Fractal boundaries are not typical

Let M be a C¹n-dimensional compact submanifold of Rn. The boundary of M, ∂M, is itself a C¹ compact (n−1)-dimensional submanifold of Rn. A carefully chosen set of deformations of ∂M defines a complete subspace consisting of boundaries of compact n-dimensional submanifolds of Rn, thus the Baire Category Theorem applies to the subspace. For the typical boundary element ∂W in this space, it is the case that ∂W is simultaneously nowhere-differentiable and of Hausdorff dimension n−1.     

Totally geodesic submanifolds of the complex quadric

In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Riemannian symmetric spaces; this is exemplified by the classification of the totally geodesic submanifolds in the complex quadric Qm:=SO(m+2)/(SO(2)×SO(m)) obtained in the second part of the article. The classification shows that the earlier classification of totally geodesic submanifolds of Qm by Chen and Nagano (see [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745-755]) is incomplete. More specifically, two types of totally geodesic submanifolds of Qm are missing from [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745-755]: The first type is constituted by manifolds isometric to CP¹×RP¹; their existence follows from the fact that Q² is (via the Segre embedding) holomorphically isometric to CP¹×CP¹. The second type consists of 2-spheres of radius which are neither complex nor totally real in Qm.     

Minimal Submanifolds in Sp+3 with Constant Scalar Curvature

Let M be a compact, minimal 3-dimensional submanifold with constant scalar curvature R immersed in the standard sphere S^3+p. In codimension 1, we know from the work that has been done on Chern’s conjecture that M is isoparametric and R = 3D0, R = 3D3 or R = 3D6. In this paper we extend this result from codimension one to compact submanifolds with a flat normal bundle and give a complete classification.     

Embedding up to homotopy type— The first obstruction

For a 2n?m connected map from an n-dimensional complex to a m-dimensional manifold, an obstruction to embedding up to homotopy type is defined. The vanishing of this obstruction is a necessary and sufficient condition (in the 2n?m connected case, 2n?m ? 2, m?n ?3) to obtain an embedding up to homotopy type. In case the target manifold is Euclidean space, it is shown that the obstruction vanishes if and only if certain Thom operations are trivial. A classification theorem is given in the 2n?m+1 connected case.     

Topological change in mean convex mean curvature flow

Consider the mean curvature flow of an (n+1)-dimensional compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the mth homotopy group of the complementary region can die only if there is a shrinking S ^k ×R ^n?k singularity for some k≤m. We also prove that for each m with 1≤m≤n, there is a nonempty open set of compact, mean convex regions K in R ⁿ⁺¹ with smooth boundary ?K for which the resulting mean curvature flow has a shrinking S ^m ×R ^n?m singularity.     

Some pinching theorems for minimal submanifolds in \mathbb{S}^m (1) \times \mathbb{R}

Let Mn be an n-dimensional compact minimal submanifolds in Sm(1)×R.We prove two pinching theorems by the Ricci curvature and the sectional curvature pinching conditions respectively.In fact,we characterize the Clifford tori and Veronese submanifolds by our pinching conditions respectively.     

Submanifolds of positive Ricci curvature in a Euclidean space

In this paper we study the role of constant vector fields on a Euclidean space R ^n+p in shaping the geometry of its compact submanifolds. For an n-dimensional compact submanifold M of the Euclidean space R ^n+p with mean curvature vector field H and a constant vector field on R ^n+p , the smooth function is used to obtain a characterization of sphere among compact submanifolds of positive Ricci curvature (cf. main Theorem).      

Einige Bemerkungen über die Regularität von stationären Punkten gewisser geometrischer Variationsintegrale

We discuss the regularity properties of weak solutions of certain degenerate elliptic systems arising from problems in geometry. For example we prove C¹-regularity of conformal m-harmonic maps from the m-ball into m-dimensional oriented submanifolds of R^m+1 as well as a regularity theorem for weak H-surfaces in R^m+1. Our arguments make strong use of a version of the isoperimetric inequality.     

Complex vector measure and integral over manifolds with locally finite variations

It is well known that any compactly supported continuous complex differential n-form can be integrated over real n-dimensional C¹ manifolds in C^m (m ≥ n). For n = 1, the integral along any locally rectifiable curve is defined. Another generalization is the theory of currents (linear functionals on the space of compactly supported C^∞ differential forms). The topic of the article is the integration of measurable complex differential (n, 0)-forms (containing no \(d{\bar z_j}\)) over real n-dimensional C⁰ manifolds in C^m with locally finite n-dimensional variations (a generalization of locally rectifiable curves to dimensions n > 1). The last result is that a real n-dimensional manifold C¹ embedded in C^m has locally finite variations, and the integral of a measurable complex differential (n, 0)-form defined in the article can be calculated by a well-known formula.     

Gelfand pairs on the Heisenberg group and Schwartz functions

Let Hn be the (2n+1)-dimensional Heisenberg group and K a compact group of automorphisms of Hn such that (K?Hn,K) is a Gelfand pair. We prove that the Gelfand transform is a topological isomorphism between the space of K-invariant Schwartz functions on Hn and the space of Schwartz function on a closed subset of Rs homeomorphic to the Gelfand spectrum of the Banach algebra of K-invariant integrable functions on Hn.     

