Volume of submanifolds

We propose a geometric definition of the volume of submanifolds using tubular neighborhoods.     

Submanifolds of Weyl Flat Manifolds

 We consider compact Weyl submanifolds of Weyl flat manifolds with special attention on compact Einstein-Weyl hypersurfaces. In particular, in the last part of the paper, we study Weyl submanifolds of special noncompact manifolds, called PC-manifolds.     

Some slant submanifolds of <Emphasis Type="Italic">S</Emphasis>-manifolds

Summary We study some special types of slant submanifolds of S-manifolds related to the second fundamental form of the immersion: totally f-geodesic and f-umbilical, pseudo-umbilical and austere submanifolds. We also give several examples of such submanifolds.     

Submanifolds of Weyl Flat Manifolds

 We consider compact Weyl submanifolds of Weyl flat manifolds with special attention on compact Einstein-Weyl hypersurfaces. In particular, in the last part of the paper, we study Weyl submanifolds of special noncompact manifolds, called PC-manifolds. Received July 16, 2001; in revised form February 6, 2002 Published online August 9, 2002     

Weighted area-minimizing submanifolds with soap-film-like singularities assembled from special Lagrangian pieces

A suffcient condition for a set of calibrated submanifolds to be area-minimizing with multiplicities,also call weighted area-minimizing under diffeomorphisms (WAMD) is stated.We construct some WAMD submanifolds by assembling pieces of special Lagrangian (SL) normal bundles including the one of three surfaces meeting at an angle of 120° along soap-film-like singularities.We also mention a symmetry property of SL submanifolds and Bjrling type problem for SL normal bundles.     

Calibrated Subbundles in Noncompact Manifolds of Special Holonomy

This paper is a continuation of Math. Res. Lett. 12 (2005), 493–512. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of Sⁿ by looking at the conormal bundle of appropriate submanifolds of Sⁿ. 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 Rⁿ in their pioneering paper, Acta Math. 148 (1982), 47–157. We also construct calibrated submanifolds in complete metrics with special holonomy G₂ and Spin(7) discovered by Bryant and Salamon (Duke Math. J. 58 (1989), 829–850) on the total spaces of appropriate bundles over self-dual Einstein four manifolds. The submanifolds are constructed as certain subbundles over immersed surfaces. We show that this construction requires the surface to be minimal in the associative and Cayley cases, and to be (properly oriented) real isotropic in the coassociative case. We also make some remarks about using these constructions as a possible local model for the intersection of compact calibrated submanifolds in a compact manifold with special holonomy. Mathematics Subject Classification (2000): 53-XX, 58-XX.     

ON SPACELIKE AUSTERE SUBMANIFOLDS IN PSEUDO-EUCLIDEAN SPACE

In this article, we construct some spacelike austere submanifolds in pseduoEuclidean spaces. We also get some indefinite special Lagrangian submanifolds by constructing twisted normal bundle of spacelike austere submanifolds in pseduo-Euclidean spaces.     

Submanifolds weakly associated with graphs

We establish an interesting link between differential geometry and graph theory by defining submanifolds weakly associated with graphs. We prove that, in a local sense, every submanifold satisfies such an association, and other general results. Finally, we study submanifolds associated with graphs either in low dimensions or belonging to some special families.     

黎曼流形在正交联络下的全脐点子流形

本文研究了正交联络下子流形基本方程以及在全脐点子流形中的应用.利用Cartan的方法将挠率张量分解成三个部分,计算得到正交联络下的三个基本方程,并考虑一个特殊的正交联络,证明了其黎曼曲率会有类似于Levi-Civita联络下的性质.利用基本方程得到常曲率空间中的全脐点子流形的性质,推广了Levi-Civita联络下的相应结果.     

Complete minimal submanifolds of compact Lie groups

We give a new method for manufacturing complete minimal submanifolds of compact Lie groups and their homogeneous quotient spaces. For this we make use of harmonic morphisms and basic representation theory of Lie groups. We then employ our method to construct many examples of compact minimal submanifolds of the special unitary groups.     

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.     

Minimal Lagrangian submanifolds of the complex hyperquadric

We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures.In particular,we define local angle functions encoding the geometry of the Lagrangian submanifold at hand.We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface.We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all local angle functions,respectively all but one,coincide.     

Geometric Aspects of Higher Order Variational Principles on Submanifolds

The geometry of jets of submanifolds is studied, with special interest in the relationship with the calculus of variations. A new intrinsic geometric formulation of the variational problem on jets of submanifolds is given. Working examples are provided.      

