Associative Cones and Integrable Systems

Abstract We identify ℝ⁷ as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S⁶. It is known that a cone over a surface M in S⁶ is an associative submanifold of ℝ⁷ if and only if M is almost complex in S⁶. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S⁶ are the equation for primitive maps associated to the 6-symmetric space G₂=T², and use this to explain some of the known results. Moreover, the equation for S¹-symmetric almost complex curves in S⁶ is the periodic Toda lattice, and a discussion of periodic solutions is given. (Dedicated to the memory of Shiing-Shen Chern) * Partially supported by NSF grant DMS-0529756.     

Moduli of Quaternionic Superminimal Immersions of 2-Spheres into Quaternionic Projective Spaces

The aim of this paper is to describe the moduli spaces of degree d quaternionic superminimal maps from 2-spheres to quaternionic projective spaces HPn. We show that such moduli spaces have the structure of projectivized fibre products and are connected quasi-projective varieties of dimension 2nd + 2n + 2. This generalizes known results for spaces of harmonic 2-spheres in S⁴.     

On almost complex curves and Hopf hypersurfaces in the nearly Khler six-sphere

We study the almost complex curves and Hopf hypersurfaces in the nearly Khler S6(1),and their relations.For Hopf hypersurfaces,we give a classification theorem under some additional conditions.For compact almost complex curves,we obtain some interesting global results with respect to Gaussian curvature,area and the genus.     

Almost complex curves and Hopf hypersurfaces in the nearly Kähler 6-sphere

We characterize Hopf hypersurfaces inS ⁶ as open parts of geodesic hyperspheres or of tubes around almost complex curves ofS ⁶.     

A new family of noncommutative 2-spheres

We give new examples of noncommutative manifolds. In particular we construct a “strong” deformation of C(S²), consisting of a family of noncommutative 2-spheres, and study their analytic and topological properties.     

Irreducibility of moduli space of harmonic 2-spheres in S4

In this paper we prove that ford3, the moduli spaces of degreed branched superminimal immersions of the 2-sphere intoS ⁴ has 2 irreducible components. Consequently, the moduli space of degreed harmonic 2-spheres inS ⁴ has 3 irreducible components.     

调和复结构

利用向量丛值微分形式的调和理论来研究近复结构, 称之为调和复结构, 它是介于复结构与 K?hler结构之间的一种新结构.特别地,证明了S⁶上不允许此种结构.     

Stable Stationary Harmonic Maps to Spheres

For k ≥ 3, we establish new estimate on Hausdorff dimensions of the singular set of stable-stationary harmonic maps to the sphere S^k. We show that the singular set of stable-stationary harmonic maps from B5 to 83 is the union of finitely many isolated singular points and finitely many HSlder continuous curves. We also discuss the minimization problem among continuous maps from B^n to S^2.     

Remarks on families of singular curves with hyperelliptic normalizations

We give restrictions on the existence of families of curves on smooth projective surfaces S of nonnegative Kodaira dimension all having constant geometric genus p_g ? 2 and hyperelliptic normalizations. In particular, we prove a Reider-like result that relies on deformation theory and bending-and-breaking of rational curves in Sym²(S). We also give examples of families of such curves.     

Stable and unstable minimal surfaces and Hopf’s problem

Hopf’s well-known conjecture is considered, which states that there exists no metric of strictly positive curvature on the topological product S² × S² of two 2-spheres. Three theorems are proved.     

On the topology of two partition posets with forbidden block sizes

We study two subposets of the partition lattice obtained by restricting block sizes. The first consists of set partitions of {1,…,n} with block size at most k, for k≤n−2. We show that the order complex has the homotopy type of a wedge of spheres, in the cases 2k+2≥n and n=3k+2. For 2k+2>n, the posets in fact have the same S_n−1-homotopy type as the order complex of Π_n−1, and the S_n-homology representation is the “tree representation” of Robinson and Whitehouse. We present similar results for the subposet of Π_n in which a unique block size k≥3 is forbidden. For 2k≥n, the order complex has the homotopy type of a wedge of (n−4)-spheres. The homology representation of S_n can be simply described in terms of the Whitehouse lifting of the homology representation of Π_n−1.     