On subharmonic functions dominated by certain functions

Given two kinds of functionsf(X) andh(y) defined on them-dimensional Euclidean spaceR ^m (m≧1) and the set of positive real numbers respectively, we give an estimation of growth of subharmonic functionsu(P) defined onR ^m+n (n≧1) such that $$u(P) \leqq f\left( X \right)h\left( {\left\| Y \right\|} \right)$$ for anyP=(X, Y),X ∈R ^m, Y ∈R ⁿ, where ‖Y ‖ denotes the usual norm ofY. Using an obtained result, we give a sharpened form of an ordinary Phragmén-Lindelöf theorem with respect to the generalized cylinderD ×R ⁿ, with a bounded domainD inR ^m.     

Calibrations and the size of Grassmann faces

Summary In the past fifteen years or so, convex geometry and the theory of calibrations have provided a deeper understanding of the behavior and singular structure ofm-dimensional area-minimizing surfaces inR ⁿ . Calibrations correspond to faces of the GrassmannianG(m,R ⁿ ) of orientedm-planes inR ⁿ , viewed as a compact submanifold of the exterior algebra ^m R ⁿ . Large faces typically provide many examples of area-minimizing surfaces. This paper studies the sizes of such faces. It also considers integrands more general than area. One result implies that form-dimensional surfaces inR ⁿ , with 2 m n – 2, for any integrand , there are -minimizing surfaces with interior singularities.     

Weak Solutions of Nonlinear Elliptic Equations with Prescribed Singular Set

GivenΩany open and bounded subset of $\text{R}$ⁿ,n⩾4, with smooth boundary and givenΣany (n−m)-dimensional compact submanifold ofΩwithout boundary,n>m>2, we prove the existence of weak solutions to the problem[formula]which are singular onΣ, whenpis a realp>m/(m−2), close to this value.     

Multidimensional analog of the two-dimensional perron effect of sign change of characteristic exponents for infinitely differentiable differential systems

We obtain a general n-dimensional analog of the two-dimensional (partial) Perron effect of sign change of all arbitrarily prescribed negative characteristic exponents of an n-dimensional differential system of the linear approximation with infinitely differentiable bounded coefficients to the positive sign for the characteristic exponents of all nontrivial solutions of a nonlinear n-dimensional differential system with infinitely differentiable perturbations of arbitrary order m > 1 of smallness in a neighborhood of the origin and growth outside it. These positive exponents take n values distributed over n arbitrarily prescribed disjoint intervals and are realized on solutions issuing from nested subspaces R ¹ ? R ² ? ... ? R ⁿ .     

On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres

The n-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the (n+1)-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2-planes and is a compact Hermitian symmetric space of rank 2. In this paper, we study geometry of compact Lagrangian submanifolds in complex hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces in spheres. From this viewpoint we provide a classification theorem of compact homogeneous Lagrangian submanifolds in complex hyperquadrics by using the moment map technique. Moreover we determine the Hamiltonian stability of compact minimal Lagrangian submanifolds embedded in complex hyperquadrics which are obtained as Gauss images of isoparametric hypersurfaces in spheres with g(=  1, 2, 3) distinct principal curvatures. Dedicated to Professor Hajime Urakawa on his sixtieth birthday. H. Ma was partially supported by NSFC grant No. 10501028, SRF for ROCS, SEM and NKBRPC No. 2006CB805905. Y. Ohnita was partially supported by JSPS Grant-in-Aid for Scientific Research (A) No. 17204006.     

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.     

The curl in seven dimensional space and its applications

In higher dimensional spacesR ⁿ(n>3) the usual curl does not have the properties as inR ³. In this paper, we established the natural concept of curl inR ⁷ via octonion O. We prove that there exists the curl inR ⁿ if and only if n=3,7. Some applications are presented, such as the new phenomenon of the differential forms inR ⁷ which is different from the ordinary de Rham cohomology and Hodge theory, grad-curl-div type Dirac operator inR ⁶, seven dimensional Maxwell equations and Navier-Stokes equations.     

On the Roter Type of Chen Ideal Submanifolds

Chen ideal submanifolds M ⁿ in Euclidean ambient spaces E ^n+m (of arbitrary dimensions n ?? 2 and codimensions m??? 1) at each of their points do realise an optimal equality between their squared mean curvature, which is their main extrinsic scalar valued curvature invariant, and their ???C(= ??(2)?C) curvature of Chen, which is one of their main intrinsic scalar valued curvature invariants. From a geometric point of view, the pseudo-symmetric Riemannian manifolds can be seen as the most natural symmetric spaces after the real space forms, i.e. the spaces of constant Riemannian sectional curvature. From an algebraic point of view, the Roter manifolds can be seen as the Riemannian manifolds whose Riemann?CChristoffel curvature tensor R has the most simple expression after the real space forms, the latter ones being characterisable as the Riemannian spaces (M ⁿ , g) for which the (0, 4) tensor R is proportional to the Nomizu?CKulkarni square of their (0, 2) metric tensor g. In the present article, for the class of the Chen ideal submanifolds M ⁿ of Euclidean spaces E ^n+m , we study the relationship between these geometric and algebraic generalisations of the real space forms.     

Soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m, n) equations

We consider the nonlinear dispersive K(m,n) equation with the generalized evolution term and derive analytical expressions for some conserved quantities. By using a solitary wave ansatz in the form of sech^p function, we obtain exact bright soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m,n) equations with the generalized evolution terms. The results are then generalized to multi-dimensional K(m,n) equations in the presence of the generalized evolution term. An extended form of the K(m,n) equation with perturbation term is investigated. Exact bright soliton solution for the proposed K(m,n) equation having higher-order nonlinear term is determined. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients.