arXiv research

Study classifies 3D self-shrinkers with constant second form norm.

We obtain an infinite family of complete non embedded rotational surfaces in $\mathbb R^3$ whose second fundamental forms have length equal to one at any point. Also we prove that a complete rotational surface with second fundamental form of constant length is either a round sphere, a circular cylinder or, up to a homo…

Researchers classify special curved spheres in a complex space.

Researchers classify 3D self-shrinkers in 4D space.

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

The complete local classification and geometric description of n-dimensional submanifolds F with recurrent nonparallel second fundamental form in the spaces of constant curvature M(c) are obtained in this article.

The paper classifies 3D self-expanders with specific properties.

The study proves properties of self-shrinkers with bounded curvature.

The paper classifies 2D complete Lagrangian self-expanders in complex 2-space.

4D self-shrinkers in 5D space are rigid.

Formula found for surfaces in Sol_3, leading to gap results.

In this note, we give a new and simple proof of a result in {\cite{DX1}} which states that any smooth complete self-shrinker in $\mathbb{R}^3$ with second fundamental form of constant length must be a generalized cylinder $\mathbb{S}^k \times \mathbb{R}^{2-k}$ for some $k\leq2$. Moreover, we prove a gap theorem for smo…

Improved eigenvalue bounds for minimal hypersurfaces in spheres.

In this paper we consider three-manifolds with weakly umbilic boundary (the Second Fundamental form of the boundary is a constant multiple of the metric). We show that if the initial manifold has positive Ricci curvature and the boundary is convex (nonnegative Second Fundamental form), its metric can be deformed via th…

In Theorem 3.1 of [12], we proved a rigidity result for self-shrinkers under the integral condition on the norm of the second fundamental form. In this paper, we relax the such bound to any finite constant (see Theorem 4.4 for details).

The paper examines biconservative hypersurfaces with constant curvature in space forms.

Paper proves a conjecture about minimal hypersurfaces in spheres.

A spacelike surface in four-dimensional Lorentz-Minkowski spacetime through the lightcone has a meaningful lightlike normal vector field $η$. Several sufficient assumptions on such a surface with non-degenerate $η$-second fundamental form are established to prove that it must be a totally umbilical round sphere. With t…

The Hopf fibration is rigid among minimal maps between spheres.

The paper generalizes a rigidity theorem for hypersurfaces with constant weighted mean curvature.

The paper studies scalar curvature of self-shrinkers and proves curvature bounds.

This is a revised version (minor changes and a deeper insight in the positive curvature case). We prove some Caccioppoli's inequalities for the traceless part of the second fundamental form of a complete, noncompact, finite index, constant mean curvature hypersurface of a Riemannian manifold, satisfying some curvature …

The study characterizes Clifford hypersurfaces in terms of curvature constants.

The paper proves rigidity for shells in non-Euclidean spaces.

The properties of Kaehler submanifolds with recurrent the second fundamental form in spaces of constant holomorphic sectional curvature are being studied in this article.

We consider minimal maps $f:M\to N$ between Riemannian manifolds $(M,\mathrm{g}_M)$ and $(N,\mathrm{g}_N)$, where $M$ is compact and where the sectional curvatures satisfy $\sec_N\le σ\le \sec_M$ for some $σ>0$. Under certain assumptions on the differential of the map and the second fundamental form of the graph $Γ(f)$…

An energy estimate is proved for the Bel--Robinson energy along a constant mean curvature foliation in a spatially compact vacuum spacetime, assuming an $L^{\infty}$ bound on the second fundamental form, and a bound on a spacetime version of Bel--Robinson energy.

Study Hamiltonian stationary Lagrangian surfaces in complex space forms.

The following results are proved: Theorem 1. A totally real semiparallel submanifold of constant curvature with parallel f-structure in the normal bundle of a Kähler manifold N is flat or a totally geodesic submanifold of N. Theorem 2. A totally real minimal semiparallel submanifold M with parallel f-structure in the n…

The study describes the structure of surfaces with constant mean curvature in 3-manifolds.

Proves a weak version of Perdomo Conjecture on minimal hypersurfaces.

The paper studies Lagrangian surfaces in a specific Riemannian product space.

The study classifies complete self-shrinkers in Euclidean space.

Study complete 3D λ-translators in Minkowski space with constant properties.

To study the Lawson-Osserman's counterexample to the Bernstein problem for minimal submanifolds of higher codimension, a new geometric concept, submanifolds in Euclidean space with constant Jordan angles(CJA), is introduced. By exploring the second fundamental form of submanifolds with CJA, we can characterize the Laws…

Study of complete space-like self-expanders in Minkovski space.

Study properties of surfaces with nonvanishing third fundamental form.

Let $Σ$ be a smooth closed hypersurface with non-negative Ricci curvature, isometrically immersed in a space form. It has been proved in \cite{P}, \cite{CZ}, and \cite{C2} that there are some $L^2$ inequalities on $Σ$ which measure the stability of closed umbilical hypersurfaces or more generally, closed hypersurfaces …

It is our purpose to study complete self-shrinkers in Euclidean space. By introducing a generalized maximum principle for $\mathcal{L}$-operator, we give estimates on supremum and infimum of the squared norm of the second fundamental form of self-shrinkers without assumption on \emph{polynomial volume growth}, which is…

The paper characterizes Whitney and contact Whitney spheres in complex and Sasakian space forms.

We concern $C^2$-compactness of the solution set of the boundary Yamabe problem on smooth compact Riemannian manifolds with boundary provided that their dimensions are $4$, $5$ or $6$. By conducting a quantitative analysis of a linear equation associated with the problem, we prove that the trace-free second fundamental…

The paper studies hypersurfaces in 5D space forms with topological and rigidity results.

In this paper we prove that an embedded and simply connected constant mean curvature surface with curvature large at a point contains a multi-valued graph around that point on the scale of $|A|^2$, where $|A|^2$ is the norm squared of the second fundamental form. This generalizes Colding and Minicozzi's result for mini…

The study shows that the second fundamental form is intrinsic under certain conditions in space forms.

We investigate the local geometry of a class of Kähler submanifolds $M \subset \R^n$ which generalize surfaces of constant mean curvature. The role of the mean curvature vector is played by the $(1,1)$-part (i.e. the $dz_id\bar z_j$-components) of the second fundamental form $α$, which we call the pluri-mean curvature.…

We consider critical points of the functionals $Π$ and $Ψ$ defined as the global $L^2$-norm of the second fundamental form and mean curvature vector of isometric immersions of compact Riemannian manifolds into a background Riemannian manifold, respectively, as functionals over the space of deformations of the immersion…

The purpose of this paper is to study complete $λ$-surfaces in Euclidean space $\mathbb R^3$. A complete classification for 2-dimensional complete $λ$-surfaces in Euclidean space $\mathbb R^3$ with constant squared norm of the second fundamental form is given.

The expression for the variation of the area functional of the second fundamental form of a hypersurface in a Euclidean space involves the so-called "mean curvature of the second fundamental form". Several new characteristic properties of (hyper)spheres, in which the mean curvature of the second fundamental form occurs…