Local calibration of mass and systolic geometry

We prove the simultaneous (k, n -- k)-systolic freedom, for a pair of adjacent integers k < 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 forms supported in such neighborhoods, to provide lower bounds for the (n -- k)-systoles. Meanwhile, the k-systoles are controlled from below by the monotonicity formula combined with the bounded geometry of the construction in a neighborhood of suitable (n -- k + 1)-dimensional submanifolds, in spite of the vanishing of the global injectivity radius. The construction is geometric, with the algebraic topology ingredient reduced to Poincaré duality and Thom's theorem on representing multiples of homology classes by submanifolds. The present result is di.erent from the proof, in collaboration with A. Suciu, and relying on rational homotopy theory, of the k-systolic freedom of X. Our results concerning systolic freedom contrast with the existence of stable systolic inequalities, studied in joint work with V. Bangert.     

On the Mean Exit Time for Compact Symmetric Spaces

Given a compact symmetric space, M, we obtain the mean exit time function from a principal orbit, for a Brownian particle starting and moving in a generalized ball whose boundary is the principal orbit. We also obtain the mean exit time flmction of a tube of radius r around special totally geodesic submanifolds P of M. Finally we give a comparison result for the mean exit time function of tubes around submanifolds in Riemannian manifolds, using these totally geodesic submanifolds in compact symmetric spaces as a model.     

Energy of Radial Vector Fields on Compact Rank One Symmetric Spaces

We consider the energy (or the total bending) of unit vector fields oncompact Riemannian manifolds for which the set of its singularitiesconsists of a finite number of isolated points and a finite number ofpairwise disjoint closed submanifolds. We determine lower bounds for theenergy of such vector fields on general compact Riemannian manifolds andin particular on compact rank one symmetric spaces. For this last classof spaces, we compute explicit expressions for the total bending whenthe unit vector field is the gradient field of the distance function toa point or to special totally geodesic submanifolds (i.e., for radialunit vector fields around this point or these submanifolds).     

Some families of special Lagrangian tori

 We give an explicit proof of the local version of Bryant's result [1], stating that any 3-dimensional real-analytic Riemannian manifold can be isometrically embedded as a special Lagrangian submanifold in a Calabi-Yau manifold. We then refine the theorem proving that a certain class of real-analytic one-parameter families of metrics on a 3-torus can be isometrically embedded in a Calabi-Yau manifold as a one-parameter family of special Lagrangian submanifolds. Two applications of these results show how the geometry of the moduli space of 3-dimesional special Lagrangian submanifolds differs considerably from the 2-dimensional one. First of all, applying Bryant's theorem and a construction due to Calabi we show that nearby elements of the local moduli space of a special Lagrangian 3-torus can intersect themselves. Secondly, we use our examples of one-parameter families to show that in dimension three (and higher) the moduli space of special Lagrangian tori is not, in general, special Lagrangian in the sense of Hitchin [13]. Received: 18 December 2001 / Revised version: 31 January 2002 / Published online: 16 October 2002 Mathematics Subject Classification (2000): 53-XX, 53C38     

Graphs associated with vector spaces of even dimension: A link with differential geometry

We define a new association between graphs and orthonormal bases of even-dimensional Euclidean vector spaces endowed with an special isomorphism motivated by the recently introduced theory of submanifolds associated with graphs. We provide several interesting examples and we analyze the shape of such graphs by proving some general results.     

Construction of Lagrangian Submanifolds in Complex Hyperquadric

In this paper, the authors present a method to construct the minimal and ${\rm H}$-minimal Lagrangian submanifolds in complex hyperquadric $Q_n$ from submanifolds with special properties in odd-dimensional spheres. The authors also provide some detailed examples.     

Simultaneous desingularizations of Calabi–Yau and special Lagrangian 3-folds with conical singularities: I. The gluing construction

In this paper we extend our previous results on resolving conically singular Calabi–Yau 3-folds (Chan, Quart. J. Math. 57:151–181, 2006; Quart. J. Math., to appear) to include the desingularizations of special Lagrangian (SL) 3-folds with conical singularities that occur at the same points of the ambient Calabi–Yau. The gluing construction of the SL 3-folds is achieved by applying Joyce’s analytic result (Joyce, Ann. Global. Anal. Geom. 26: 1–58, 2004, Thm. 5.3) on deforming Lagrangian submanifolds to nearby special Lagrangian submanifolds. Our result will in principle be able to construct more examples of compact SL submanifolds in compact Calabi–Yau manifolds. Various explicit examples and applications illustrating the result in this paper can be found in the sequel (Chan, Ann. Global. Anal. Geom., to appear).