arXiv research

We prove some stability results for smooth H-minimal hypersurfaces immersed in a sub-Riemannian k-step Carnot group G. The main tools are the formulas for the 1st and 2nd variation of the H-perimeter measure.

Estimates eigenvalues on weighted manifolds with curvature.

Let S be a C^2 H-minimal noncharacteristic hypersurface in the first Heisenberg group. We show that if S contains a graphical strip, then it is not a stable minimal surface. Moreover, we show that if S is a C^2 H-minimal noncharacteristic entire graph which is not itself a vertical plane, then S contains a graphical st…

In this paper we investigate H-minimal graphs of lower regularity. We show that noncharactersitic C^1 H-minimal graphs whose components of the unit horizontal Gauss map are in W^{1,1} are ruled surfaces with C^2 seed curves. In a different direction, we investigate ways in which patches of C^1 H-minimal graphs can be g…

Formula proves almost monotonicity for H-minimal surfaces in Heisenberg group.

H-minimal Lagrangian submanifolds in general Kähler manifolds generalize special Lagrangian submanifolds in Calabi-Yau manifolds. In this paper we will use the deformation theory of H-minimal Lagrangian submanifolds in Kähler manifolds to construct minimal Lagrangian torus in certain Kähler-Einstein manifolds with nega…

Hamiltonian minimality (H-minimality) for Lagrangian submanifolds is a symplectic analogue of Riemannian minimality. A Lagrangian submanifold is called H-minimal if the variations of its volume along all Hamiltonian vector fields are zero. This notion was introduced in the work of Y.-G. Oh in connection with the celebr…

The paper studies harmonic graphs in the Heisenberg group and their properties.

Corrected Monti's blow-up analysis for H-minimizing sets in Heisenberg group.

Let L be a Lagrangian submanifold of a pseudo- or para-Kähler manifold which is H-minimal, i.e. a critical point of the volume functional restricted to Hamiltonian variations. We derive the second variation of the volume of L with respect to Hamiltonian variations. We apply this formula to several cases. In particular …

We establish the following theorem of Bernstein type for the first Heisenberg group: Let S be a C^2 connected H-minimal surface which is a graph over some plane P, then S is either a non-characteristic vertical plane, or its generalized seed curve satisfies a type of constant curvature condition.

We describe several families of Lagrangian submanifolds in the complex Euclidean space which are H-minimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving planar, spherical and hyperbolic curves, as well as Legendrian submanifolds of th…

In this paper we investigate a family of Hamiltonian-minimal Lagrangian submanifolds in ${\mathbb C}^m$, ${\mathbb C}P^m$ and other symplectic toric manifolds constructed from intersections of real quadrics. In particular, we explain the nature of this phenomenon by proving H-minimality in a more conceptual way, and pr…

Let $P$ be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight $e^h$. The aim of this paper is twofold. First, by assuming certain control on the $h$-mean curvature of $P$, we establish comparisons for the $h$-capacity of extrinsic balls in $P$, from which we ded…

For Lorentzian 2-manifolds $(Σ_1,g_1)$ and $(Σ_2,g_2)$ we consider the two product para-Kähler structures $(G^ε,J,Ω^ε)$ defined on the product four manifold $Σ_1\timesΣ_2$, with $ε=\pm 1$. We show that the metric $G^ε$ is locally conformally flat (resp. Einstein) if and only if the Gauss curvatures $κ_1,κ_2$ of $g_1,g_…

Let (M,w) be a compact symplectic 2n-manifold, and g a Riemannian metric on M compatible with w. For instance, g could be Kahler, with Kahler form w. Consider compact Lagrangian submanifolds L of M. We call L Hamiltonian stationary, or H-minimal, if it is a critical point of the volume functional under Hamiltonian defo…

The paper classifies Codazzi hypersurfaces and characterizes minimal hypersurfaces in Nil^4.

Classification of hypersurfaces in homogeneous spaces with specific properties.

Study on biconservative hypersurfaces with constant scalar curvature in space forms.

The paper classifies various types of hypersurfaces in a product space.

The study of Laguerre isotropic hypersurfaces with rigidity and isoparametric properties.

Classifies hypersurfaces with constant isotropic curvature in space forms.

We study the constant mean curvature (CMC) hypersurfaces in hyperbolic space whose asymptotic boundaries are closed codimension-1 submanifolds in sphere at infinity. We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area m…

New global section found for geodesic flows on convex hypersurfaces.

New tensors capture intrinsic embedding data of conformal hypersurfaces.

The study characterizes canal hypersurfaces in Euclidean spaces and their curvature properties.

Survey on Dupin hypersurfaces and isoparametric hypersurfaces in spheres.

In this paper we introduce radical transversal lightlike hypersurfaces of almost complex manifolds with Norden metric. The study of these hypersurfaces is motivated by the fact that for indefinite almost Hermitian manifolds this class of lightlike hypersurfaces does not exist. We also establish that radical transversal…

Study on Dirac operators on lightlike hypersurfaces in 4D Lorentzian manifolds.

In this paper we show that a Dupin hypersurface with constant Möbius curvatures is Möbius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat L…

New compact mean convex hypersurfaces found for positive λ.

In this paper, we study generalized constant ratio (GCR) hypersurfaces in Euclidean spaces. We mainly focus on the hypersurfaces in $\mathbb E^4$. First, we deal with $δ(2)$-ideal GCR hypersurfaces. Then, we study on hypersurfaces with constant (first) mean curvature. Finally, we obtain the complete classification of G…

Study on minimal hypersurfaces in a special normed space.

The paper proves rigidity results for capillary hypersurfaces in hyperbolic space.

Classifies and describes hypersurfaces in Siklos spacetimes.

Proves convexity of certain hypersurfaces with negative λ.

Study on rotational hypersurfaces with constant Gauss-Kronecker curvature.

Minimal hypersurfaces are the only $H$-tensional in 4D space forms.

Study shows constant curvature convex hypersurfaces on hyperboloids are parts of hyperboloids.

Paper characterizes a special hypersurface in 5D sphere.

The problem of determining the {\it Bonnet hypersurfaces in} $R^{n+1}$, for $n>1$, is studied here. These hypersurfaces are by definition those that can be isometrically mapped to another hypersurface or to itself (as locus) by at least one nontrivial isometry preserving the mean curvature. The other hypersurface and/o…

The paper classifies isoparametric hypersurfaces in conic Finsler spaces.

Paper introduces new hypersurface types on statistical manifolds.

Study on properties of tangential hypersurfaces in product-like manifolds.

In this paper, geometric characterizations of conformally flat and radially flat hypersurfaces in $\mathbb{S}^n \times \mathbb{R}$ and $\mathbb{H}^n \times \mathbb{R}$ are given by means of their extrinsic geometry. Under suitable conditions on the shape operator, we classify conformally flat hypersurfaces in terms of …

Study on isoparametric and constant-curvature hypersurfaces in Finsler spaces.

Similar to the definition of Dupin hypersurface in Riemannian space forms, we define the spacelike Dupin hypersurface in Lorentzian space forms. As conformal invariant objects, spacelike Dupin hypersurfaces are studied in this paper using the framework of conformal geometry. Further we classify the spacelike Dupin hype…

Classifies isoparametric hypersurfaces in 3D manifolds.