arXiv research

It is proved, that if M is a connected, complete submanifold of a complex space form N and each geodesic of M lies in an 1-dimensional totally geodesic complex submanifold of N, then M is totally geodesic in N and is a real space form or a complex space form.

New method uses manifold learning to infer latent positions of 1D submanifolds in random dot product graphs.

The paper studies how certain submanifolds collapse to lower-dimensional ones.

We classify the biharmonic Legendre curves in a Sasakian space form, and obtain their explicit parametric equations in the $(2n+1)$-dimensional unit sphere endowed with the canonical and deformed Sasakian structures defined by Tanno. Then, composing with the flow of the Reeb vector field, we transform a biharmonic inte…

Galatius, Madsen, Tillmann and Weiss have identified the homotopy type of the classifying space of the cobordism category with objects (d-1)-dimensional manifolds embedded in R^\infty. In this paper we apply the techniques of spaces of manifolds, as developed by the author and Galatius, to identify the homotopy type of…

In this paper we prove a symmetry result on submanifolds of codimension one in a n + 1-dimensional space form, related to the geodesic distance function and to the normal curvature of some fixed vector field. As applications we will prove sphere characterization type theorems for Kahler manifolds endowed with a toric g…

The set N of all null geodesics of a globally hyperbolic (d+1)-dimensional spacetime (M,g) is naturally a smooth (2d-1)-dimensional contact manifold. The sky of an event is the subset of N defined by all null geodesics through that event, and is an embedded Legendrian submanifold of N diffeomorphic to a (d-1)-dimension…

In this paper we construct new examples of minimal Lagrangian submanifolds in the complex hyperbolic space with large symmetry groups, obtaining three 1-parameter families with cohomegeneity one. We characterize them as the only minimal Lagrangian submanifolds in CH^n foliated by umbilical hypersurfaces of Lagrangian s…

We establish a Cauchy type inequality for the geometric intersection number between two 1-dimensional submanifolds in a surface. Some of the basic results in Thurston's theory of measured laminations on surfaces are derived from the Cauchy inequality.

We use pinched smooth hyperbolization to show that every closed, nonpositively curved $n$-dimensional manifold $M$ can be embedded as a totally geodesic submanifold of a closed, nonpositively curved $(n+1)$-dimensional manifold $\hat{M}$ of geometric rank one.

We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measu…

Constructs special Lagrangian submanifolds in Calabi-Yau 3-folds.

The total diameter of a closed planar curve $C\subset R^2$ is the integral of its antipodal chord lengths. We show that this quantity is bounded below by twice the area of $C$. Furthermore, when $C$ is convex or centrally symmetric, the lower bound is twice as large. Both inequalities are sharp and the equality holds i…

Study shows no closed trapped submanifolds can be tangent to certain spacelike hypersurfaces.

The $n$-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the $(n+1)$-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2- planes and is a compact Hermitian symmetric space of rank 2. In this paper we study…

In this note, we investigate conformally flat submanifolds of Euclidean space with positive index of relative nullity. Let $M^n$ be a complete conformally flat manifold and let $f\colon M^n\to \R^m$ be an isometric immersion. We prove the following results: (1) If the index of relative nullity is at least two, then $M^…

Given an d-dimensional manifold with two commuting Killing vectors, together with an d - 1 dimensional submanifold in which one of the Killing vectors lies, then the lapse and shift of the second Killing vector, relative to this slice, remain constant along the orbits of the `surface' Killing vector. Alternatively, the…

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

Plateau's soap film problem is to find a surface of least area spanning a given boundary. We begin with a compact orientable $(n-2)$-dimensional submanifold $M$ of $\R^n$. If $M$ is connected, we say a compact set $X$ "spans" $M$ if $X$ intersects every Jordan curve whose linking number with $M$ is 1. Picture a soap fi…

In this paper, we prove a classification theorem for the stable compact minimal submanifolds of the Riemannian product of an $m_1$-dimensional ($m_1\geq3$) hypersurface $M_1$ in the Euclidean space and any Riemannian manifold $M_2$, when the sectional curvature $K_{M_1}$ of $M_1$ satisfies $\frac{1}{\sqrt{m_1-1}}\leq K…

Classifies horocycle flow closures in hyperbolic 3-manifolds.

On a Riemannian manifold $\bar{M}^{m+n}$ with an $(m+1)$-calibration $Ω$, we prove that an $m$-submanifold $M$ with constant mean curvature $H$ and calibrated extended tangent space $\mathbb{R}H\oplus TM$ is a critical point of the area functional for variations that preserve the enclosed $Ω$-volume. This recovers the …

A {\it wrinkled embedding} $f:V^n\to W^m$ is a topological embedding which is a smooth embedding everywhere on $V$ except a set of $(n-1)$-dimensional spheres, where $f$ has cuspidal corners. In this paper we prove that any rotation of the tangent plane field $TV\subset TW$ of a {\it smoothly embedded} submanifold $V\s…

Given a piecewise smooth submanifold $Γ^{n-1} \subset \R^m$ and $p \in \R^m$, we define the {\em vision angle} $Π_p(Γ)$ to be the $(n-1)$-dimensional volume of the radial projection of $Γ$ to the unit sphere centered at $p$. If $p$ is a point on a stationary $n$-rectifiable set $Σ\subset \R^m$ with boundary $Γ$, then w…

Paper defines linking numbers for periodic tangles.

