arXiv research

We show that immersed minimal surfaces of $\mathbb{R}^{3}$ with bounded curvature and proper self intersections are proper. We also show that the restriction of the immersing map to a wide component is always proper. When the immersing map is injective the whole surface is a wide component. Prior to these results it wa…

We show that many 3-manifold groups have no nonabelian surface subgroups. For example, any link of an isolated complex surface singularity has this property. In fact, we determine the exact class of closed graph-manifolds which have no immersed pi_1-injective surface of negative Euler characteristic. We also determine …

Among other things, we prove the following two topologcal statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by an oriented connected closed $π_1$-injectively immersed quasi-Fuchsian subsurface. Second…

In this article we introduce conformal Riemannian morphisms. The idea of conformal Riemannian morphism generalizes the notions of an isometric immersion, a Riemannian submersion, an isometry, a Riemannian map and a conformal Riemannian map. We show that every injective conformal Riemannian morphism is an injective conf…

The paper bounds radii and curvatures in Riemannian manifolds.

We study a recent general criterion for the injectivity of the conformal immersion of a Riemannian manifold into higher dimensional Euclidean space, and show how it gives rise to important conditions for Weierstrass-Ennerper lifts defined in the unit disk $\mathbb{D}$ endowed with a conformal metric. Among the corollar…

Any generalized distance-squared mapping of equidimensional case has singularities, and their singularity types are wrapped into mystery in higher dimensional cases. Any generalized distance-squared mapping of equidimensional case is not injective. Nevertheless, in this paper, it is shown that the non-singular property…

Let M be a closed hyperbolic three manifold. We construct closed surfaces which map by immersions into M so that for each one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding induced mapping on fundamental groups is an injection.

We give an easy example showing that sections of a singular Riemannian foliation on a simply connected space neither have to be isometric nor injectively immersed.

Let $N$ (resp., $U$) be a manifold (resp., an open subset of $\mathbb{R}^m$). Let $f:N\to U$ and $F:U\to \mathbb{R}^\ell$ be an immersion and a $C^{\infty}$ mapping, respectively. Generally, the composition $F\circ f$ does not necessarily yield a mapping transverse to a given subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$…

Let $M$ be a symplectic symmetric space, and let $\imath : M \to V$ be an extrinsic symplectic symmetric immersion, i.e., $(V, Ω)$ is a symplectic vector space and $\imath$ is an injective symplectic immersion such that for each point $p \in M$, the geodesic symmetry in $p$ is compatible with the reflection in the affi…

The simple loop conjecture for 3-manifolds states that every 2-sided immersion of a closed surface into a 3-manifold is either injective on fundamental groups or admits a compression. This can be viewed as a generalization of the Loop Theorem to immersed surfaces. We prove the conjecture in the case that the target 3-m…

We study the embedded Calabi-Yau problem for complete embedded constant mean curvature surfaces of finite topology or of positive injectivity radius in a simply-connected three-dimensional Lie group X endowed with a left-invariant Riemannian metric. We first prove a half-space theorem for constant mean curvature surfac…

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

In this paper we will prove the Calabi-Yau conjectures for embedded surfaces. In fact, we will prove considerably more. The Calabi-Yau conjectures about surfaces date back to the 1960s. Much work has been done on them over the past four decades. In particular, examples of Jorge-Xavier from 1980 and Nadirashvili from 19…

In this paper we develop a global correspondence between immersed horospherically convex hypersurfaces in hyperbolic space and complete conformal metrics on domains in the sphere. We establish results on when the hyperbolic Gauss map is injective and when an immersed horospherically convex hypersurface can be unfolded …

The paper extends Huber's theorem to higher dimensions using n-Laplace equations.

We show that there are Haken 3-manifolds whose fundamental groups do not satisfy the engulfing property. In particular one can construct a pi_1-injective immersion of a surface into a graph manifold which does not factor through any proper finite cover of the 3-manifold.

The paper studies a relative version of non-positive immersion for 2-complex pairs and shows conditions under which a transitivity law holds.

We consider properties of the total absolute geodesic curvature functional on circle immersions into a Riemann surface. In particular, we study its behavior under regular homotopies, its infima in regular homotopy classes, and the homotopy types of spaces of its local minima. We consider properties of the total curvatu…

A translation structure on a surface is an atlas of charts to the plane so that the transition functions are translations. We allow our surfaces to be non-compact and infinite genus. We endow the space of all pointed surfaces equipped with a translation structure with a topology, which we call the immersive topology be…

We consider the following properties of compact oriented irreducible graph-manifolds: to contain a $π_1$-injective surface (immersed, virtually embedded or embedded), be (virtually) fibered over $S^1$, and to carry a metric of nonpositive sectional curvature. It turns out that all these properties can be described from…

Every nonflat conformal minimal surface is homotopic to a proper one.

The Bergman kernels of holomorphic vector bundles are studied to extend the Fubini-Study map.

