On Einstein Hermitian manifolds

We show that every compact Einstein Hermitian surface with constant *–scalar curvature is a K?hler surface. In contrast to the 4-dimensional case, it is shown that there exists a compact Einstein Hermitian (4n + 2)-dimensional manifold with constant *–scalar curvature which is not K?hler. This study is supported by Kangwon National University.     

Curvature and distance function from a manifold

This paper is concerned with the relations between the differential invariants of a smooth manifold embedded in the Euclidean space and the square of the distance function from the manifold. In particular, we are interested in curvature invariants like the mean curvature vector and the second fundamental form. We find that these invariants can be computed in a very simple way using the third order derivatives of the squared distance function. Moreover, we study a general class of functionals depending on the derivatives up to a given order γ of the squared distance function and we find an algorithm for the computation of the Euler equation. Our class of functionals includes as particular cases the well-known area functional (γ = 2), the integral of the square of the quadratic norm of the second fundamental form (γ = 3), and the Willmore functional.     

The mean curvature of cylindrically bounded submanifolds

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder in a product Riemannian manifold . It follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in Euclidean space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to a conjecture of Calabi on complete minimal hypersurfaces cannot be proper. As another application of our method, we derive a result about the stochastic incompleteness of submanifolds with sufficiently small mean curvature. Dedicated to Professor Manfredo P. do Carmo on the occasion of his 80th birthday.     

The Spectrum of the Martin-Morales-Nadirashvili Minimal Surfaces Is Discrete

We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of ℝ³ is discrete. This gives a positive answer to a question of Yau (Asian J. Math. 4:235–278, 2000).     

Bundle stabilisation and positive Ricci curvature

If E is the total space of a vector bundle over a compact Ricci non-negative manifold, it is known that E×Rp admits a complete metric of positive Ricci curvature for all sufficiently large p. In this paper we establish a small, explicit lower bound for the dimension p.     

Antiplectic Structure, Diffeomorphism and Generalized KdV Family

In this paper we show that the generalized KdV, generalized Camassa–Holm equations and the corresponding Möbius invariant generalized Schwarzian KdV, Schwarzian CH equations can be realized in terms of flows induced by on the space of differential operators and on the space of immersion curves, respectively. These are Euler–Poincaré type flows, and one of the flow takes place on an infinite-dimensional Poisson manifold and the other on a slightly degenerate infinite-dimensional Symplectic manifold. They form an Antiplectic pair. We also study Euler–Poincaré flow with respect to metric, and this induces generalized Camassa–Holm equation. In the final section we discuss the Antiplectic pair in dimensions.Dedicated to Professor George Wilson on his 65th birthday with great respect and admiration.     

A Construction of Constant Scalar Curvature Manifolds with Delaunay-type Ends

It has been showed by Byde (Indiana Univ. Math. J. 52(5):1147–1199, 2003) that it is possible to attach a Delaunay-type end to a compact nondegenerate manifold of positive constant scalar curvature, provided it is locally conformally flat in a neighborhood of the attaching point. The resulting manifold is noncompact with the same constant scalar curvature. The main goal of this paper is to generalize this result. We will construct a one-parameter family of solutions to the positive singular Yamabe problem for any compact non-degenerate manifold with Weyl tensor vanishing to sufficiently high order at the singular point. If the dimension is at most 5, no condition on the Weyl tensor is needed. We will use perturbation techniques and gluing methods.     

Closed manifolds with small excess

In this article, we study closed Riemannian manifolds with small excess. We show that a closed connected Riemannian manifold with Ricci curvature and injectivity radius bounded from below is homeomorphic to a sphere if it has sufficiently small excess. We also show that a closed connected Riemannian manifold with weakly bounded geometry is a homotopy sphere if its excess is small enough.     

Constant mean curvature spheres in Riemannian manifolds

We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points.     

Collapsing manifolds with boundary

This paper studies manifolds-with-boundary collapsing in the Gromov– Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The first group of results provide a fiber bundle structure to the manifolds-with-boundary. One of the main theorems establishes a disc bundle structure for any manifold-with-boundary having two-sided bounds on sectional curvature and second fundamental form, and a lower bound on intrinsic injectivity radius, which is sufficiently close in the Gromov–Hausdorff topology to a closed manifold. Another result is a rough version of Toponogov’s Splitting Theorem. The second group of results identify Gromov–Hausdorff limits of certain sequences of manifolds with non-convex boundaries as Alexandrov spaces of curvature bounded below.     

Constant Mean Curvature Surfaces with Boundaryin Hyperbolic Space

 We study constant mean curvature compact surfaces immersed in hyperbolic space with non-empty boundary (=H-surfaces). We prove that the only H-surfaces with boundary circular and 0≤∣H∣≤1, are the umbilical examples. When the surface is embedded, conditions to be umbilical are given. Finally, we characterize umbilical surfaces bounded by a circle among all H-discs with small area. Received 27 March 1997; in final form 11 June 1998     

Hypersurfaces in statistical manifolds

The condition for the curvature of a statistical manifold to admit a kind of standard hypersurface is given as a first step of the statistical submanifold theory. A complex version of the notion of statistical structures is also introduced.     

Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of a Riemannian Manifold

In this paper we study a Riemannian metric on the tangent bundle T(M) of a Riemannian manifold M which generalizes Sasaki metric and Cheeger–Gromoll metric and a compatible almost complex structure which confers a structure of locally conformal almost K?hlerian manifold to T(M) together with the metric. This is the natural generalization of the well known almost K?hlerian structure on T(M). We found conditions under which T(M) is almost K?hlerian, locally conformal K?hlerian or K?hlerian or when T(M) has constant sectional curvature or constant scalar curvature. Then we will restrict to the unit tangent bundle and we find an isometry with the tangent sphere bundle (not necessary unitary) endowed with the restriction of the Sasaki metric from T(M). Moreover, we found that this map preserves also the natural contact structures obtained from the almost Hermitian ambient structures on the unit tangent bundle and the tangent sphere bundle, respectively. This work was also partially supported by Grant CEEX 5883/2006–2008, ANCS, Romania.     

Affine umbilical surfaces inR 4

Summary A surface in ℝ⁴ is called affine umbilical if for each vector belonging to the affine normal plane the corresponding shape operator is a multiple of the identity. We will classify affine umbilical definite surfaces which either have constant curvature or which satisfy ∇^⊥ g ^⊥. Furthermore, it will be shown that for an affine umbilical definite surface, the affine mean curvature vector can not have constant non-zero length. The last author is a Senior Research Assistant of the National Fund for Scientific Research (Belgium) This article was processed by the author using the Springer-Verlag TEX PJour1g macro package 1991.     

On families of Lagrangian submanifolds

We study the stability of a compact Lagrangian submanifold of a symplectic manifold under perturbation of the symplectic structure. If X is a compact manifold and the ω _t are cohomologous symplectic forms on X, then by a well-known theorem of Moser there exists a family Φ _t of diffeomorphisms of X such that ω _t =Φ _t ^*(ω₀). If L⊂X is a Lagrangian submanifold for (X,ω₀), L _t =Φ _t ^-1(L) is thus a Lagrangian submanifold for (X,ω _t ). Here we show that if we simply assume that L is compact and ω _t | _L is exact for every t, a family L _t as above still exists, for sufficiently small t. Similar results are proved concerning the stability of special Lagrangian and Bohr–Sommerfeld special Lagrangian submanifolds, under perturbation of the ambient Calabi–Yau structure. Received: 29 May 2001/ Revised version: 17 October 2001     

The prescribed curvature problem in dimension four

Necessary and sufficient conditions for a g-valued differential 2-form on a 4-dimensional manifold to be, locally, a curvature form, are given. The dimension four is exceptional for the problem of prescribed curvature as, in this dimension, Bianchi's identities can be eliminated for a large class of Lie algebras, including semisimple algebras. Hence, the curvature forms are characterized as the solutions to a second-order partial differential system, which is proved to be formally integrable.     

Averaging of Jacobi fields along geodesics on manifolds of random curvature

The paper considers the Jacobi field along a geodesic on a Riemannian manifold on which the curvature is a stochastic process. The author introduces the concept of linearizing tensor of the Jacobi field on the basis of which a sufficiently universal averaging algorithm is constructed. The equations for higher-order means 〈y ^p 〉 for p = 2, 3, 4 are deduced. It is shown that these statistical means, as well as the expectation of the Jacobi field, exponentially grow even in the case where the mean value of the curvature vanishes. The growth exponents of higher statistical moments of the Jacobi field obtained analytically with the corresponding exponents obtained from the numerical experiment are compared. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 57, Suzdal Conference–2006, Part 3, 2008.     

Asymptotically hyperbolic metrics on the unit ball with horizons

In this paper, we construct a family of three-dimensional asymptotically hyperbolic manifolds with horizons and with scalar curvature equal to −6. The manifolds we construct can be arbitrarily close to anti-de Sitter-Schwarzschild manifolds at infinity. Hence, the mass of our manifolds can be very large or very small. The main arguments we use in this paper are gluing methods which are used by Miao in (Proc Am Math Soc 132(1):217–222, 2004).     

The lightlike flat geometry on spacelike submanifolds of codimension two in Minkowski space

We introduce the notion of the lightcone Gauss–Kronecker curvature for a spacelike submanifold of codimension two in Minkowski space, which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. In the local sense, this curvature describes the contact of such submanifolds with lightlike hyperplanes. We study geometric properties of such curvatures and show a Gauss–Bonnet type theorem. As examples we have hypersurfaces in hyperbolic space, spacelike hypersurfaces in the lightcone and spacelike hypersurfaces in de Sitter space.     

Nonexistence Results of Minimal Immersions

Let M be a Riemannian m-dimensional manifold with m ≥ 3, endowed with non zero parallel p-form. We prove that there is no minimal isometric immersions of M in a Riemannian manifold N with constant strictly negative sectional curvature. Next we show that, under the conform flatness of the manifold N and some assumptions on the Ricci curvature of N, there is no α-pluriharmonic isometric immersion.