Patterson-Sullivan-type measures on Riccati foliations with quasi-Fuchsian holonomy

We consider Riccati foliations ?_ρ with hyperbolic leaves, over a finite hyperbolic Riemann Surface S, constructed by suspending a representation ρ: π ₁(S) → PSL(2,?) in a quasi-Fuchsian group. The foliated geodesic flow has a repeller-attractor dynamic with generic statistics µ⁺ and µ^? for positive and negative times, respectively. These measures have a common projection to a harmonic measure μ_ρ for the Riccati foliation. We describe μ _ρ ⁺ , μ _ρ ^- and μ_ρ in terms of the Patterson-Sullivan construction, and we show that the measures μ_ρ provide examples of the conformal harmonic measures introduced by M. Brunella.     

Exterior Monge-Ampère solutions

We discuss the Siciak-Zaharjuta extremal function of a real convex body in Cn, a solution of the homogeneous complex Monge-Ampère equation on the exterior of the convex body. We determine several conditions under which a foliation by holomorphic curves can be found in the complement of the convex body along which the extremal function is harmonic. We study a variational problem for holomorphic disks in projective space passing through prescribed points at infinity. The extremal curves are all complex quadratic curves, and the geometry of such curves allows for the determination of the leaves of the foliation by simple geometric criteria. As a byproduct we encounter a new invariant of an exterior domain, the Robin indicatrix, which is in some cases the dual of the Kobayashi indicatrix for a bounded domain. Finally, we construct extremal curves for two non-convex bodies in R².     

Minimal 2-spheres in a complex projective space

In this paper conformal minimal 2-spheres immersed in a complex projective space are studied by applying Lie theory and moving frames. We give differential equations of Kähler angle and square length of the second fundamental form. By applying these differential equations we give characteristics of conformal minimal 2-spheres of constant Kähler angle and obtain pinching theorems for curvature. We also discuss conformal minimal 2-spheres of constant normal curvature and prove that there does not exist any linearly full minimal 2-sphere immersed in a complex projective space CPn (n>2) with non-positive constant normal curvature. We also prove that a linearly full minimal 2-sphere immersed in a complex projective space CPn (n>2) with constant normal curvature and constant Kähler angle is of constant curvature.     

Polytopal and nonpolytopal spheres an algorithmic approach

The convexity theory for oriented matroids, first developed by Las Vergnas [17], provides the framework for a new computational approach to the Steinitz problem [13]. We describe an algorithm which, for a given combinatorial (d − 2)-sphereS withn vertices, determines the setC _d,n(S) of rankd oriented matroids withn points and face latticeS. SinceS is polytopal if and only if there is a realizableM εC _d,n(S), this method together with the coordinatizability test for oriented matroids in [10] yields a decision procedure for the polytopality of a large class of spheres. As main new result we prove that there exist 431 combinatorial types of neighborly 5-polytopes with 10 vertices by establishing coordinates for 98 “doubted polytopes” in the classification of Altshuler [1]. We show that for alln ≧k + 5 ≧8 there exist simplicialk-spheres withn vertices which are non-polytopal due to the simple fact that they fail to be matroid spheres. On the other hand, we show that the 3-sphereM ₉₆₃ ⁹ with 9 vertices in [2] is the smallest non-polytopal matroid sphere, and non-polytopal matroidk-spheres withn vertices exist for alln ≧k + 6 ≧ 9.     

