arXiv research

Paper constructs a new type of hypersurface in Euclidean spaces.

We construct an immersed and non-embedded $S^2$ self-shrinker.

New proof shows no Hölder embeddings into Heisenberg group.

We show existence of constant mean curvature 1 surfaces in both hyperbolic 3-space and de Sitter 3-space with two complete embedded ends and any positive genus up to genus twenty. We also find another such family of surfaces in de Sitter 3-space, but with a different non-embedded end behavior.

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

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…

We prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the RCD(0,N), with N > 1. In fact, we can prove that a subRiemannian manifold whose generic degree of nonholonomy is not smaller…

Let $(M,ω)$ be a geometrically bounded symplectic manifold, $N\subseteq M$ a closed, regular (i.e. "fibering") coisotropic submanifold, and $φ:M\to M$ a Hamiltonian diffeomorphism. The main result of this article is that the number of leafwise fixed points of $φ$ is bounded below by the sum of the $Z_2$-Betti numbers o…

Develops analysis of Hölder continuous mappings on Heisenberg groups.

New theorem shows embedding impossibility for certain 3-manifolds.

In the present paper we give a geometric proof for the existence of cylinders with constant mean curvature $H>H(X)$ in certain simply connected homogeneous three-manifolds $X$ diffeomorphic to $\mathbb{R}^3$, which always admit a Lie group structure. Here, $H(X)$ denotes the critical value for which constant mean curva…

Study tied links in various 3-manifolds, introducing new groups and proving theorems.

Volume gaps for minimal submanifolds in spheres are proven.

Let M^3 be a closed CR 3-manifold. In this paper we derive a Bochner formula for the Kohn Laplacian in which the pseudo-hermitian torsion plays no role. By means of this formula we show that the non-zero eigenvalues of the Kohn Laplacian are bounded below by a positive constant provided the CR Paneitz operator is non-n…

The paper constructs surfaces with constant mean curvature in a specific space and explores their properties.

The paper proves the existence of hypersurfaces with prescribed mean curvature.

For $0<k<1$, a finite-type $k$-surface in $3$-dimensional hyperbolic space is a complete, immersed surface of finite area and of constant extrinsic curvature equal to $k$. In [32], we showed that such surfaces have finite genus and finitely many cusp-like ends. Each of these cusps is asymptotic to an immersed cylinder …