Complete hypersurfaces with constant laguerre scalar curvature in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let x: M ^n?1 → R ⁿ be an umbilical free hypersurface with non-zero principal curvatures. Two basic invariants of M under the Laguerre transformation group of R ⁿ are Laguerre form C and Laguerre tensor L. In this paper, n > 3) complete hypersurface with vanishing Laguerre form and with constant Laguerre scalar curvature R in R ⁿ , denote the trace-free Laguerre tensor by ?\(\widetilde L = L - \frac{1}{{n - 1}}tr\left( L \right)\) · Id. If \(\widetilde L = L - \frac{1}{{n - 1}}tr\left( L \right)\), then M is Laguerre equivalent to a Laguerre isotropic hypersurface; and if \({\sup _M}\left\| {\widetilde L} \right\| = \frac{{\sqrt {\left( {n - 1} \right)\left( {n - 2} \right)} R}}{{\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)}},\), M is Laguerre equivalent to the hypersurface ?x: H ¹ × S ^n?2 → R ⁿ .     

The scalar curvature flow on <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>—perturbation theorem revisited

For the problem of finding a geometry on S ⁿ , for n≥3, with a prescribed scalar curvature, there is a well-known result which is called the perturbation theorem; it is due to Chang and Yang (Duke Math. J. 64, 27–69, 1991). Their key assumption is that the candidate f for the prescribed scalar curvature is sufficiently near the scalar curvature of the standard metric in the sup norm. It is important to know how large that difference in sup norm can possibly be. Here we consider prescribing scalar curvature problem using the scalar curvature flow.     

Classification of hypersurfaces with parallel Möbius second fundamental form in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript>

Let M ⁿ(n ≥ 2) be an immersed umbilic-free hypersurface in the (n+1)-dimensional unit sphere S ⁿ⁺¹. Then M ⁿ is associated with a so-called Möbius metric g, and a Möbius second fundamental form B which are invariants of M ⁿunder the Möbius transformation group of S ⁿ⁺¹. In this paper, we classify all umbilic-free hypersurfaces with parallel Möbius second fundamental form.     

Regular<Emphasis Type="Italic">n</Emphasis>-simplices in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with vertices in ℤ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper theI andII regularn-simplices are introduced. We prove that the sufficient and necessary conditions for existence of anI regularn-simplex in ℝ ⁿ are that ifn is even thenn = 4m(m + 1), and ifn is odd thenn = 4m + 1 with thatn + 1 can be expressed as a sum of two integral squares orn = 4m - 1, and that the sufficient and necessary condition for existence of aII regularn-simplex in ℝ ⁿ isn = 2m ² - 1 orn = 4m(m + 1)(m ∈ ℕ). The connection between regularn-simplex in ℝ ⁿ and combinational design is given.     

Free involutions on <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Topological free involutions on S ¹ × S ⁿ are classified up to conjugation. We prove that this is the same as classifying quotient manifolds up to homeomorphism. There are exactly four possible homotopy types of such quotients, and surgery theory is used to classify all manifolds within each homotopy type.     

The smooth structure set of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of S ^p × S ^q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general \({\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}\) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful map \({\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}\) is not in general a subgroup of the topological structure set.     

Quantization of curvature for compact surfaces in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n⋆</Emphasis></Superscript>

For minimal surfaces in spheres, there is a well known conjecture about the quantization of intrinsic curvature which has been solved only in special cases so far. We recall an intrinsic and an extrinsic version for the known results and extend them to compact non-minimal surfaces in spheres. In particular we discuss special classes like Willmore surfaces and surfaces with parallel mean curvature vector. Mathematics Subject Classification (2000):53C42, 53A10.H.Li is partially supported by a research fellowship of the Alexander von Humboldt Stiftung 2001/2002 and the Zhongdian grant of NSFC. U. Simon is partially supported by DFG 163/Si-7-2 and a Chinese–German research cooperation of NSFC and DFG.     

Difference sets with <Emphasis Type="Italic">n</Emphasis> = 5<Emphasis Type="Italic"> p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>

Let D be a (v, k, λ)-difference set in an abelian group G, and (v, 31) = 1. If n = 5p ^r with p a prime not dividing v and r a positive integer, then p is a multiplier of D. In the case 31|v, we get restrictions on the parameters of such difference sets D for which p may not be a multiplier.      

