The Gradient Flow of <Emphasis Type="Italic">∫</Emphasis><Subscript><Emphasis Type="Italic">M</Emphasis></Subscript>|Rm|<Superscript>2</Superscript>

We study the gradient flow of the Riemannian functional ℱ(g):=∫ _M |Rm|². This flow corresponds to a fourth-order degenerate parabolic equation for a Riemannian metric. We prove that the degeneracies may be accounted for entirely by diffeomorphism flow, and hence we show short-time existence using the DeTurck method. We prove L ² derivative estimates of Bernstein-Bando-Shi type and use these to give a basic obstruction to long time existence and prove a compactness theorem.      

关于退化中立型微分方程的周期解

讨论了退化中立型微分方程的周期解问题,给出了周期解存在性的条件和二维退化中立型微分方程周期解存在的代数判据,并且举例说明了其应用.     

A Sobolev-like inequality for the Dirac operator

In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We finally specify a case where this equation can be solved.     

THE V-FUNCTIONAL METHOD FOR THE STABILITY OF DEGENERATE DIFFERENTIAL SYSTEMS WITH DELAYS

1 IntroductionStability is an important perfermance index of concrete systems such asmanagement systems, power systems, engineering systems and so on. Forthe stability V--functional criterion method of functional differential systems,have had many results in recent years, see) for example, [1] and [2]. Andfor degenerate systems, papers [41-[81 have carefully discussed and given someimportant results. But we note that (see [9]) many concrete systems, besidescontain delay, are degenerate. So, f…     

The Riemannian <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> topology on the manifold of Riemannian metrics

We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate the topology on the manifold of metrics induced by the distance function of the L ² Riemannian metric—so-called because it induces an L ² topology on each tangent space. It turns out that this topology on the tangent spaces gives rise to an L ¹-type topology on the manifold of metrics itself. We study this new topology and its completion, which agrees homeomorphically with the completion of the L ² metric. We also give a user-friendly criterion for convergence (with respect to the L ² metric) in the manifold of metrics.     

On the existence of almost contact structure and the contact magnetic field

We give a simple proof of the existence of an almost contact metric structure on any orientable 3-dimensional Riemannian manifold (M ³, g) with the prescribed metric g as the adapted metric of the almost contact metric structure. By using the key formula for the structure tensor obtained in the proof this theorem, we give an application which allows us to completely determine the magnetic flow of the contact magnetic field in any 3-dimensional Sasakian manifold.     

Special metrics and triality

We investigate a new 8-dimensional Riemannian geometry defined by a generic closed and coclosed 3-form with stabiliser PSU(3), and which arises as a critical point of Hitchin's variational principle. We give a Riemannian characterisation of this structure in terms of invariant spinor-valued 1-forms, which are harmonic with respect to the twisted Dirac operator ? on Δ⊗Λ¹. We establish various obstructions to the existence of topological reductions to PSU(3). For compact manifolds, we also give sufficient conditions for topological PSU(3)-structures that can be lifted to topological SU(3)-structures. We also construct the first known compact example of an integrable non-symmetric PSU(3)-structure. In the same vein, we give a new Riemannian characterisation for topological quaternionic Kähler structures which are defined by an Sp(1)⋅Sp(2)-invariant self-dual 4-form. Again, we show that this form is closed if and only if the corresponding spinor-valued 1-form is harmonic for ? and that these equivalent conditions produce constraints on the Ricci tensor.     

Strichartz Estimates for Non-Elliptic Schrödinger Equations on Compact Manifolds

In this note we consider the Schrödinger equation on compact manifolds equipped with possibly degenerate metrics. We prove Strichartz estimates with a loss of derivatives. The rate of loss of derivatives depends on the degeneracy of metrics. For the non-degenerate case we obtain, as an application of the main result, the same Strichartz estimates as that in the elliptic case. This extends Strichartz estimates for Riemannian metrics proved by Burq-Gérard-Tzvetkov to the non-elliptic case and improves the result by Salort for the degenerate case. We also investigate the optimality of the result for the case on 𝕊³ × 𝕊³.     

Compactification of moduli space of harmonic mappings

We introduce the notion of harmonic nodal maps from the stratified Riemann surfaces into any compact Riemannian manifolds and prove that the space of the energy minimizing nodal maps is sequentially compact. We also give an existence result for the energy minimizing nodal maps. As an application, we obtain a general existence theorem for minimal surfaces with arbitrary genus in any compact Riemannian manifolds. Received: 1 April 1997; revised: 15 April 1998.     