On finite groups acting on ℤ<Subscript>2</Subscript>-homology 3-spheres

We consider finite groups G admitting orientation-preserving actions on homology 3-spheres (arbitrary, i.e. not necessarily free actions), concentrating on the case of nonsolvable groups. It is known that every finite group G admits actions on rational homology 3-spheres (and even free actions). On the other hand, the class of groups admitting actions on integer homology 3-spheres is very restricted (and close to the class of finite subgroups of the orthogonal group SO(4), acting on the 3-sphere). In the present paper, we consider the intermediate case of ₂-homology 3-spheres (i.e., with the ₂-homology of the 3-sphere where ₂ denote the integers mod two; we note that these occur much more frequently in 3-dimensional topology than the integer ones). Our main result is a list of finite nonsolvable groups G which are the candidates for orientation-preserving actions on ₂-homology 3-spheres. From this we deduce a corresponding list for the case of integer homology 3-spheres. In the integer case, the groups of the list are closely related to the dodecahedral group or the binary dodecahedral group most of these groups are subgroups of the orthogonal group SO(4) and hence admit actions on S³. Roughly, in the case of ₂-homology 3-spheres the groups PSL(2,5) and SL(2,5) get replaced by the groups PSL(2,q) and SL(2,q), for an arbitrary odd prime power q. We have many examples of actions of the groups PSL(2,q) and SL(2,q) on ₂-homology 3-spheres, for various small values of q (constructed as regular coverings of suitable hyperbolic 3-orbifolds and 3-manifolds, using computer-supported methods to calculate the homology of the coverings). We think that all of them occur but have no method to prove this at present (in particular, the exact classification of the finite nonsolvable groups admitting actions on ₂-homology 3-spheres remains still open).     

Commuting dual Toeplitz operators on the harmonic Dirichlet space

In this paper,we characterize the symbols for(semi-)commuting dual Toeplitz operators on the orthogonal complement of the harmonic Dirichlet space.We show that for φ,ψ∈W~(1,∞),S_φS_ψ=S_ψ Sφ on(D_h)~⊥ if and only if φ and ψ satisfy one of the following conditions:(1) Both φ and ψ are harmonic functions;(2) There exist complex constants α and β,not both 0,such that φ=αψ +β.     

From the Icosahedron to Natural Triangulations of??P 2 and S 2��S 2

We present two constructions in this paper: (a) a 10-vertex triangulation $\mathbb{C}P^{2}_{10}$ of the complex projective plane ?P ² as a subcomplex of the join of the standard sphere ( $S^{2}_{4}$ ) and the standard real projective plane ( $\mathbb{R}P^{2}_{6}$ , the decahedron), its automorphism group is A ₄; (b) a 12-vertex triangulation (S ²×S ²)₁₂ of S ²×S ² with automorphism group 2S ₅, the Schur double cover of the symmetric group S ₅. It is obtained by generalized bistellar moves from a simplicial subdivision of the standard cell structure of S ²×S ². Both constructions have surprising and intimate relationships with the icosahedron. It is well known that ?P ² has S ²×S ² as a two-fold branched cover; we construct the triangulation $\mathbb{C}P^{2}_{10}$ of ?P ² by presenting a simplicial realization of this covering map S ²×S ²???P ². The domain of this simplicial map is a simplicial subdivision of the standard cell structure of S ²×S ², different from the triangulation alluded to in (b). This gives a new proof that Kühnel??s $\mathbb{C}P^{2}_{9}$ triangulates ?P ². It is also shown that $\mathbb{C}P^{2}_{10}$ and (S ²×S ²)₁₂ induce the standard piecewise linear structure on ?P ² and S ²×S ² respectively.     

Joachimsthal surfaces inS 3

A representation of the Joachimsthal surfaces (having a family of curvature lines that lie in totally geodesic 2-spheres) in the sphereS ³ is obtained. It is proved that, if a surface of constant mean curvature inS ³ has one family of curvature lines lying in totally geodesic 2-spheres, then it is a surface of rotation. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 221–229, February, 2000.     

Non-linear circle actions on the 4-sphere and twisting spun knots

CIRCLE actions on spheres form one of the most important problems in transformation groups. The aim of this paper is to study this problem in dimension 4. We answer a question of Montgomery and Yang[7], and show that there are infinitely many non-linear circle actions on S⁴. Moreover, if the 3-dimensional Poincaré conjecture is true, these actions plus the linear ones are the only possible circle actions on S⁴. The proof of this assertion involves identifying some homotopy 4-spheres. It is closely related to the work, twisting spun knots, of Zeeman[14]. We give a different treatment of this subject. This new setting yields new proofs and substantial strengthenings of some known results. In particular, we answer two questions of Zeeman[14, pp. 493–494, Questions 3 and 4].