On the regularity of scalar <Emphasis Type="Italic">p</Emphasis>-Harmonic functions in <Emphasis Type="Italic">n</Emphasis> dimensions with 2 ≤ <Emphasis Type="Italic">p</Emphasis> < <Emphasis Type="Italic">n</Emphasis>

Using Tilli’s technique [Cal Var 25(3):395–401, 2006], we shall give a new proof of the regularity of the local minima of the functional

$J\left( u\right) =\int\limits_{\Omega } \left\vert \partial u\right\vert^{p}\,dx$

with Ω a domain of class C ^{0, 1} in \({\mathbb{R}^{n}}\) and 2 ≤ p < n.

     

Tight Lagrangian surfaces in <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript>

We determine all tight Lagrangian surfaces in S ² × S ². In particular, globally tight Lagrangian surfaces in S ² × S ² are nothing but real forms of this symmetric space.     

Hamiltonian Gromov–Witten invariants on ℂ<Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript> with <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>-action

Consider a Hamiltonian action of S1 on (C ⁿ⁺¹, ω _std), we shown that the Hamiltonian Gromov–Witten invariants of it are well-defined. After computing the Hamiltonian Gromov–Witten invariants of it, we construct a ring homomorphism from \(H_{{S^1},CR}^*\left( {X,R} \right)\) to the small orbifold quantum cohomology of X// _τ S ¹ and obtain a simpler formula of the Gromov–Witten invariants for weighted projective space.     

Dense subgroups of <Emphasis Type="Italic">n</Emphasis>-dimensional Möbius groups

We will derive a new discreteness condition for n-dimensional M?bius subgroups as well as obtain some results concerning classification of such groups. We will also discuss dense subgroups of n-dimensional M?bius groups. The main result is that any dense group of an n-dimensional M?bius group contains a dense subgroup which is generated by at most n elements if . Received: 5 June 2001 / Published online: 24 February 2003 RID="*" ID="*" The research was partly supported by FNS of China, grant number 19801011     

On 2 – (<Emphasis Type="Italic">n</Emphasis><Superscript>2</Superscript>, 2<Emphasis Type="Italic">n</Emphasis>, 2<Emphasis Type="Italic">n</Emphasis>–1) designs with three intersection numbers

The simple incidence structure , formed by the points and the unordered pairs of distinct parallel lines of a finite affine plane of order n > 4, is a 2 – (n ²,2n,2n–1) design with intersection numbers 0,4,n. In this paper, we show that the converse is true, when n ≥ 5 is an odd integer. Supported by M.I.U.R., Università di Palermo.     

The hot spots conjecture on a class of domains in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with <Emphasis Type="Italic">n</Emphasis> ⩾ 3

In this paper, we define a class of domains in R ⁿ . Using the synchronous coupling of reflecting Brownian motion, we obtain the monotonicity property of the solution of the heat equation with the Neumann boundary conditions. We then show that the hot spots conjecture holds for this class of domains.     

<Emphasis Type="Italic">C</Emphasis><Superscript>1+<Emphasis Type="Italic">α</Emphasis></Superscript>-Regularity for Two-Dimensional Almost-Minimal Sets in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We give a new proof and a partial generalization of Jean Taylor’s result (Ann. Math. (2) 103(3), 489–539, 1976) that says that Almgren almost-minimal sets of dimension 2 in ℝ³ are locally C ^1+α -equivalent to minimal cones. The proof is rather elementary, but uses a local separation result proved in Ann. Fac. Sci. Toulouse 18(1), 65–246, 2009 and an extension of Reifenberg’s parameterization theorem (David et al. in Geom. Funct. Anal. 18, 1168–1235, 2008). The key idea is still that if X is the cone over an arc of small Lipschitz graph in the unit sphere, but X is not contained in a disk, we can use the graph of a harmonic function to deform X and substantially diminish its area. The local separation result is used to reduce to unions of cones over arcs of Lipschitz graphs. A good part of the proof extends to minimal sets of dimension 2 in ℝ ⁿ , but in this setting our final regularity result on E may depend on the list of minimal cones obtained as blow-up limits of E at a point.     

On Hamiltonian-Minimal Isotropic Homogeneous Tori in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> and ℂP<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We study isotropic homogeneous tori in ?ⁿ and ?Pⁿ. We find necessary and sufficient conditions for their Hamiltonian minimality.     