We study complete finite topology immersed surfaces $Σ$ in complete Riemannian $3$-manifolds $N$ with sectional curvature $K_N\leq -a^2\leq 0$, such that the absolute mean curvature function of $Σ$ is bounded from above by $a$ and its injectivity radius function is not bounded away from zero on each of its annular end …

Study immersions and embeddings of manifolds with trivial normal bundles.

Let $g \colon S \looparrowright N$ be a properly immersed $π_1$--injective surface in a non-geometric $3$--manifold $N$. We compute the distortion of $π_1(S)$ in $π_1(N)$ and show that how it is related to separability of $π_1(S)$ in $π_1(N)$. The only possibility of the distortion is linear, quadratic, exponential, an…

Diffeological submanifolds are a new type of submanifold in manifold theory.

The Bieberbach estimate, a pivotal result in the classical theory of univalent functions, states that any injective holomorphic function $f$ on the open unit disc $D$ satisfies $|f"(0)|\leq 4 |f'(0)|$. We generalize the Bieberbach estimate by proving a version of the inequality that applies to all injective smooth conf…

We approximate the spectral data (eigenvalues and eigenfunctions) of compact Riemannian manifold by the spectral data of a sequence of (computable) discrete Laplace operators associated to some graphs immersed in the manifold. We give an upper bound on the error that depends on upper bounds on the diameter and the sect…

Paper extends cobordism maps in Khovanov and instanton homologies.

It is proved that if a Paley-Wiener family of eigenfunctions of the Laplace operator in $\mathbb R^3$ vanishes on a real analytically ruled two-dimensional surface $S \subset \mathbb R^3$ then $S$ is a union of cones, each of which is contained in a translate of the zero set of a nonzero harmonic homogeneous polynomial…

We define pointwise partial differential relations for holomorphic discs. Given a relative homotopy class, a relation, and a generic almost complex structure we provide the moduli space of discs which have an injective point with the structure of a smooth manifold. Applications to the local behaviour are given and an a…

Symplectic manifold rays can be removed without changing the manifold's structure.

Let N be a complete, homogeneously regular Riemannian manifold of dimension greater than 2 and let M be a compact submanifold of N. Let $Σ$ be a compact orientable surface with boundary. We show that for any continuous $f: (Σ, \partial Σ) \rightarrow (N, M)$ for which the induced homomorphism on certain fundamental gro…

In this paper, we give a complete characterization on which finitely generated subgroups of finitely generated $3$-manifold groups are separable. Our characterization generalizes Liu's spirality character on $π_1$-injective immersed surface subgroups of closed $3$-manifold groups. A consequence of our characterization …

This is an exposition of results on the existence problem of $π_1$-injective immersed and embedded surfaces in graph-manifolds, and also of nonpositively curved metrics on graph-manifolds, obtained by different authors. The results are represented from a unified point of view based on the notion of compatible cohomolog…

Avoids noncompact hypersurfaces from touching in evolving flows.

We show that a map with Hölder exponent bigger than $1/2$ from a quasi-convex metric space with vanishing first Lipschitz homology into the Sub-Riemannian Heisenberg group factors through a tree. In particular, if the domain contains a disk, such a map can't be injective. This gives an answer to a question of Gromov fo…

In this paper we construct a complete injective holomorphic immersion $\mathbb{C}\to\mathbb{C}^2$ whose image is dense in $\mathbb{C}^2$. The analogous result is obtained for any closed complex submanifold $X\subset \mathbb{C}^n$ for $n>1$ in place of $\mathbb{C}\subset\mathbb{C}^2$. We also show that, if $X$ intersect…

We study curvature functionals for immersed 2-spheres in non-compact, three-dimensional Riemannian manifold $(M,h)$ without boundary. First, under the assumption that $(M,h)$ is the euclidean 3-space endowed with a semi-perturbed metric with perturbation small in $C^1$ norm and of compact support, we prove that if ther…

Any closed, connected Riemannian manifold $M$ can be smoothly embedded by its Laplacian eigenfunction maps into $\mathbb{R}^m$ for some $m$. We call the smallest such $m$ the maximal embedding dimension of $M$. We show that the maximal embedding dimension of $M$ is bounded from above by a constant depending only on the…

Certain classes of 3-manifolds, following Thurston, give rise to a 'skinning map', a self-map of the Teichmüller space of the boundary. This paper examines the skinning map of a 3-manifold M, a genus-2 handlebody with two rank-1 cusps. We exploit an orientation-reversing isometry of M to conclude that the skinning map …

We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng's eigenvalue comparison theorem for the Laplace-Beltrami operator. For manifolds with bounded sectional curvature, we p…

In this paper, we consider the high order geometric flows of a submanifolds $M$ in a complete Riemannian manifold $N$ with $\dim(N)=\dim(M)+1=n+1$, which were introduced by Mantegazza in the case the ambient space is an Euclidean space, and extend some results due to Mantegazza to the present situation under some assum…

Injectivity of ReLU networks is characterized for generative models and inverse problems.

Classifies $π_1$-injective maps between non-compact surfaces.

Positive injectivity radius for manifolds with Lie structure at infinity.