Flat Normal Bundle Research Articles

Self-shrinkers for the mean curvature flow in arbitrary codimension

In this paper, we generalize Colding–Minicozzi’s recent results about codimension-1 self-shrinkers for the mean curvature flow to higher codimension. In particular, we prove that the sphere $${bf S}^{n}(\sqrt{2n})$$ is the only complete embedded connected $$F$$ -stable self-shrinker in $$\mathbf{R}^{n+k}$$ with $$\mathbf{H}\ne 0$$ , polynomial volume growth, flat normal bundle and bounded geometry. We also discuss some properties of symplectic self-shrinkers, proving that any complete symplectic self-shrinker in $$\mathbf{R}^4$$ with polynomial volume growth and bounded second fundamental form is a plane. As a corollary, we show that there is no finite time Type I singularity for symplectic mean curvature flow, which has been proved by Chen–Li using different method. We also study Lagrangian self-shrinkers and prove that for Lagrangian mean curvature flow, the blow-up limit of the singularity may be not $$F$$ -stable.

Conformally flat submanifolds in spheres and integrable systems

É. Cartan proved that conformally flat hypersurfaces in $S^{n+1}$ for $n>3$ have at most two distinct principal curvatures and locally envelop a one-parameter family of $(n-1)$-spheres. We prove that the Gauss-Codazzi equation for conformally flat hypersurfaces in $S^4$ is a soliton equation, and use a dressing action from soliton theory to construct geometric Ribaucour transforms of these hypersurfaces. We describe the moduli of these hypersurfaces in $S^4$ and their loop group symmetries. We also generalise these results to conformally flat $n$-immersions in $(2n-2)$-spheres with flat and non-degenerate normal bundle.

Contact properties of codimension 2 submanifolds with flat normal bundle

Given an immersed submanifold M^n\subset\mathbb{R}^{n+2} , we characterize the vanishing of the normal curvature R_D at a point p \in M in terms of the behaviour of the asymptotic directions and the curvature locus at p . We relate the affine properties of codimension 2 submanifolds with flat normal bundle with the conformal properties of hypersurfaces in Euclidean space. We also characterize the semiumbilical, hypespherical and conformally flat submanifolds of codimension 2 in terms of their curvature loci.

Rigidity of minimal submanifolds with flat normal bundle

Let Mn (n ≥ 3) be an n-dimensional complete immersed \( \frac{{n - 2}} {n} \)-super-stable minimal submanifold in an (n + p)-dimensional Euclidean space ℝn+p with flat normal bundle. We prove that if the second fundamental form of M satisfies some decay conditions, then M is an affine plane or a catenoid in some Euclidean subspace.

The push-out space of transnormal immersions

Here we study the push-out space of a transnormal immersion of an m-dimensional compact manifold M without boundary into the Euclidean (m + k)-space and show that for an r-transnormal immersion with flat normal bundle, the push-out space has at most r components and this number can be attained in some cases, and any two components are isometric to each other.

Minimal submanifolds with flat normal bundle

Let Mn (n ≤ 7) be an n-dimensional complete immersed super stable minimal submanifold in an (n + p)-dimensional Euclidean space Rn+p with flat normal bundle. We prove that if the second fundamental form A of M satisfies ∫M |A|3 < ∞, then M is an affine n-dimensional plane.

On a class of hypersurfaces in % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqipC0xg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Wqpe0x % c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-dir-f0-yqaqVeLsFr0-vr % 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 % uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab-jj8tbaa!456F! $$ \mathbb{S} $$ n × ℝ and ℍ n × ℝ

We give a complete description of all hypersurfaces of the product spaces \( \mathbb{S} \)n × ℝ and ℍn × ℝ that have flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces ℝn+2 ⊃ \( \mathbb{S} \)n × ℝ and \( \mathbb{L} \)n+2 ⊃ ℍn × ℝ. We prove that any such hypersurface in \( \mathbb{S} \)n × ℝ (respectively, ℍn × ℝ) can be constructed by means of a family of parallel hypersurfaces in \( \mathbb{S} \)n (respectively, ℍn) and a smooth function of one variable. Then we show that constant mean curvature hypersurfaces in this class correspond to an isoparametric family in the base space and a smooth function that is explicitly determined in terms of the mean curvature function of the isoparametric family. As another consequence of our general result, we classify the constant angle hypersurfaces of \( \mathbb{S} \)n × ℝ and ℍn × ℝ, that is, hypersurfaces with the property that its unit normal vector field makes a constant angle with the unit vector field spanning the second factor ℝ. This extends previous results by Dillen, Fastenakels, Van der Veken, Vrancken and Munteanu for surfaces in \( \mathbb{S} \)n × ℝ and ℍn × ℝ Our method also yields a classification of all Euclidean hypersurfaces with the property that the tangent component of a constant vector field in the ambient space is a principal direction, in particular of all Euclidean hypersurfaces whose unit normal vector field makes a constant angle with a fixed direction.

