arXiv research

Formal normal form created for real-smooth hypersurfaces.

Lower bounds for delta invariant of weighted hypersurfaces proved for K-stability.

New invariant for 4D hypersurfaces ensures smooth critical points.

We prove that, for a generic set of smooth prescription functions $h$ on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature $h$. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersur…

Minimal surfaces in 8D smooth and nondegenerate.

Smooth minimizing hypersurfaces in 11D are generic, with singularities in higher dimensions.

Generic smooth minimal hypersurfaces exist in 8D manifolds.

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…

Proof shows smooth minimal hypersurfaces for perimeter-minimizing sets in low-dimensional Riemannian manifolds.

The paper studies deformations of singular minimal hypersurfaces in dimensions 7 and above.

Given a globally hyperbolic spacetime $M$, we show the existence of a {\em smooth spacelike} Cauchy hypersurface $S$ and, thus, a global diffeomorphism between $M$ and $\R \times S$.

We prove that every smooth rigid spherical hypersurface in ${\mathbb C}^2$ is in fact real-analytic. As an application of this result, it follows that the classification of real-analytic rigid spherical hypersurfaces in ${\mathbb C}^2$ found by V. Ezhov and G. Schmalz applies in the smooth case.

Constructs CMC hypersurfaces in S^4 from piecewise-smooth unions of spheres.

In this paper we prove that every smooth complete closed complex hypersurface in the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\ge 2)$ is a level set of a noncritical holomorphic function on $\mathbb{B}_n$ all of whose level sets are complete. This shows that $\mathbb{B}_n$ admits a nonsingular holomorphic fol…

We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. We prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero. We apply this result to …

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

Curvature estimates prove existence of smooth hypersurfaces in hyperbolic space.

We study a variational problem for piecewise-smooth hypersurfaces in the (n+1)-dimensional Euclidean space with an anisotropic energy. An anisotropic energy is the integral of an energy density that depends on the normal at each point over the considered hypersurface. The minimizer of such an energy among all closed hy…

In this paper we construct all smooth torus fibres of the generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties near the large complex limit.

In this paper we show that every area minimizing cone C^{n-1} in R^n can be approximated by entirely smooth area minimizing hypersurfaces. This extensively uses hyperbolic unfoldings of such hypersurfaces and the resulting potential theory for the Jacobi field operator. Applications include the splitting theorem in sca…

Smooth approximations near singularities of constant mean curvature surfaces are found.

The paper proves the existence of specific spacelike hypersurfaces in Minkowski space.

Recently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M, have been solved. Here we give further results, applicable to several problems: (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface …

Convex hypersurfaces in hyperbolic space evolve to geodesic spheres.

We study $λ$-hypersurfaces that are critical points of a Gaussian weighted area functional $\int_Σ e^{-\frac{|x|^2}{4}}dA$ for compact variations that preserve weighted volume. First, we prove various gap and rigidity theorems for complete $λ$-hypersurfaces in terms of the norm of the second fundamental form $|A|$. Sec…

In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities. In the first part of the paper, we consider volume preserving flows by a family of curvature functions including positive powers of $k$-th mean curvatures with $k=1,\cdots,n$, and positive powers of $p$-t…

Study on smoothness of 4D Willmore-type hypersurfaces.

Let $(M^{n+1},g,e^{-f}dμ)$ be a complete smooth metric measure space with $2\leq n\leq 6$ and Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded $f$-minimal hypersurfaces in $M$ with uniform upper bounds on $f$-index and weighted vo…

Singularities of area minimizing hypersurfaces can be smoothed in dimensions 9 and 10.

Smooth convergence to an enveloping cylinder proved for mean curvature flow of complete graphical hypersurfaces.

The paper constructs metrics on spheres with families of minimal hypersurfaces.

Study smooth hypersurfaces with prescribed curvature in Minkowski space.

Inverse curvature flows shape star-shaped hypersurfaces into spheres.

The paper proves smooth convergence of evolving hypersurfaces to critical points.

Manifolds can be dominated by hypersurfaces in a sphere.

Navigation in Lorentz Finsler geometry induces isoparametric hypersurfaces.

Smooth metrics satisfying Penrose inequality are necessarily smooth.

For any $n$-dimensional smooth manifold $Σ$, we show that all the singularities of the mean curvature flow with any initial mean convex hypersurface in $Σ$ are cylindrical (of convex type) if the flow converges to a smooth hypersurface $M_{\infty}$ (maybe empty) at infinity. Previously this was shown (i) for $n\leq 7$,…

Study on convex capillary hypersurfaces with Lp curvature in half-space.

Smooth approximations bound dihedral angles of convex polytopes.

In this paper we discuss the smoothness conditions for metrics on a cohomogeneity one manifold, i.e. metrics invariant under a Lie group whose generic orbits are hypersurfaces. Along these hypersurfaces one describes the metrics in terms of a collection of functions defined along a geodesic normal to the hypersurfaces.…

For an $(m+1)$-dimensional space-time $(X^{m+1}, g),$ define a mapped null hypersurface to be a smooth map $ν:N^{m}\to X^{m+1}$ (that is not necessarily an immersion) such that there exists a smooth field of null lines along $ν$ that are both tangent and $g$-orthogonal to $ν.$ We study relations between mapped null hyp…

Let $N$ be a Riemannian manifold and consider a stationary union of three or more $C^{1,μ}$ hypersurfaces-with-boundary $M_k$ in $N$ with a common boundary $Γ$. We show that if $N$ is smooth, then $Γ$ is smooth and each $M_k$ is smooth up to $Γ$ (real analytic in the case $N$ is real analytic). Consequently we strength…

Symmetric hypersurfaces with constant mean curvature are spheres.

Given 2 points of a smooth hypersurface, their mid-hyperplane is the hyperplane passing through their mid-point and the intersection of their tangent spaces. In this paper we study the envelope of these mid-hyperplanes (EMH) at pairs whose tangent spaces are transversal. We prove that this envelope consists of centers …

The present paper describes a way to relate Martin boundaries on spaces of varying topology. This enables us to approach some detailed inductive analysis of the eigenfunctions of conformal Laplacians on minimal hypersurfaces near their singularities. This can directly be used resp. translated to understand the way how …

For a given smooth $2$-knot in $S^4$, we relate the existence of a smooth Seifert hypersurface of a certain class to the existence of irreducible $SU(2)$-representations of its knot group. For example, we see that any smooth $2$-knot having the Poincaré homology $3$-sphere as a Seifert hypersurface has at least four i…

We solve spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in Schwarzschild spacetimes and analyze their asymptotic behavior near the coordinate singularity r = 2M. Furthermore, we join SS-CMC hypersurfaces in the Kruskal extension to obtain complete ones and discuss the smooth properties.