Pinching of the first eigenvalue of the Laplacian and almost-Einstein hypersurfaces of the Euclidean space

In this article, we prove new pinching theorems for the first eigenvalue λ₁(M) of the Laplacian on compact hypersurfaces of the Euclidean space. These pinching results are associated with the upper bound for λ₁(M) in terms of higher order mean curvatures H _k . We show that under a suitable pinching condition, the hypersurface is diffeomorpic and almost-isometric to a standard sphere. Moreover, as a corollary, we show that a hypersurface of the Euclidean space which is almost-Einstein is diffeomorpic and almost-isometric to a standard sphere.      

Maximal surfaces and the universal Teichmüller space

We show that any element of the universal Teichmüller space is realized by a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in AdS _n+1, any subset E of the boundary at infinity which is the boundary at infinity of a space-like hypersurface bounds a maximal space-like hypersurface. In AdS₃, if E is the graph of a quasi-symmetric homeomorphism, then this maximal surface is unique, and it has negative sectional curvature. As a by-product, we find a simple characterization of quasi-symmetric homeomorphisms of the circle in terms of 3-dimensional projective geometry.     

Pappus type theorems for hypersurfaces in a space form

In order to get further insight on the Weyl’s formula for the volume of a tubular hypersurface, we consider the following situation. Letc(t) be a curve in a space formM _λ ⁿ of sectional curvature λ. LetP ₀ be a totally geodesic hypersurface ofM _λ ⁿ throughc(0) and orthogonal toc(t). LetC ₀ be a hypersurface ofP ₀. LetC be the hypersurface ofM _λ ⁿ obtained by a motion ofC ₀ alongc(t). We shall denote it byC ^PorC ^Fif it is obtained by a parallel or Frenet motion, respectively. We get a formula for volume(C). Among other consequences of this formula we get that, ifc(0) is the centre of mass ofC ₀, then volume(C) ≥ volume(C),^P),and the equality holds whenC ₀ is contained in a geodesic sphere or the motion corresponds to a curve contained in a hyperplane of the Lie algebraO(n−1) (whenn=3, the only motion with these properties is the parallel motion). Work partially supported by a DGES Grant No. PB97-1425 and a AGIGV Grant No. GR0052.     

Dupin hypersurfaces with four principal Curvatures,II

If M is an isoparametric hypersurface in a sphere S ⁿ with four distinct principal curvatures, then the principal curvatures κ₁, . . . , κ₄ can be ordered so that their multiplicities satisfy m ₁ = m ₂ and m ₃ = m ₄, and the cross-ratio r of the principal curvatures (the Lie curvature) equals −1. In this paper, we prove that if M is an irreducible connected proper Dupin hypersurface in R ⁿ (or S ⁿ ) with four distinct principal curvatures with multiplicities m ₁ = m ₂ ≥ 1 and m ₃ = m ₄ = 1, and constant Lie curvature r = −1, then M is equivalent by Lie sphere transformation to an isoparametric hypersurface in a sphere. This result remains true if the assumption of irreducibility is replaced by compactness and r is merely assumed to be constant.      

Volume-preserving mean curvature flow of revolution hypersurfaces in a Rotationally Symmetric Space

In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces (tgh for short). Requiring that, for each time t ≥ 0, the evolving hypersurface M _t meets such tgh orthogonally, we prove that: (a) the flow exists while M _t does not touch the axis of rotation; (b) throughout the time interval of existence, (b1) the generating curve of M _t remains a graph, and (b2) the averaged mean curvature is double side bounded by positive constants; (c) the singularity set (if non-empty) is finite and lies on the axis; (d) under a suitable hypothesis relating the enclosed volume to the n-volume of M, we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature.     

有关不能全含于半球中的一些曲面的性质讨论

本文主要研究了不能全含于开半球中的一些特殊曲面.利用Lr算子的相关性质,证明了对S~(n+1)中紧致r-极小超曲面,如果第二基本形式的秩rank(h_(ij))r,则其不全含在S~(n+1)的一个开半球中.     

Fourier Inversion of Distributions Supported by a Hypersurface

Let μ _Σ be the natural measure on R ^N (N≥3) supported by a compact oriented analytic hypersurface Σ, ψ a smooth function on R ^N and P(D) a differential operator in N variables of order m. We determine a sufficient condition on the number λ such that the Fourier integral of the distribution P(D)ψ μ _Σ be summable by Cesàro means of order λ to zero in a point outside the hypersurface. This condition depends on m and on the position of the point with respect to the caustic of the hypersurface.     

