arXiv research

Study submanifolds in hyperbolic space, focusing on their boundary and Laplace operator.

Upper bounds for essential spectrum of minimal submanifolds linked to volume growth.

The paper studies essential spectra of submanifolds in Euclidean spaces.

The submanifold quantum mechanics was opened by Jensen and Koppe (Ann. Phys. {\bf 63} (1971) 586-591) and has been studied for these three decades. This article gives its more algebraic definition and show what is the essential of the submanifold quantum mechanics from an algebraic viewpoint.

Minimal real Kähler submanifolds in codimension 6 are holomorphic.

The connected components of the zero set of any conformal vector field $v$, in a pseudo-Riemannian manifold $(M,g)$ of arbitrary signature, are of two types, which may be called `essential' and `nonessential'. The former consist of points at which $v$ is essential, that is, cannot be turned into a Killing field by a lo…

We give lower bounds for the fundamental tone of open sets in minimal submanifolds immersed into warped product spaces of type $N^n \times_f Q^q$, where $f \in C^\infty(N)$. We also study the essential spectrum of these minimal submanifolds.

We extend the classical theory of isothermic surfaces in conformal 3-space, due to Bour, Christoffel, Darboux, Bianchi and others, to the more general context of submanifolds of symmetric $R$-spaces with essentially no loss of integrable structure.

Holomorphic Riemannian geometry connects submanifolds in different spaces.

In this paper we prove the Conley conjecture and the almost existence theorem in a neighborhood of a closed nowhere coisotropic submanifold under certain natural assumptions on the ambient symplectic manifold. Essential to the proofs is a displacement principle for such submanifolds. Namely, we show that a topologicall…

It is proved that a germ of a real analytic CR map from a smooth real-analytic minimal CR manifold M to an essentially finite real-algebraic generic submanifold M' of P^N of the same CR-dimension extends as a holomorphic correspondence along M. Applications are given for pseudoconcave submanifolds of P^N.

Characterizes submanifolds with minimum ratio of diameter to focal radius.

Two regularization methods show similar results for Riesz energies of submanifolds.

Study sharp geometric and topological properties of pinched 4D submanifolds.

The braneworld theory appear with the purpose of solving the problem of the hierarchy of the fundamental interactions. The perspectives of the theory emerge as a new physics, for example, deviation of the law of Newton's gravity. One of the principles of the theory is to suppose that the braneworld is local submanifold…

We study isoparametric submanifolds of rank at least two in a separable Hilbert space, which are known to be homogeneous by a result of Heintze and Liu, and associate to such a submanifold M and a point x in M a canonical homogeneous structure (a certain bilinear map on the tangent space). We prove that the homogeneous…

Maps obstructing positive scalar curvature and inessentialness.

We show that two of the Bryant-Salamon G_2-manifolds have a simple topology ; homeomorphic to the complement of some submanifolds of the 7-dimensional sphere. In this connection, we show there exists a complete Ricci-flat (non-flat) metric on the complement of an m-dimensional sphere in an n-dimensional sphere for some…

We prove, under a certain boundedness condition at infinity on the $(\bar{X}^{\top}, \bar{X}^{\bot})$-component of the second fundamental form, the vanishing of the essential spectrum of a complete minimal $\bar{X}$-bounded and $\bar{X}$-properly immersed submanifold on a Riemannian manifold endowed with a strongly con…

This paper removes the log loss for eigenfunction restrictions to certain submanifolds.

We prove a normal form theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds), which simultaneously generalizes Weinstein's symplectic neighborhood theorem from symplectic geometry and Weinstein's splitting theorem. Our approach turns out to be essentially canonical, and as a…

One of the most useful tools for studying the geometry of the mapping class group has been the subsurface projections of Masur and Minsky. Here we propose an analogue for the study of the geometry of Out(F_n) called submanifold projection. We use the doubled handlebody M_n = #^n S^2 \times S^1 as a geometric model of F…

New methods prove non-squeezing in locally conformal symplectic geometry.

Paper introduces an invariant to distinguish handlebody-knot exteriors.

We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of $\mathbb{R}^{3}$ is discrete. This give…

The paper proves a rigidity theorem for minimal submanifolds in spheres with flat normal bundle.

Haken n-manifolds are aspherical manifolds, defined and studied by B. Foozwell and H. Rubinstein, that can be successively cut open along essential codimension-one submanifolds until a disjoint union of n-cells is obtained. Such manifolds come equipped with a boundary pattern, a particular kind of decomposition of the …

On a real analytic 5-dimensional CR-generic submanifold M^5 in C^4 of codimension 3, hence of CR dimension 1, which enjoys the generically satisfied nondegeneracy condition that Lie brackets up to length 3 of T^{1,0}M generate CTM, a canonical Cartan connection is constructed after reduction to a certain partially expl…

A 3D area-minimizing current in R^5 has a 2-fold essential singularity.

The study examines properties of projection onto differentiable manifolds.

Floer theory for Lagrangians in open symplectic manifolds with smooth divisors.

Suppose M_t is a smooth family of compact connected two dimensional submanifolds of Euclidean space E^3 without boundary varying isometrically in their induced Riemannian metrics. Then we show that the mean curvature integrals over M_t are constant. It is unknown whether there are nontrivial such bendings. The estimate…

Survey of rigidity and gap phenomena in sphere-ball submanifolds.

New examples show non-commensurable 3-manifolds with similar geodesic lengths.

We study the symplectic topology of some finite algebraic quotients of the An Milnor fibre which are diffeomorphic to the rational homology balls that appear in Fintushel and Stern's rational blowdown construction. We prove that these affine surfaces have no closed exact Lagrangian submanifolds by using the already ava…

New calculus for pseudodifferential operators on manifolds with cylindrical ends.

Constructs odd Euler characteristic 4-manifolds.

Framework for reconstructing knot exterior surface complexes.

The paper studies curvatures and austere properties of orbits in symmetric spaces.

The paper studies symplectic geometry of cotangent bundles using sheaves theory.

Let M be a compact, orientable, irreducible, atoroidal 3-manifold with boundary an incompressible torus. Techniques based on the characteristic submanifold theory are used to bound the intersection number of two slopes αand βon the boundary of M. The method applies when βis the boundary slope of an essential surface F …

This paper surveys parallel submanifolds in Riemannian and pseudo-Riemannian manifolds.

The study characterizes submanifolds in product spaces.

Researchers describe a specific type of submanifolds in Euclidean space.

Study on hemi-slant submanifolds in nearly Kaehler manifolds.

The paper finds many infinite-dimensional weakly reflective PF submanifolds in Hilbert spaces.

We introduce the notions of pointwise almost h-slant submanifolds and pointwise almost h-semi-slant submanifolds as a generalization of slant submanifolds, pointwise slant submanifolds, semi-slant submanifolds, and pointwise semi-slant submanifolds. We have characterizations and investigate the integrability of distrib…

Study of warped product pointwise bi-slant submanifolds in Kaehler manifolds.