformly convex spaces.

New compact mean convex hypersurfaces found for positive λ.

Paper solves Dirichlet problem for $p$-convex hypersurfaces with curvature constraints.

Compactness proven for manifolds with nonnegative Ricci curvature and uniformly convex boundary.

Lower bounds for surface area and volume of convex hypersurfaces.

The study finds large Betti numbers in minimal hypersurfaces with positive Ricci curvature.

Analyzes singularities of convex hypersurfaces in hyperbolic space.

The paper studies critical sections of the Allen-Cahn functional and their relation to minimal hypersurfaces.

Paper establishes an optimal inequality for convex hypersurfaces.

We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold of dimension at most seven, can degenerate. Loosely speaking, our results show …

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

The study extends Huisken's theorem to nonconvex surfaces that shrink to round points.

2-convex translating solitons are locally strictly convex.

Paper proves rigidity of convex hypersurfaces in various spaces.

In this paper, we consider minimal hypersurfaces in the product space $\mathbb{H}^n \times \mathbb{R}$. We begin by studying examples of rotation hypersurfaces and hypersurfaces invariant under hyperbolic translations. We then consider minimal hypersurfaces with finite total curvature. This assumption implies that the …

Optimal inequality for free boundary hypersurfaces in convex domains.