Given a calibration $α$ whose stabilizer acts transitively on the Grassmanian of calibrated planes, we introduce a nontrivial Lie-theoretic condition on $α$, which we call compliancy, and show that this condition holds for many interesting geometric calibrations, including Kähler, special Lagrangian, associative, coassociative, and Cayley. We determine a sufficient condition that ensures compliancy of $α$, we completely characterize compliancy in terms of properties of a natural involution determined by a calibrated plane, and we relate compliancy to the geometry of the calibrated Grassmanian.
The condition that a Riemannian immersion $ι\colon L \to M$ be calibrated is a first order condition. By contrast, its extrinsic geometry, given by the second fundamental form $A$ and the induced tangent and normal connections $\nabla$ on $TL$ and $D$ on $NL$, respectively, is second order information. We characterize the conditions imposed on the extrinsic geometric data $(A, \nabla, D)$ when the Riemannian immersion $ι\colon L \to M$ is calibrated with respect to a calibration $α$ on $M$ which is both parallel and compliant. This motivate the definition of an infinitesimally calibrated Riemannian immersion, generalizing the classical notion of a superminimal surface in $\mathbb R^4$.
We prove that complete submanifolds, on which the Omori-Yau weak maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower the codimension is, the richer such points are. The smaller the radius is, the larger such curvatures are. This work unifies and generalizes several previous results on submanifolds with nonpositive extrinsic curvature.
We prove Reilly-type upper bounds for divergence-type operators of the second order as well as for Steklov problems on submanifolds of Riemannian manifolds of bounded sectional curvature endowed with a weighted measure.
We establish some inequalities of Chen's type between certain intrinsic invariants (involving sectional, Ricci and scalar curvatures) and the squared mean curvature of submanifolds tangent to the structure vector fields of a generalized S-space-form and we discuss the equality cases of them. We apply the obtained results to slant submanifolds.
We provide a parametric construction in terms of minimal surfaces of the Euclidean submanifolds of codimension two and arbitrary dimension that attain equality in an inequality due to De Smet, Dillen, Verstraelen and Vrancken. The latter involves the scalar curvature, the norm of the normal curvature tensor and the length of the mean curvature vector.
We study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the comparison of those volumes and capacities with the corresponding entities in a rotationally symmetric model manifold. Using the asymptotic behavior of the volumes and capacities we then obtain upper bounds for the number of ends as well as estimates for the fundamental tone of the submanifolds in question.
We study, from the extrinsic point of view, the structure at infinity of open submanifolds isometrically immersed in the real space forms of constant sectional curvature $κ\leq 0$. We shall use the decay of the second fundamental form of the the so-called tamed immersions to obtain a description at infinity of the submanifold in the line of the structural results in the papers Internat. Math. Res. Notices 1994, no. 9, authored by R. E. Greene, P. Petersen and S. Zhou and Math. Ann. 2001, 321 (4), authored by A. Petrunin and W. Tuschmann. We shall obtain too an estimation from below of the numb…
From the basic geometry of submanifolds will be recalled what are the extrinsic principal tangential directions, (first studied by Camille Jordan in the $18$seventies), and what are the principal first normal directions, (first studied by Kostadin Trencevski in the $19$nineties), and what are their corresponding Casorati curvatures. For reasons of simplicity of exposition only, hereafter this will merely be done explicitly in the case of arbitrary submanifolds in Euclidean spaces. Then, for the special case of Lagrangian submanifolds in complex Euclidean spaces, the natural relationships betwe…
We consider parallel submanifolds $M$ of a Riemannian symmetric space $N$ and study the question whether $M$ is extrinsically homogeneous in $N$\,, i.e.\ whether there exists a subgroup of the isometry group of $N$ which acts transitively on $M$\,. First, given a "2-jet" $(W,b)$ at some point $p\in N$ (i.e. $W\subset T_pN$ is a linear space and $b:W\times W\to W^\bot$ is a symmetric bilinear form)\,, we derive necessary and sufficient conditions for the existence of a parallel submanifold with extrinsically homogeneous tangent holonomy bundle which passes through $p$ and whose 2-jet at $p$ is …
The aim of this paper is to obtain the fundamental tone for minimal submanifolds of the Euclidean or hyperbolic space under certain restrictions on the extrinsic curvature. We show some sufficient conditions on the norm of the second fundamental form that allow us to obtain the same upper and lower bound for the fundamental tone of minimal submanifolds in a Cartan-Hadamard ambient manifold. As an intrinsic result, we obtain a sufficient condition on the volume growth of a Cartan-Hadamard manifold to achieve the lowest bound for the fundamental tone given by McKean.
We find calibrated submanifolds in neck manifolds. Particularly, we obtain a calibrated submanifold in the Lagrangian self-expander constructed by Joyce, Lee and Tsui.
