Submanifold Research Articles

Flexible and inflexible $CR$ submanifolds

In this paper we prove new embedding results for compactly supported deformations of $CR$ submanifolds of $\mathbb{C}^{n+d}$: We show that if $M$ is a $2$-pseudoconcave $CR$ submanifold of type $(n,d)$ in $\mathbb{C}^{n+d}$, then any compactly supported $CR$ deformation stays in the space of globally $CR$ embeddable in $\mathbb{C}^{n+d}$ manifolds. This improves an earlier result, where $M$ was assumed to be a quadratic $2$-pseudoconcave $CR$ submanifold of $\mathbb{C}^{n+d}$. We also give examples of weakly $2$-pseudoconcave $CR$ manifolds admitting compactly supported $CR$ deformations that are not even locally $CR$ embeddable.

Mean Curvature Flow of Submanifolds With Small Traceless Second Fundamental Form

Consider a family of smooth immersions F(; t) : Mn ! Rn+k of submanifolds in Rn+k moving by mean curvature ow @F @t = ~H, where ~H is the mean curvature vector for the evolving submanifold. We prove that for any n 2 and k 1, the ow starting from a closed submanifold with small L2-norm of the traceless second fundamental form contracts to a round point in nite time, and the corresponding normalized ow converges exponentially in the C1-topology, to an n-sphere in some subspace Rn+1 of Rn+k.

Ridegs and ravine on Chen sub manifolds in R^4

Ridegs and ravine on Chen sub manifolds in R^4

Some classification results on totally umbilical proper slant and hemi-slant submanifolds of a nearly Kenmotsu manifold

This paper is concerned with the study of slant and hemi-slant submanifolds of a nearly Kenmotsu manifold. We prove that every totally umbilical proper slant submanifold of a nearly Kenmotsu manifold is totally geodesic and derive the integrability conditions of involved distributions in the definition of a hemi-slant submanifold. Finally, we obtain a classification theorem on a totally umbilical hemi-slant submanifold of a nearly Kenmotsu manifold.

On a Theorem Due to Birkhoff

The manifold M being closed and connected, we prove that every submanifold of T*M that is Hamiltonianly isotopic to the zero-section and that is invariant by a Tonelli flow is a graph.

The quantum query complexity of elliptic PDE

The query complexity of the following numerical problem is studied in the quantum model of computation: consider a general elliptic partial differential equation of order 2m in a smooth, bounded domain Q ⊂ Rd with smooth coefficients and homogeneous boundary conditions. We seek to approximate the solution on a smooth submanifold M ⊆ Q of dimension 0 ≤ d1 ≤ d. With the right-hand side belonging to Cr (Q), and the error being measured in the L∞(M) norm, we prove that the nth minimal quantum error is (up to logarithmic factors) of order n-min((r+2m)/d1,r/d+1). For comparison, in the classical deterministic setting the nth minimal error is known to be of order n-r/d, for all d1, while in the classical randomized setting it is (up to logarithmic factors) n-min((r+2m)/d1,r/d+1/2).

Types of integrability on a submanifold and generalizations of Gordon’s theorem

Types of integrability on a submanifold and generalizations of Gordon’s theorem

A rigidity theorem for submanifolds inSn+p with constant scalar curvature

LetM1 be a closed submanifold isometrically immersed in a unit sphereSn+p. Denote byR, H andS, the normalized scalar curvature, the mean curvature, and the square of the length of the second fundamental form ofM1, respectively. SupposeR is constant and ≥1. We study the pinching problem onS and prove a rigidity theorem forM1 immersed inSn+p with parallel normalized mean curvature vector field. Whenn≥8 or,n=7 andp≤2, the pinching constant is best.