On the bore radius for minimal surfaces

A least upper bound for the inner radiusR of an opening in a complete minimal hypersurface contained in a parallel layer is given. Namely, if Δ is the width of this layer, thenR≤Δ/(2c _p), wherec _p is an absolute constant depending only on the dimensionp of the minimal hypersurface. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 909–913, June, 1996. I thank V. M. Milyukov for useful discussions of this work. This research was supported by the “Culture Initiative. Mathematics” Foundation.     

Mean Curvature Flow Closes Open Ends of Noncompact Surfaces of Rotation

We discuss the motion of noncompact axisymmetric hypersurfaces Γ _t evolved by mean curvature flow. Our study provides a class of hypersurfaces that share the same quenching time with the shrinking cylinder evolved by the flow and prove that they tend to a smooth hypersurface having no pinching neck and having closed ends at infinity of the axis of rotation as the quenching time is approached. Moreover, they are completely characterized by a condition on initial hypersurface.     

Möbius Isoparametric Hypersurfaces in S n+1 with Two Distinct Principal Curvatures

A hypersurface x : M → S ⁿ⁺¹ without umbilic point is called a Möbius isoparametric hypersurface if its Möbius form Φ = ?ρ ^?2∑ _i (e _i (H) + ∑ _j (h _ij ?Hδ _ij )e _j (log ρ))θ _i vanishes and its Möbius shape operator $ {\Bbb {S}}A hypersurface x : M → S ^{n +1} without umbilic point is called a M?bius isoparametric hypersurface if its M?bius form Φ = −ρ⁻²∑ _i (e _i (H) + ∑ _j (h _ij −Hδ _ij )e _j (log ρ))θ _i vanishes and its M?bius shape operator ? = ρ⁻¹(S−Hid) has constant eigenvalues. Here {e _i } is a local orthonormal basis for I = dx·dx with dual basis {θ _i }, II = ∑ _ij h _ij θ _i ⊗θ _i is the second fundamental form, and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in S ^{n +1} is a M?bius isoparametric hypersurface, but the converse is not true. In this paper we classify all M?bius isoparametric hypersurfaces in S ^{n +1} with two distinct principal curvatures up to M?bius transformations. By using a theorem of Thorbergsson [1] we also show that the number of distinct principal curvatures of a compact M?bius isoparametric hypersurface embedded in S ^{n +1} can take only the values 2, 3, 4, 6. Received September 7, 2001, Accepted January 30, 2002     

Two questions on the exterior geometry of the plücker embeddings of the Grassmann manifolds

The Plücker model of the Grassmann manifold G _p,p+q ⁺ is considered. The structure of intersections of G _p,p+q ⁺ with tangent spaces of G _p,p+q ⁺ regarded as subspaces of the ambient exterior algebra is described. An explicit formula for the second fundamental form of G _2,4 ⁺ as of a hypersurface in the five-dimensional sphere is given. The level sets of the normal curvature functions for this hypersurface are studied. Bibliography: 3 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 5–12. Translated by N. Yu. Netsvetaev.     

A characterization of totally <Emphasis Type="Italic">η</Emphasis>-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form

We give a characterization of totally η-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator A of a real hypersurface M of a complex space form M ⁿ (c), c ≠ 0, n ⩾ 3, satisfies g(AX, Y) = ag(X, Y) for any X, Y ∈ T ₀(x), a being a function, where T ₀ is the holomorphic distribution on M, then M is a totally η-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of η-umbilical real hypersurfaces.     

On the problem of isometry of a hypersurface preserving mean curvature

The problem of determining the Bonnet hypersurfaces in R ⁿ⁺¹, for n > 1, is studied here. These hypersurfaces are by definition those that can be isometrically mapped to another hypersurface or to itself (as locus) by at least one nontrivial isometry preserving the mean curvature. The other hypersurface and/or (the locus of) itself is called Bonnet associate of the initial hypersurface. The orthogonal net which is called A-net is special and very important for our study and it is described on a hypersurface. It is proved that, non-minimal hypersurface in R ⁿ⁺¹ with no umbilical points is a Bonnet hypersurface if and only if it has an A-net.     

First nonzero eigenvalue of a pseudo-umbilical hypersurface in the unit sphere

S. Deshmukh has obtained interesting results for first nonzero eigenvalue of a minimal hypersurface in the unit sphere. In the present article, we generalize these results to pseudoumbilical hypersurface and prove: What conditions are satisfied by the first nonzero eigenvalue λ ₁ of the Laplacian operator on a compact immersed pseudo-umbilical hypersurface M in the unit sphere S ⁿ⁺¹. We also show that a compact immersed pseudo-umbilical hypersurface of the unit sphere S ⁿ⁺¹ with λ ₁ = n is either isometric to the sphere S ⁿ or for this hypersurface an inequaluity is fulfilled in which sectional curvatures of the hypersuface M participate.     

