Some remarks on factoriality of certain hypersurfaces in \mathbb{P}^4

Let V be a reduced and irreducible hypersurface of degree k 3. In this paper we prove that if the singular locus of V consists of ₂ ordinary double points, ₃ ordinary triple points and if ₂ + 4₃ < (k – 1)², then any smooth surface contained in V is a complete intersection on V.Received: 7 January 2004     

The Hilbert functions ofk-configurations in\mathbb{P}^2 and\mathbb{P}^3

In this paper, we proved the set of points which are the vertices of then-gon in $\mathbb{P}^2 $ (n ≥ 3) has the Uniform Position Property and what the graded free resolutions of the ideals ofk-configurations in $\mathbb{P}^3 $ are.     

Laguerre geometry of hypersurfaces in $$\mathbb{R}^{n}$$

Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ , and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC.     

Classification of rational holomorphic maps from \mathbb{B}^2 into \mathbb{B}^\mathbb{N} with degree 2

Rational proper holomorphic maps from the unit ball in ?² into the unit ball ? ^N with degree 2 are classified, up to automorphisms of balls.     

Classifying ACM sets of points in $${\mathbb{P}^{1} \times \mathbb{P}^{1}}$$ via separators

The purpose of this note is to give a new, short proof of a classification of ACM sets of points in in terms of separators.     

Hopf hypersurfaces of small Hopf principal curvature in $${\mathbb{C}{\rm H}^2}$$

Using the methods of moving frames and exterior differential systems, we show that there exist Hopf hypersurfaces in complex hyperbolic space with any specified value of the Hopf principal curvature α less than or equal to the corresponding value for the horosphere. We give a construction for all such hypersurfaces in terms of Weierstrass-type data, and also obtain a classification of pseudo-Einstein hypersurfaces in .      

Metrics with nonnegative curvature on {S^2 \times \mathbb{R}^4}

We study nonnegatively curved metrics on S²×\mathbbR⁴{S^2\times\mathbb{R}^4}. First, we prove rigidity theorems for connection metrics; for example, the holonomy group of the normal bundle of the soul must lie in a maximal torus of SO(4). Next, we prove that Wilking’s almost-positively curved metric on S ² × S ³ extends to a nonnegatively curved metric on S²×\mathbbR⁴{S^2\times\mathbb{R}^4} (so that Wilking’s space becomes the distance sphere of radius 1 about the soul). We describe in detail the geometry of this extended metric.     

On closed minimal hypersurfaces with constant scalar curvature in \mathbb{S}^7

We consider minimal closed hypersurfaces ${M \subset \mathbb{S}^7(1)}$ with constant scalar curvature. We prove that if M fulfills particular additional assumptions, then it is isoparametric. This gives a partial answer to the question made by S.-S. Chern about the pinching of the scalar curvature for closed minimal hypersurfaces in dimension 6.     

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$      

Biharmonic Submanifolds with Parallel Mean Curvature in \mathbb{S}^{n}\times\mathbb{R}

We find a Simons type formula for submanifolds with parallel mean curvature vector (pmc submanifolds) in product spaces M ⁿ (c)×?, where M ⁿ (c) is a space form with constant sectional curvature c, and then we use it to prove a gap theorem for the mean curvature of certain complete proper-biharmonic pmc submanifolds, and classify proper-biharmonic pmc surfaces in $\mathbb{S}^{n}(c)\times\mathbb{R}$ .     

On Discrete groups of automorphisms of \mathbb P ^2_\mathbb{C }

We study the geometry and dynamics of discrete subgroups $\Gamma $ of $\mathrm{PSL}(3,\mathbb C )$ with an open invariant set $\Omega \subset \mathbb P _\mathbb{C }^2$ where the action is properly discontinuous and the quotient $\Omega /\Gamma $ contains a connected component whicis compact. We call such groups quasi-cocompact. In this case $\Omega /\Gamma $ is a compact complex projective orbifold and $\Omega $ is a divisible set. Our first theorem refines classical work by Kobayashi–Ochiai and others about complex surfaces with a projective structure: We prove that every such group is either virtually affine or complex hyperbolic. We then classify the divisible sets that appear in this way, the corresponding quasi-cocompact groups and the orbifolds $\Omega /\Gamma $ . We also prove that excluding a few exceptional cases, the Kulkarni region of discontinuity coincides with the equicontinuity region and is the largest open invariant set where the action is properly discontinuous.     

$L^p$ Harmonic $k$-Forms on Complete Noncompact Hypersurfaces in $\mathbb{S}^{n+1}$ with Finite Total Curvature

In general, the space of $L^p$ harmonic forms $\mathcal{H}^k(L^p(M))$ and reduced $L^p$ cohomology $H^k(L^p(M))$ might be not isomorphic on a complete Riemannian manifold $M$, except for $p=2$. Nevertheless, one can consider whether $\mathrm{dim}\mathcal{H}^k(L^p(M))<+\infty$ are equivalent to $\mathrm{dim}H^k(L^p(M))<+\infty$. In order to study such kind of problems, this paper obtains that dimension of space of $L^p$ harmonic forms on a hypersurface in unit sphere with finite total curvature is finite, which is also a generalization of the previous work by Zhu. The next step will be the investigation of dimension of the reduced $L^p$ cohomology on such hypersurfaces.     

Example of a Five-Sheeted Exotic Covering over \mathbb{C}^2

In the paper, an example of a five-sheeted covering which is a topological counterexample in the same sense as Vitushkin's example is proposed. Its topological characteristics such as generators of the two-dimensional homology and values of canonical classes are computed.     

Linearity and Second Fundamental Forms for Proper Holomorphic Maps from \mathbb{B}^{n+1} to \mathbb{B}^{4n-3}

For a holomorphic proper map F from the ball $\mathbb{B}^{n+1}$ into $\mathbb{B}^{N+1}$ that is C ³ smooth up to the boundary, the image $M=F(\partial\mathbb{B}^{n})$ is an immersed CR submanifold in the sphere $\partial \mathbb{B}^{N+1}$ on which some second fundamental forms II _M and $\mathit{II}^{CR}_{M}$ can be defined. It is shown that when 4??n+1<N+1??4n?3, F is linear fractional if and only if $\mathit{II}_{M} - \mathit{II}_{M}^{CR} \equiv 0$ .     

Rank-one convex hulls in \mathbb{R}^{2\times2}

We study the rank-one convex hull of compact sets . We show that if K contains no two matrices whose difference has rank one, and if K contains no four matrices forming a T ₄ configuration, then the rank-one convex hull K ^rc is equal to K. Furthermore, we give a simple numerical criterion for testing for T ₄ configurations. Received: 20 August 2003, Accepted: 3 March 2004, Published online: 12 May 2004 Mathematics Subject Classification (2000): 49J45, 52A30 An erratum to this article can be found at     

A note on compact Weingarten hypersurfaces embedded in $$\mathbb {R}^{n + 1}$$

We prove that the round sphere is the only compact Weingarten hypersurface embedded in the Euclidean space such that \(H_r = aH + b\), for constants \(a, b \in \mathbb {R}\). Here, \(H_r\) stands for the r-th mean curvature and H denotes the standard mean curvature of the hypersurface.