Isometric immersions of $${{\mathbb R}^2}$$ into $${{\mathbb R}^4}$$ and perturbation of Hopf tori

We produce a new general family of flat tori in \({{\mathbb R}^4}\) , the first one since Bianchi’s classical works in the nineteenth century. To construct these flat tori, obtained via small perturbation of certain Hopf tori in \({{\mathbb S}^3}\) , we first present a global description of all isometric immersions of \({{\mathbb R}^2}\) into \({{\mathbb R}^4}\) with flat normal bundle.

A LOOP GROUP FORMULATION FOR CONSTANT CURVATURE SUBMANIFOLDS OF PSEUDO-EUCLIDEAN SPACE

We give a loop group formulation for the problem of isometric immersions with flat normal bundle of a simply connected pseudo-Riemannian manifold $M_{c,r}^m$, of dimension $m$, constant sectional curvature $c \neq 0$, and signature $r$, into the pseudo-Euclidean space $\bf R_s^{m+k}$, of signature $s\geq r$. In fact these immersions are obtained canonically from the loop group maps corresponding to isometric immersions of the same manifold into a pseudo-Riemannian sphere or hyperbolic space $S_s^{m+k}$ or $H_s^{m+k}$, which have been known for some time. A simple formula is given for obtaining these immersions from those loop group maps.

Transformations between surfaces in $\mathbb{R}^4$ with flat normal and/or tangent bundles

We exhibit several transformations of surfaces M in \mathbb{R}^4 : a transformation of flat surfaces that gives surfaces with flat normal bundle (semiumbilical surfaces); and its inverse that from a semiumbilical surface obtains a flat surface; then a one-parameter family of transformations f on flat semiumbilical immersed surfaces (FSIS), such that df(T_pM) is totally orthogonal to T_pM, and that give FSIS. This family satisfies a Bianchi type of permutability property.

Submanifolds for which a lower bound of the Ricci curvature is achieved

Sharp estimates for the Ricci curvature of a submanifold Mn of an arbitrary Riemannian manifold Nn+p are established. It is shown that the equality in the lower estimate of the Ricci curvature of Mn in a space form Nn+p(c) is achieved only when Mn is quasiumbilical with a flat normal bundle. In the case when the codimension p satisfies 1 ≤ p ≤ n − 3, the only submanifolds in Nn+p(c) on which the Ricci curvature is minimal are the conformally flat ones with a flat normal bundle.

A LOOP GROUP FORMULATION FOR CONSTANT CURVATURE SUBMANIFOLDS OF PSEUDO-EUCLIDEAN SPACE

We give a loop group formulation for the problem of isometric immersions with flat normal bundle of a simply connected pseudo-Riemannian manifold $M_{c,r}^m$, of dimension $m$, constant sectional curvature $c \neq 0$, and signature $r$, into the pseudo-Euclidean space $\bf R_s^{m+k}$, of signature $s\geq r$. In fact these immersions are obtained canonically from the loop group maps corresponding to isometric immersions of the same manifold into a pseudo-Riemannian sphere or hyperbolic space $S_s^{m+k}$ or $H_s^{m+k}$, which have been known for some time. A simple formula is given for obtaining these immersions from those loop group maps.

The vectorial Ribaucour transformation for submanifolds and applications

In this paper we develop the vectorial Ribaucour transformation for Euclidean submanifolds. We prove a general decomposition theorem showing that under appropriate conditions the composition of two or more vectorial Ribaucour transformations is again a vectorial Ribaucour transformation. An immediate consequence of this result is the classical permutability of Ribaucour transformations. Our main application is to provide an explicit local construction of an arbitrary Euclidean n n -dimensional submanifold with flat normal bundle and codimension m m by means of a commuting family of m m Hessian matrices on an open subset of Euclidean space R n \mathbb {R}^n . Actually, this is a particular case of a more general result. Namely, we obtain a similar local construction of all Euclidean submanifolds carrying a parallel flat normal subbundle, in particular of all those that carry a parallel normal vector field. Finally, we describe all submanifolds carrying a Dupin principal curvature normal vector field with integrable conullity, a concept that has proven to be crucial in the study of reducibility of Dupin submanifolds.