Poincaré series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity

A relation is proved between the Poincaré series of the coordinate algebra of a two-dimensional quasihomogeneous isolated hypersurface singularity and the characteristic polynomial of its monodromy operator. For a Kleinian singularity not of type A _{2 n} , this amounts to the statement that the Poincaré series is the quotient of the characteristic polynomial of the Coxeter element by the characteristic polynomial of the affine Coxeter element of the corresponding root system. We show that this result also follows from the McKay correspondence. Received: Received: 25 October 2001 / Revised version: 19 November 2001     

Dual Riemannian spaces of constant curvature on a normalized hypersurface

In this paper, the following results are obtained: 1) It is proved that, in the fourth order differential neighborhood, a regular hypersurface V _n−1 embedded into a projective-metric space K _n , n ≥ 3, intrinsically induces a dual projective-metric space $ \bar K_n $ \bar K_n . 2) An invariant analytical condition is established under which a normalization of a hypersurface V _n−1 ⊂ K _n (a tangential hypersurface $ \bar V_{n - 1} $ \bar V_{n - 1} ⊂ $ \bar K_n $ \bar K_n ) by quasitensor fields H _n ⁱ , H _i ($ \bar H_n^i $ \bar H_n^i , $ \bar H_i $ \bar H_i ) induces a Riemannian space of constant curvature. If the two conditions are fulfilled simultaneously, the spaces R _n−1 and $ \bar R_{n - 1} $ \bar R_{n - 1} are spaces of the same constant curvature $ K = - \tfrac{1} {c} $ K = - \tfrac{1} {c} . 3) Geometric interpretations of the obtained analytical conditions are given.     

Compact minimal hypersurfaces with index one in the real projective space

Let M ⁿ be a compact (two-sided) minimal hypersurface in a Riemannian manifold . It is a simple fact that if has positive Ricci curvature then M cannot be stable (i.e. its Jacobi operator L has index at least one). If is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator.?We prove that if is the real projective space , obtained as a metric quotient of the unit sphere, and the Jacobi operator of M has index one, then M is either a totally geodesic sphere or the quotient to the projective space of the hypersurface obtained as the product of two spheres of dimensions n ₁, n ₂ and radius R ₁, R ₂, with and . Received: June 6, 1998     

Holomorphic Curves on Hyperplane Sections of 3-Folds

In this paper we prove a conjectured height inequality of Lang and Vojta for holomorphic curves lying on generic hyperplane sections of 3-folds. As a consequence we deduce a conjecture of Kobayashi that a generic hypersurface in \Bbb P³_{\Bbb C} {\Bbb P}^3_{\Bbb C} of sufficiently high degree is hyperbolic.     

Gaussian mean curvature flow

In the Euclidean Space \mathbb Rⁿ⁺¹{\mathbb {R}^{n+1}} with a density e^{e\frac12 n m2 |x|2}, (e = ±1){e^{\varepsilon \frac12 n \mu^2 |x|^2},} {(\varepsilon =\pm1}), we consider the flow of a hypersurface driven by its mean curvature associated to this density. We give a detailed account of the evolution of a convex hypersurface under this flow. In particular, when e = -1{ \varepsilon=-1} (Gaussian density), the hypersurface can expand to infinity or contract to a convex hypersurface (not necessarily a sphere) depending on the relation between the bound of its principal curvatures and μ.     

Higher order Willmore hypersurfaces in Euclidean space

Let x ： Mn^n→ R^n＋1 be an n（≥2）-dimensional hypersurface immersed in Euclidean space Rn＋1. Let σi（0≤ i≤ n） be the ith mean curvature and Qn = ∑i=0^n（-1）^i＋1 （n^i）σ1^n-iσi. Recently, the author showed that Wn（x） = ∫M QndM is a conformal invariant under conformal group of R^n＋1 and called it the nth Willmore functional of x. An extremal hypersurface of conformal invariant functional Wn is called an nth order Willmore hypersurface. The purpose of this paper is to construct concrete examples of the 3rd order Willmore hypersurfaces in Ra which have good geometric behaviors. The ordinary differential equation characterizing the revolutionary 3rd Willmore hypersurfaces is established and some interesting explicit examples are found in this paper.