We study forgetful maps between Deligne-Mostow moduli spaces of weighted points on P^1, and classify the forgetful maps that extend to a map of orbifolds between the stable completions. The cases where this happens include the Livné fibrations and the Mostow/Toledo maps between complex hyperbolic surfaces. They also in…

The following Theorem is proved: Let M be an n-dimensional (n>2) submanifold of a Riemannian manifold N. Suppose that through each point p of M there exist two (n-1)-dimensional extrinsic spheres of N, which are contained in M in a neighbourhood of p and are tangent to each other at p. Then M is totally geodesic in N o…

Study optimizes decay estimates for minimizing currents in submanifolds.

Formula for renormalized area of hypersurfaces in hyperbolic spaces.

Researchers extend topological classification to knotted semimetals in 3D.

Let $(\MM ,{\tilde g})$ be an $N$-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen-Cahn equation $$ ε^2Δ_{ {\tilde g}} {u}\,+\, (1 - {u}^2)u \,=\,0\quad \mbox{in } \MM, $$ where $ε$ is a small parameter. Let $\KK\subset \MM$ be an $(N-1)$-dimensional smooth minimal submanifold …

We prove the simultaneous (k,n-k)-systolic freedom, for a pair of adjacent integers k smaller than n/2, of a simply connected n-manifold X. Our construction, related to recent results of I. Babenko, is concentrated in a neighborhood of suitable k-dimensional submanifolds of X. We employ calibration by differential form…

We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call "Morse 2-functions", and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is "fiber-connected", and to avoid local extrema over 1-dimensional submanifolds of th…

Paper proves conditions for 3D submanifolds to embed in 4D space.

Let $Z \to Y^{2n+1}$ be the bundle of Legendrian $n$-planes over a contact manifold $Y$. We consider a foliation of $Z$ by canonical lifts of Legendrian submanifolds, called \emph{Legendrian submanifold path geometry}, whose flat model is \[ Sp(n+1, R) \to RP^{2n+1}. \] The equivalence problem provides an $sp(n+1, R)$ …

Perturbs area-minimizing hypersurfaces to reduce singular set's dimension.

We provide a measure based topology for certain unions of C2 rectifiable submanifolds of mixed dimensions in Rn. In this topology lower dimensional sets remain in the limit as measures when higher dimensional sets collapse down to them. For example a decreasing sequence of spheres may have a limit consisting of just a …

This paper gives some examples of hypersurfaces $φ_t(M^n)$ evolving in time with speed determined by functions of the normal curvatures in an $(n+1)$-dimensional hyperbolic manifold; we emphasize the case of flow by harmonic mean curvature. The examples converge to a totally geodesic submanifold of any dimension from 1…

In this short note, we hope to give a rapid induction for non-experts into the world of Differential Harnack inequalities, which have been so influential in geometric analysis and probability theory over the past few decades. At the coarsest level, these are often mysterious-looking inequalities that hold for `positive…

This work forms a foundational study of factorization homology, or topological chiral homology, at the generality of stratified spaces with tangential structures. Examples of such factorization homology theories include intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with c…

The Einstein/Abelian-Yang-Mills Equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities $\p\colon\R^3\smΣ\to\H^{k+1}_\C$ into the $(k+1)$-dimensional complex hyperbolic space. In this paper, we prove the existence and uniqueness of harmonic maps with prescribed sing…

The phase space of relativistic particle mechanics is defined as the 1st jet space of motions regarded as timelike 1-dimensional submanifolds of spacetime. A Lorentzian metric and an electromagnetic 2-form define naturally on the odd-dimensional phase space a generalized contact structure. In the paper infinitesimal sy…

This paper is concerned with basic geometric properties of the phase space of a classical general relativistic particle, regarded as the 1st jet space of motions, i.e. as the 1st jet space of timelike 1--dimensional submanifolds of spacetime. This setting allows us to skip constraints. Our main goal is to determine the…

We study the family $Ω^1(-1^s)$ of rational 1--forms on the Riemann sphere, having exactly $-s \leq -2$ simple poles. Three equivalent $(2s-1)$--dimensional complex atlases on $Ω^1(-1^s)$, using coefficients, zeros--poles and residues--poles of the 1--forms, are recognized. A rational 1--form is isochronous when all th…

We characterize those planar Peano continua that are homotopy equivalent to 1-dimensional sets. While many planar Peano continua are not homotopically 1-dimensional, we prove that each has fundamental group that embeds in the fundamental group of a 1-dimensional planar Peano continuum. We leave open the following quest…

Study computes Cheeger constants for specific submanifolds in asymptotically hyperbolic spaces.

This paper proves the manifold hypothesis for lower embedding dimensions using osculating hyperspheres.

The purpose of this paper is to study the validity of Stokes' Theorem for singular submanifolds and differential forms with singularities in Euclidean space. The results are presented in the context of Lebesgue Integration, but their proofs involve techniques from gauge integration in the spirit of R.~Henstock, J.~Kurz…