We review some results concerning the deformations of calibrated minimal submanifolds which occur in Riemannian manifolds with special holonomy. The calibrated submanifolds are assumed compact with a non-empty boundary which is constrained to move in a particular fixed submanifold. The results extend McLean's deformation theory previously developed for closed compact submanifolds.
We investigate parallel submanifolds of a Riemannian symmetric space $N$. The special case of a symmetric submanifold has been investigated by many authors before and is well understood. We observe that there is an intrinsic property of the second fundamental form which distinguishes full symmetric submanifolds from arbitrary full parallel submanifolds of $N$, usually called "1-fullness of $M$". Furthermore, for every parallel submanifold $M$ of $N$ we consider the pullback bundle $TN|M$ with its induced connection, which admits a distinguished parallel subbundle $osc M$, usually called the "s…
Due to its length and several inaccuracies, this article is no longer suggested for reading. Moreover, in the meantime certain results presented herein could be improved by the author in a non-trivial fashion. Instead, the reader is referred to the following two articles: The extrinsic holonomy Lie algebra of a parallel submanifold (arXiv:0904.2611), Extrinsic homogeneity of parallel submanifolds (arXiv:0904.2636).
We construct calibrated submanifolds of R^7 and R^8 by viewing them as total spaces of vector bundles and taking appropriate sub-bundles which are naturally defined using certain surfaces in R^4. We construct examples of associative and coassociative submanifolds of R^7 and of Cayley submanifolds of R^8. This construction is a generalization of the Harvey-Lawson bundle construction of special Lagrangian submanifolds of R^{2n}.
`Gluing' is a technique of constructing solutions to non-linear (elliptic) partial differential equations such as Yang--Mills equations, minimal surface equations and Einstein equations. Calibrated submanifolds are a certain class of minimal surfaces, and there are various examples of them constructed by the gluing technique. We have existence theorems in that sense, but there seems to have been no uniqueness theory for higher-dimensional ones such as special Lagrangian submanifolds, which we discuss in the present paper.
We use techniques based on the splitting tensor to explicitly integrate the Codazzi equation along the relative nullity distribution and express the second fundamental form in terms of the Jacobi tensor of the ambient space. This approach allows us to easily recover several important results in the literature on complete submanifolds with relative nullity of the sphere as well as derive new strong consequences in hyperbolic and Euclidean spaces. Among the consequences of our main theorem are results on submanifolds with sufficiently high index of relative nullity, submanifolds with nonpositive…
We provide an introduction to the theory of calibrated submanifolds through the key examples related with special holonomy. We focus on calibrated geometry in Calabi-Yau, G$_2$ and Spin(7) manifolds, and describe fundamental results and techniques in the field.
In this note, we prove lower and upper bounds for Dirac operators of submanifolds in certain ambient manifolds in terms of conformal and extrinsic quantities.
For an element $Ψ$ in the graded vector space $Ω^*(M, TM)$ of tangent bundle valued forms on a smooth manifold $M$, a $Ψ$-submanifold is defined as a submanifold $N$ of $M$ such that $Ψ_{|N} \in Ω^*(N, TN)$. The class of $Ψ$-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups in compact Lie groups. The graded vector space $Ω^*(M, TM)$ carries a natural graded Lie algebra structure, given by the Frölicher-Nijenhuis bracket $[-,- ]^{FN}$. When $Ψ$ is an odd degree element with $[ Ψ, Ψ]^{FN} =0$, we associate to a $Ψ$-submanifold $N$ a strongly homotopy Li…
The principal theory of this paper comprises a technique for constructing associative, coassociative and Cayley submanifolds of Euclidean space with symmetries, using first-order ordinary differential equations. Explicit examples of U(1)-invariant associative cones in R^7 and SU(2)-invariant Cayley 4-folds in R^8 are then produced using this method. Further examples of associative 3-folds are presented, which are ruled, and other systems of differential equations defining calibrated submanifolds in R^7 and R^8 are given.
We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the intrinsic unit normals at the boundaries of the extrinsic spheres, respectively. In the same vein we also establish lower bounds on the mean exit time for Brownian motion in the extrinsic balls. In those cases, where we may extend our analysis to hold all the way to infinity, we…
Given a complete isometric immersion $ϕ: P^m \longrightarrow N^n$ in an ambient Riemannian manifold $N^n$ with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space $M^n_w$, we determine a set of conditions on the extrinsic curvatures of $P$ that guarantees that the immersion is proper and that $P$ has finite topology, in the line of the paper "On Submanifolds With Tamed Second Fundamental Form", (Glasgow Mathematical Journal, 51, 2009), authored by G. Pacelli Bessa and M. Silvana Costa. When the ambient ma…
On real hypersurfaces in complex space forms many results are proven. In this paper we generalize some results concerning extrinsic geometry of real hypersurfaces, to CR submanifolds of maximal CR dimension in complex space forms.