Best constant in critical Sobolev inequalities of second-order in the presence of symmetries

Let (M,g) be a smooth compact Riemannian manifold. We first give the value of the best first constant for the critical embedding H²(M)?L^2?(M) for second-order Sobolev spaces of functions invariant by some subgroup of the isometry group of (M,g). We also prove that we can take ?=0 in the corresponding inequality under some geometric assumptions. As an application we give a sufficient condition for the existence of a smooth positive symmetric solution to a critical equation with a symmetric Paneitz-Branson-type operator. A sufficient condition for the existence of a nodal solution to such an equation is also derived. We eventually prove a multiplicity result for such an equation.     

On uniruled degenerations of algebraic varieties with trivial canonical divisor

We give a uniruledness criterion on degenerate fibers in a family of varieties whose general fiber has a numerically trivial canonical divisor. The criterion is related with the so-called non-nef locus of the canonical divisor of the total space.      

A Note on the <Emphasis Type="Italic">p</Emphasis>-Parabolicity of Submanifolds

We give a geometric criterion which shows p-parabolicity of a class of submanifolds in a Riemannian manifold, with controlled second fundamental form, for p ≥ 2.     

Existence of Ricci Flows of Incomplete Surfaces

We prove a general existence result for instantaneously complete Ricci flows starting at an arbitrary Riemannian surface which may be incomplete and may have unbounded curvature. We give an explicit formula for the maximal existence time, and describe the asymptotic behaviour in most cases.     

Transnormal functions on a Riemannian manifold

We extend theorems of É. Cartan, Nomizu, Münzner, Q.M. Wang, and Ge–Tang on isoparametric functions to transnormal functions on a general Riemannian manifold. We show that if a complete Riemannian manifold M admits a transnormal function, then M is diffeomorphic to either a vector bundle over a submanifold, or a union of two disk bundles over two submanifolds. Moreover, a singular level set Q is austere and minimal, if exists, and generic level sets are tubes over Q. We give a criterion for a transnormal function to be an isoparametric function.     

Nonlocal Interaction Equations in Environments with Heterogeneities and Boundaries

We study well-posedness of a class of nonlocal interaction equations with spatially dependent mobility. We also allow for the presence of boundaries and external potentials. Such systems lead to the study of nonlocal interaction equations on subsets ? of ? ^d endowed with a Riemannian metric g. We obtain conditions, relating the interaction potential and the geometry, which imply existence, uniqueness and stability of solutions. We study the equations in the setting of gradient flows in the space of probability measures on ? endowed with Riemannian 2-Wasserstein metric.     

On geometric vector fields of Minkowski spaces and their applications

As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F⁻¹(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian.     

Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields

In this paper, we study the global existence and the asymptotic behavior of classical solution of the Cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields. We prove that the global C¹ solution exists uniquely if the BV norm of the initial data is sufficiently small. Based on the existence result on the global classical solution, we show that, when the time t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions. Finally, we give an application to the equation for time-like extremal surfaces in the Minkowski space-time R1+n.     

Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature

We give an estimate of the smallest spectral value of the Laplace operator on a complete noncompact stable minimal hypersurface M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface M has sufficiently small total scalar curvature then M has only one end. We also obtain a vanishing theorem for L ² harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover, we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable.     

Nonlinear Bound States on Weakly Homogeneous Spaces

We prove the existence of ground state solutions for a class of nonlinear elliptic equations, arising in the production of standing wave solutions to an associated family of nonlinear Schrödinger equations. We examine two constrained minimization problems, which give rise to such solutions. One yields what we call F _λ-minimizers, the other energy minimizers. We produce such ground state solutions on a class of Riemannian manifolds called weakly homogeneous spaces, and establish smoothness, positivity, and decay properties. We also identify classes of Riemannian manifolds with no such minimizers, and classes for which essential uniqueness of positive solutions to the associated elliptic PDE fails.     

Classes of submersions of Riemannian manifolds with compact fibers

In Riemannian geometry and its applications, the most popular is the class of Riemannian submersions (and foliations) [1–4] which are characterized by simplest mutual disposition of fibers. The purpose of the present article is to introduce other, more general, classes of submersions of Riemannian manifolds which, as well as the class of Riemannian submersions, are described by simple local properties of configuration tensors and to begin their study.Given a submersion :MM of differentiable manifolds with compact connected fibers and any metric onM, we define a metric on the base with the help of theL ²-norm of horizontal fields. In this caseT¯ M becomes a subbundle of some larger bundleM. The main class of totally geodesic submersions introduced in the article (Definition 1) corresponds to the metrics onM with simplest disposition ofT¯ M inM. In the article we obtain a criterion for such submersions (Corollary 1); existence is proved by means of the product with a metric varying along fibers (Example 2). To study totally geodesic submersions, we use ideas from the theory of Riemannian submersions and submanifolds with degenerate second form (Theorems 1 and 2 and Corollary 4).Foliations modeled by totally geodesic submersions (see equality (13)) are of interest too, but we leave them beyond the scope of the article.This work was supported by the Russian Foundation for Fundamental Research (Grant 94-01-00271).Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 1154–1164, September–October, 1994.