arXiv research

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.

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.

The paper studies a flow for convex capillary hypersurfaces in a ball, proving smooth convergence to a spherical cap.

Abstract properties of hypersurface data analyzed in spherical symmetry.

Study convex capillary hypersurfaces with prescribed curvature in a spherical cap.

As this is for the Bulletin of the A.M.S., it is not only a review of Alexander Isaev's Spherical Tube Hypersurfaces but also a brief introduction to CR geometry.

New flow for capillary surfaces converges to spherical caps.

The study finds regular null hypersurfaces in a perturbed Schwarzschild black hole exterior.

Study shows how curved surfaces evolve smoothly to spherical shapes.

Paper solves inequalities for capillary hypersurfaces in half-spaces.

The paper classifies 3D spherical Sasakian manifolds using geometric and algebraic methods.

The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion. In this work, we extend these investigations to characterize curves that lie on a…

New capillary Christoffel-Minkowski problem solved for half-space.

The paper extends Liebmann's Theorem to convex hypersurfaces with boundary.

New inequalities for convex hypersurfaces in various spaces.

Unified treatment of stability problems in geometry and analysis.

The paper proves inequalities for convex capillary hypersurfaces in a half-space.

The techniques developed by Butscher in arXiv:math/0703469 for constructing constant mean curvature (CMC) hypersurfaces in the (n+1)-sphere by gluing together spherical building blocks are generalized to handle less symmetric initial configurations. The outcome is that the approximately CMC hypersurface obtained by glu…

We provide an explicit description of all rigid hypersurfaces that are equivalent to a Heisenberg sphere. These hypersurfaces are determined by 4 real parameters. The defining equations of the rigid spheres can also be viewed as the complete solution of a non-linear PDE that expresses the vanishing Cartan curvature con…

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…

We show that every spherical 2-Dupin submanifold that is not a hypersurface is conformally congruent to the standard embedding of the real, complex, quaternionic or octonionic projective plane. We also classify 2-CPC, 2-umbilical and weakly 2-umbilical submanifolds in space forms.

A Levi nondegenerate real analytic hypersurface M of C^2 represented in local coordinates (z, w) in C^2 by a complex defining equation of the form w = Theta (z, \bar z, \bar w) which satisfies an appropriate reality condition, is spherical if and only if its complex graphing function Theta satisfies an explicitly writt…

Study stable capillary hypersurfaces with planar boundaries in half-spaces and domains.

We study the stability of capillary hypersurfaces in a unit Euclidean ball. It is proved that if the mass center of the generalized body enclosed by the immersed capillary hypersurface and the wetted part of the sphere is located at the origin, then the hypersurface is unstable. An immediate result is that all known ex…

Tight isoparametric hypersurfaces in spheres have minimal critical points.

The paper classifies hypersurfaces with special curvature properties in various spaces.

We show that for a real-analytic connected holomorphically nondegenerate 5-dimensional CR-hypersurface $M$ and its symmetry algebra $\mathfrak{s}$ one has either: (i) $\dim\mathfrak{s}=15$ and $M$ is spherical (with Levi form of signature either $(2,0)$ or $(1,1)$ everywhere), or (ii) $\dim\mathfrak{s}\le11$ where $\di…

We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Be…

Low-entropy surfaces can be flowed into spheres and cylinders.

We first summarize the characterization of smooth spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in the Schwarzschild spacetime and Kruskal extension. Then use the characterization to prove special SS-CMC foliation property, and verify part of the conjecture by Malec and Ó Murchadha in t…

We prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is diffeomorphic to a sphere or to a quotient of a sphere by a group action. We also…

It is still an open question whether a compact embedded hypersurface in the Euclidean space R^{n+1} with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of surfaces in R^3. In a recent paper the first and third authors have shown that this i…

In this paper we introduce a Guan-Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan-Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, pro…

A spherical set is called convex if for every pair of its points there is at least one minimal geodesic segment that joins these points and lies in the set. We prove that for n >= 3 a complete locally-convex (topological) immersion of a connected (n-1)-manifold into the n-sphere is a surjection onto the boundary of a c…

We use purely topological methods to prove the semicontinuity of the mod 2 spectrum of local isolated hypersurface singularities in $\mathbb{C}^{n+1}$, using Seifert forms of high-dimensional non-spherical links, the Levine--Tristram signatures and the generalized Murasugi--Kawauchi inequality obtained in earlier work …

We construct a $T$-axisymmetric, spacelike, spherically symmetric, constant mean curvature hypersurfaces foliation in the Kruskal extension with properties that the mean curvature varies in each slice and ranges from minus infinity to plus infinity. This family of hypersurfaces extends the CMC foliation discussions pos…

The paper proves new inequalities for convex hypersurfaces in hyperbolic and spherical spaces.

Study of spacelike submanifolds in spherical RW spacetime, proving a Lorentzian Takahashi theorem.

We prove that in Euclidean space $R^{n+1}$ any compact immersed nonnegatively curved hypersurface $M$ with free boundary on the sphere $S^n$ is an embedded convex topological disk. In particular, when the $m^{th}$ mean curvature of $M$ is constant, for any $1\leq m\leq n$, $M$ is a spherical cap or an equatorial disk.

In this paper, in Euclidean n -space, we investigate the relation between slant helices and spherical helices. Moreover, in E n, we show that a slant helix and the tangent indicatrix of the slant helix have the same axis (or direction). Also, we give the important relations between slant helices, spherical helices in E…

Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.

We give an explicit estimate of the distance of a closed, connected, oriented and immersed hypersurface of a space form to a geodesic sphere and show that the spherical closeness can be controlled by a power of an integral norm of the traceless second fundamental form, whenever the latter is sufficiently small. Further…

In this paper, firstly, inspired by Natário's recent work \cite{Na}, we use the isoperimetric inequality to derive some Alexandrov-Fenchel type inequalities for closed convex hypersurfaces in the hyperbolic space $\H^{n+1}$ and in the sphere $\SS^{n+1}$. We also get the rigidity in the spherical case. Secondly, we use …

For a Riemannian manifold $M^{n+1}$ and a compact domain $Ω\subset M^{n+1}$ bounded by a hypersurface $\partial Ω$ with normal curvature bounded below, estimates are obtained in terms of the distance from $O$ to $\partial Ω$ for the angle between the geodesic line joining a fixed interior point $O$ in $Ω$ to a point on…

For a convex domain $D$ bounded by the hypersurface $\partial D$ in a space of constant curvature we give sharp bounds on the width $R-r$ of a spherical shell with radii $R$ and $r$ that can enclose $\partial D$, provided that normal curvatures of $\partial D$ are pinched by two positive constants. Furthermore, in the …

In this paper, we study a mean curvature type flow with capillary boundary in the unit ball. Our flow preserves the volume of the bounded domain enclosed by the hypersurface, and monotonically decreases an energy functional $E$. We show that it has the longtime existence and subconverges to spherical caps. As an applic…

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

We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasi-linear e…