For a sequence of immersed connected closed Hamiltonian stationary Lagrangian submaniolds in $\mathbb{C}^{n}$ with uniform bounds on their volumes and the total extrinsic curvatures, we prove that a subsequence converges either to a point or to a Hamiltonian stationary Lagrangian $n$-varifold locally uniformly in $C^{k}$ for any nonnegative integer $k$ away from a finite set of points, and the limit is Hamiltonian stationary in ${\mathbb{C}}^{n}$. We also obtain a theorem on extending Hamiltonian stationary Lagrangian submanifolds $L$ across a compact set $N$ of Hausdorff codimension at least …
We consider a unit normal vector field of (local) hyperfoliation on a given Riemannian manifold as a submanifold in the unit tangent bundle with Sasaki metric. We give an explicit expression of the second fundamental form for this submanifold and a rather simple condition its totally geodesic property in the case of a totally umbilic hyperfoliation. A corresponding example shows the non-triviality of this condition. In the 2-dimensional case, we give a complete description of Riemannian manifolds admitting a geodesic unit vector field with totally geodesic property.
We extend the results given by Colbois, Dryden and El Soufi on the relationships between the eigenvalues of the Laplacian and an extrinsic invariant called intersection index, in two directions. First, we replace this intersection index by invariants of the same nature which are stable under small perturbations. Second, we consider complex submanifolds of the complex projective space $\mathbb{C} P^N$ instead of submanifolds of $\mathbb{R}^N$ and we obtain an eigenvalue upper bound depending only on the dimension of the submanifold which is sharp for the first non-zero eigenvalue.
We give estimates on the intrinsic and the extrinsic curvature of manifolds that are isometrically immersed as cylindrically bounded submanifolds of warped products. We also address extensions of the results in the case of submanifolds of the total space of a Riemannian submersion.
We investigate the geometry and topology of submanifolds under a sharp pinching condition involving extrinsic invariants like the mean curvature and the length of the second fundamental form. Several homology vanishing results are given. Moreover, an integral bound is provided for the Bochner operator of compact Euclidean submanifolds in terms of the Betti numbers.
The famous Nash embedding theorem published in 1956 was aiming for the opportunity to use extrinsic help in the study of (intrinsic) Riemannian geometry, if Riemannian manifolds could be regarded as Riemannian submanifolds. However, this hope had not been materialized yet according to \cite{G}. The main reason for this was the lack of control of the extrinsic properties of the submanifolds by the known intrinsic invariants. In order to overcome such difficulties as well as to provide answers to an open question on minimal immersions, the first author introduced in the early 1990's new types of…
This paper is a continuation of math.DG/0408005. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of S^n by looking at the conormal bundle of appropriate submanifolds of S^n. We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey-Lawson for submanifolds in R^n in their pioneering paper. We also construct calibrated submanifolds in complete metrics with special holonomy G_2 and Spin(7) discovered by Bryant and Salamon on the total spaces of appropriate bun…
The famous Nash embedding theorem was aimed for in the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. However, as late as 1985 (see \cite{G}) this hope had not been materialized. The main reason for this is due to the lack of controls of the extrinsic properties of the submanifolds by the known intrinsic invariants. In order to overcome such difficulties as well as to provide answers to an open question on minimal immersions, we introduced in the early 1990's new types of Riemannian invariants, known …
An extrinsic representation of a Ricci flow on a differentiable n-manifold M is a family of submanifolds S(t), each smoothly embedded in R^{n+k}, evolving as a function of time t such that the metrics induced on the submanifolds S(t) by the ambient Euclidean metric yield the Ricci flow on M. When does such a representation exist? We formulate this question precisely and describe a new, comprehensive way of addressing it for surfaces of revolution in R^3. Of special interest is the Ricci flow on a toroidal surface of revolution, that is, a surface of revolution whose profile curve is an immerse…
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 or an extrinsic sphere of N.
A conformal structure on a manifold $M^n$ induces natural second order conformally invariant operators, called Möbius and Laplace structures, acting on specific weight bundles of $M$, provided that $n\ge 3$. By extending the notions of Möbius and Laplace structures to the case of surfaces and curves, we develop here the theory of extrinsic conformal geometry for submanifolds, find tensorial invariants of a conformal embedding, and use these invariants to characterize various forms of geodesic submanifolds.
We address the study of some curvature equations for distinguished submanifolds in para-Kähler geometry. We first observe that a para-complex submanifold of a para-Kähler manifold is minimal. Next we describe the extrinsic geometry of Lagrangian submanifolds in the para-complex Euclidean space D^n and discuss a number of examples, such as graphs and normal bundles. We also characterize those Lagrangian surfaces of D^2 which are minimal and have indefinite metric. Finally we describe the Lagrangian self-similar solutions of the Mean Curvature Flow which are SO(n)-equivariant.