Ample vector bundles with zero loci having a bielliptic curve section

Let X be a smooth complex projective variety and let $Z \subset X$ be a smooth submanifold of dimension $\geq 2$, which is the zero locus of a section of an ample vector bundle $\mathcal{E}$ of rank dim $X$ - dim $Z \geq 2$ on $X$. Let $H$ be an ample line bundle on $X$ whose restriction $H_Z$ to $Z$ is very ample. Triplets $(X, \mathcal{E}, H)$ as above are studied and classified under the assumption that $Z$ is a projective manifold of high degree with respect to $H_Z$, dmitting a curve section which is a double cover of an elliptic curve.

Global geometric properties of AdS space and the AdS/CFT correspondence

The Poisson kernels and relations between them for a massive scaler field in a unit ball B-n with Hua's metric and conformal flat metric are obtained by describing the B-n as a submanifold of an (n + 1)-dimensional embedding space. Global geometric properties of the AdS space are discussed. We show that the (n + 1)-dimensional AdS space AdS(n+1) is isomorphic to RP1 x B-n and boundary of the AdS is isomorphic to RP1 x Sn-1. Bulk-boundary propagator and the AdS/CFT like correspondence are demonstrated based on these global geometric properties of the RP1 x B-n.

Minimal symmetric submani folds in compact Riemannian symmetric spaces

A necessary and sufficient condition for a symmetric submanifold to be minimal in its ambient compact Riemannian symmetric space is given. By this condition a complete classification of nonminimal symmetric submanifolds in compact symmetric spaces is obtained.

An integral formula for the heat kernel of tubular neighborhoods of complete (connected) Riemannian manifolds

Let M be a complete connected Riemannian manifold and let N be a submanifold of M. Let v: Ev»N be the normal bundle of N and expv: Ev»M its exponential map.

Microlocalisation of $\cal{D}$-modules along a submanifold

Soit X une variete analytique complexe. Dans [K-S1], Kashiwara et Schapira ont defini et etudie un bifoncteur μhom dans D b (X) qui generalise le foncteur de microlocalisation de [SKK]. A peu pres au meme moment, Kashiwara et Kawai on introduit dans [K-K3] un bifoncteur dans la categorie des systemes holonomes reguliers. Si l'on se donne un couple (M,N) de tels systemes, ce foncteur consiste a microlocaliser le produit formel de M et N le long de la diagonale de X x X, consideree comme un D T*X -module. Le but principal de cet article est de mettre en rapport les fonctorialites de la specialisation et de la microlocalisation; nous montrons en particulier que le bifoncteur de [K-K3], que nous notons μhom * , est l'analogue du bifoncteur de [K-S1] via le foncteur de De Rham (Theoreme 3.2) dans le cadre des D-modules. Nous n'exigeons pas que M et N soient holonomes reguliers puisque la propriete essentielle de μhom * (M,N) est la regularite du produit formel de M et N le long de Δ.

On the mean exit time from a minimal submanifold

Soit M m une sous variete minimale immergee d'une variete de Riemann N n et on considere le mouvement brownien sur une boule reguliere Ω R CM de rayon exterieur R. Le temps de sortie moyen pour le mouvement a partir d'un point xeΩ R est appele E R (x). On trouve des fonctions support pointues pour E R (x) dans plusieurs cas distincts

Condition of holonomicity of characteristic directions of a submanifold

Condition of holonomicity of characteristic directions of a submanifold

A submanifold which contains many extrinsic circles

A submanifold which contains many extrinsic circles

A Note on Hilbert $C^*$-Modules Associated with a Foliation

in a foliatedmanifold M, and then reduced the stability of foliation C*-algebrasto that of Hilbert C*-modules ([5] Th. 2). In the course cf thisreduction, they proved the relation, Jf (E$) = C* (G{£), with T a faith-ful transversal submanifold (Jf (£?) denotes the C*-algebra cf 'com-pact' operators in Ep

A criterion for a submanifold to be quasiumbilical

A criterion for a submanifold to be quasiumbilical

Certain properties of a submanifold in a sphere

Certain properties of a submanifold in a sphere

The normal singularities of a submanifold

The normal singularities of a submanifold