Curved flats, pluriharmonic maps and constant curvature immersions into pseudo-Riemannian space forms

We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among different space forms and pseudo-Riemannian space forms. As a corollary, we obtain a non-immersibility theorem for spheres into certain pseudo-Riemannian spheres and hyperbolic spaces. The second aspect pursued is to clarify the relationship between the loop group formulation of isometric immersions of space forms and that of pluriharmonic maps into symmetric spaces. We show that the objects in the first class are, in the real analytic case, extended pluriharmonic maps into certain symmetric spaces which satisfy an extra reality condition along a totally real submanifold. We show how to construct such pluriharmonic maps for general symmetric spaces from curved flats, using a generalised DPW method.

Curvature estimates for graphs with prescribed mean curvature and flat normal bundle

We consider graphs $${{\Sigma^n \subset \mathbb{R}^m}}$$ with prescribed mean curvature and flat normal bundle. Using techniques of Schoen et al. (Acta Math 134:275–288, 1975) and Ecker and Huisken (Ann Inst H Poincaré Anal Non Linèaire 6:251–260, 1989), we derive the interior curvature estimate $$ {\sup_{\Sigma \cap B_R} |A|^2 \leq \frac{C}{R^2}} $$ up to dimension n ≤ 5, where C is a constant depending on natural geometric data of Σ only. This generalizes previous results of Smoczyk et al. (Calc Var Partial Differ Equs 2006) and Wang (Preprint, 2004) for minimal graphs with flat normal bundle.

Grassmann Geometries in Infinite Dimensional Homogeneous Spaces and an Application to Reflective Submanifolds

Let U be a real form of a complex semisimple Lie group, and (�, �) a pair of commuting involutions on U. This data corresponds to a reflective submanifold of a symmetric space, U/K. We define an associated integrable system, and describe how to produce solutions from curved flats. This gives many new examples of submanifolds as integrable systems. The solutions are shown to correspond to various special submani- folds, depending on which homogeneous space, U/L, one projects to. We apply the construction to a question which generalizes, to the context of reflective submanifolds of arbitrary symmetric spaces, the problem of isometric immersions of space forms with negative extrinsic curva- ture and flat normal bundle. For this problem, we prove that the only cases where local solutions exist are the previously known cases of space forms, in addition to our new example of constant curvature Lagrangian immersions into complex projective and complex hyperbolic spaces. We also prove non-existence of global solutions in the compact case. For other reflective submanifolds, lower dimensional solutions exist, and can be described in terms of Grassmann geometries. We consider one example in detail, associated to the group G2, obtaining a special class of surfaces in S 6 .

Isometric immersions of [formula omitted] into [formula omitted

We give a global Weierstrass representation for isometric immersions with flat normal bundle from domains of the Lorentz plane L 2 into L 4 . This representation relies on the choice of two holomorphic data on a Riemann surface, and the integration of a hyperbolic linear differential system. As applications, we give classification results and construct complete examples in explicit coordinates by exact integration of the differential system for some particular choices of the holomorphic data.

PSEUDO-PARALLEL SUBMANIFOLDS WITH FLAT NORMAL BUNDLE OF SPACE FORMS

We provide a complete local classification of pseudoparallel submanifolds with flat normal bundle of space forms, extending the classification by Dillen-Nolker for the semi-parallel case.

Bernstein type theorems with flat normal bundle

We prove Bernstein type theorems for minimal n-submanifolds in ℝn+p with flat normal bundle. Those are natural generalizations of the corresponding results of Ecker-Huisken and Schoen-Simon-Yau for minimal hypersurfaces.

Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies, and the associativity equations

We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is equivalent to describing all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. We prove that every such Hamiltonian operator (or the corresponding submanifold) specifies a pencil of compatible Poisson brackets, generates bihamiltonian integrable hierarchies of hydrodynamic type, and also defines a family of integrals in involution. We prove that there is a natural special class of such Hamiltonian operators (submanifolds) exactly described by the associativity equations of two-dimensional topological quantum field theory (the Witten-Dijkgraaf-Verlinde-Verlinde and Dubrovin equations). We show that each N-dimensional Frobenius manifold can locally be represented by a special flat N-dimensional submanifold with flat normal bundle in a 2N-dimensional pseudo-Euclidean space. This submanifold is uniquely determined up to motions.