In this paper we will investigate torus actions on complete manifolds with calibrations. For Calabi-Yau manifolds M^2n with a Hamiltonian structure-preserving k-torus action we show that any symplectic reduction has a natural holomorphic volume form. Moreover Special Lagrangian (SLag) submanifolds of the reduction lift to SLag submanifolds of M, invariant under the torus action. If k=n-1 and the first cohomology of M is trivial, then we prove that M is a fibration with generic fiber being a SLag submanifold. As an application we will see that crepant resolutions of singularities of a finite Ab…
We investigate minimal helix submanifolds of any dimension and codimension immersed in Euclidean space. Our main result proves that a ruled minimal helix submanifold is a cylinder. As an application we classify complex helix submanifolds of $\mathbb{C}^n$: They are extrinsic products with a complex line as a factor. The key tool is Corollary 1.3 which allows us to classify Riemannian foliations of open subsets of the Euclidean space with minimal leaves. Finally, we consider the case of a helix hypersurface with constant mean curvature and prove that it is either a cylinder or an open part of a…
Anisotropic calibrations, Fueter maps and mirror symmetry
Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）；Department of Mathematics, Osaka Metropolitan University（大阪公立大学数学系）；Department of Mathematics, University of Torino（都灵大学数学系）
Let $M$ be a submanifold of a Riemannian manifold $(N,g)$. $M$ induces a subbundle $O(M,N)$ of adapted frames over $M$ of the bundle of orthonormal frames $O(N)$. Riemannian metric $g$ induces natural metric on $O(N)$. We study the geometry of a submanifold $O(M,N)$ in $O(N)$. We characterize the horizontal distribution of $O(M,N)$ and state its correspondence with the horizontal lift in $O(N)$ induced by the Levi--Civita connection on $N$. In the case of extrinsic geometry, we show that minimality is equivalent to harmonicity of the Gauss map of the submanifold $M$ with deformed Riemannian me…
Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivial can be isometrically embedded as a coassociative submanifold in a G_2-manifold, even as the fixed locus of an anti-G_2 involution. These results, when coupled with McLean's analysis of the moduli spaces of such calibrated submanifolds, yield a plentiful supply of examples of c…
In this paper, the first Chen inequality is proved for general warped product submanifolds in Riemannian space forms, this inequality involves intrinsic invariants ($δ$-invariant and sectional curvature) controlled by an extrinsic one (the mean curvature vector), which provides an answer for Problem 1. As a geometric application, this inequality is applied to derive a necessary condition for the immersed submanifold to be minimal in Riemannian space forms, which presents a partial answer for the well-known problem proposed by S.S. Chern, Problem 2. For further research directions, we address a…
Let H be the hyperbolic space of dimension n+1. A geodesic foliation of H is given by a smooth unit vector field on H all of whose integral curves are geodesics. Each geodesic foliation of H determines an n-dimensional submanifold M of the 2n-dimensional manifold L of all the oriented geodesics of H (up to orientation preserving reparametrizations). The space L has a canonical split semi-Riemannian metric induced by the Killing form of the isometry group of H. Using a split special Lagrangian calibration, we study the volume maximization problem for a certain class of geometrically distinguish…
A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden-Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. Parallel submanifolds form an important class of Riemannian submanifolds since extrinsic invariants of a parallel submanifold do not vary from point to point. In this paper we provide a comprehensive survey on this important class of submanifolds.
Given a parallel calibration $ϕ\in Ω^p(M)$ on a Riemannian manifold $M$, I prove that the $ϕ$--critical submanifolds with nonzero critical value are minimal submanifolds. I also show that the $ϕ$--critical submanifolds are precisely the integral manifolds of a $\mathscr{C}^\infty(M)$--linear subspace $\sP \subset Ω^p(M)$. In particular, the calibrated submanifolds are necessarily integral submanifolds of the system. (Examples of parallel calibrations include the special Lagrangian calibration on Calabi-Yau manifolds, (co)associative calibrations on $G_2$--manifolds, and the Cayley calibration …
In this paper we investigate the geometry of Calibrated submanifolds and study relations between their moduli-space and geometry of the ambient manifold. In particular for a Calabi-Yau manifold we define Special Lagrangian submanifolds for any Kahler metric on it. We show that for a choice of Kahler metric the Borcea-Voisin threefold has a fibration structure with generic fiber being a Special Lagrangian torus. Moreover we construct a mirror to this fibration. Also for any closed G_2 form on a 7-manifold we study coassociative submanifolds and we show that one example of a G_2 manifold constru…