Minimal helix surfaces in <Emphasis Type="Italic">N</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×ℝ

An immersed surface M in N ⁿ ×ℝ is a helix if its tangent planes make constant angle with ∂ _t . We prove that a minimal helix surface M, of arbitrary codimension is flat. If the codimension is one, it is totally geodesic. If the sectional curvature of N is positive, a minimal helix surfaces in N ⁿ ×ℝ is not necessarily totally geodesic. When the sectional curvature of N is nonpositive, then M is totally geodesic. In particular, minimal helix surfaces in Euclidean n-space are planes. We also investigate the case when M has parallel mean curvature vector: A complete helix surface with parallel mean curvature vector in Euclidean n-space is a plane or a cylinder of revolution. Finally, we use Eikonal f functions to construct locally any helix surface. In particular every minimal one can be constructed taking f with zero Hessian.     

Cycles in <Emphasis Type="Italic">G</Emphasis>-orbits in <Emphasis Type="Italic">G</Emphasis><Superscript><Emphasis Type="Italic">ℂ</Emphasis></Superscript>-flag manifolds

There is a natural duality between orbits of a real form G of a complex semisimple group G on a homogeneous rational manifold Z=G /P and those of the complexification K of any of its maximal compact subgroups K: (,) is a dual pair if is a K-orbit. The cycle space C() is defined to be the connected component containing the identity of the interior of {g:g() is non-empty and compact}. Using methods which were recently developed for the case of open G-orbits, geometric properties of cycles are proved, and it is shown that C() is contained in a domain defined by incidence geometry. In the non-Hermitian case this is a key ingredient for proving that C() is a certain explicitly computable universal domain.Research of the first author partially supported by Schwerpunkt Global methods in complex geometry and SFB-237 of the Deutsche Forschungsgemeinschaft.The second author was supported by a stipend of the Deutsche Akademische Austauschdienst.     

Asymptotic purity for very general hypersurfaces of ℙ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> × ℙ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> of bidegree (<Emphasis Type="Italic">k,k</Emphasis>)

For a complex irreducible projective variety, the volume function and the higher asymptotic cohomological functions have proven to be useful in understanding the positivity of divisors as well as other geometric properties of the variety. In this paper, we study the vanishing properties of these functions on hypersurfaces of ℙ ⁿ × ℙ ⁿ . In particular, we show that very general hypersurfaces of bidegree (k, k) obey a very strong vanishing property, which we define as asymptotic purity: at most one asymptotic cohomological function is nonzero for each divisor. This provides evidence for the truth of a conjecture of Bogomolov and also suggests some general conditions for asymptotic purity.     

The <Emphasis Type="Italic">ψ</Emphasis>-associate <Emphasis Type="Italic">X</Emphasis><Superscript>#</Superscript> of a flat <Emphasis Type="Italic">X</Emphasis> in PG(<Emphasis Type="Italic">n</Emphasis>, 2)

For a given hypersurface ψ in PG(n, 2), with equation Q(x) = 0, where Q is a polynomial of (reduced) degree d > 1, a definition is given of the ψ-associate X ^# of a flat X in PG(n, 2). The definition involves the fully polarized form of the polynomial Q; in the cubic case d = 3 it reads: X ^# = {z ∈ PG(n, 2) | T(x, y, z) = 0 for all x, y ∈ X}, where T(x, y, z) denotes the alternating trilinear form obtained by completely polarizing Q. Some general results, valid for any degree d and projective dimension n, are first expounded. Thereafter several choices of ψ are visited, but for each choice just a few aspects are highlighted. Despite the partial nature of the survey quite a variety of behaviours of the ψ-associate are uncovered. Many of the choices of ψ which are considered are of cubic hypersurfaces in PG(5, 2). If ψ is the Segre variety it is shown that the 48 planes external to fall into eight pairs of ordered triplets {(P ₁, R ₁, S ₁), (P ₂, R ₂, S ₂)} such that and . Further those lines L of PG(5, 2) which are singular, satisfying that is L ^# = PG(5.2), are in this case shown to form a complete spread of 21 lines. Another result of note arises in the case where ψ is the underlying 35-set of a non-maximal partial spread Σ₅ of five planes in PG(5, 2), where it is shown that one plane is singled out by the property that every line is singular.