Geometric Heat Comparison Criteria for Riemannian Manifolds

The main results of this paper are small-time heat comparison results for two points in two manifolds with characteristic functions as initial temperature distributions (Theorems 1 and 2). These results are based on the geometric concepts of (essential) distance from the complement and spherical area function. We also discuss some other geometric results about the heat development and illustrate them by examples. Mathematics subject classifications (2000): 58J35, 35K05     

正则局部环的判别法（英）

本文给出了非Artin的Noether局部环(A, m)为正则环的充要条件.设n为正整数，则(A, m)为正则环的充要条件为A/mⁿ的投射维数或mⁿ的内射维数是有限的.     

Local Convexity on Smooth Manifolds

Some properties of the spaces of paths are studied in order to define and characterize the local convexity of sets belonging to smooth manifolds and the local convexity of functions defined on local convex sets of smooth manifolds. This paper is dedicated to the memory of Guido Stampacchia. This research was supported in part by the Hungarian Scientific Research Fund, Grants OTKA-T043276 and OTKA-T043241, and by CNR, Rome, Italy.     

A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

We consider the problem of finding a singularity of a differentiable vector field X defined on a complete Riemannian manifold. We prove a unified result for theexistence and local uniqueness of the solution, and for the local convergence of a Riemannian version of Newton's method. Our approach relies on Kantorovich's majorant principle: under suitable conditions, we construct an auxiliary scalar equation φ(r) = 0 which dominates the original equation X(p) = 0 in the sense that the Riemannian-Newton method for the latter inherits several features of the real Newton method applied to the former. The majorant φ is derived from an adequate radial parametrization of a Lipschitz-type continuity property of the covariant derivative of X, a technique inspired by the previous work of Zabrejko and Nguen on Newton's method in Banach spaces. We show how different specializations of the main result recover Riemannian versions of Kantorovich's theorem and Smale's α-theorem, and, at least partially, the Euclidean self-concordant theory of Nesterov and Nemirovskii. In the specific case of analytic vector fields, we improve recent developments inthis area by Dedieu et al. . Some Riemannian techniques used here were previously introduced by Ferreira and Svaiter in the context of Kantorovich's theorem for vector fields with Lipschitz continuous covariant derivatives.     

Levi-Flat Hypersurfaces Tangent to Projective Foliations

Let M?? ⁿ be a singular real-analytic Levi-flat hypersurface tangent to a codimension-one holomorphic foliation \(\mathcal{F}\) on ? ⁿ . For n≥3, we give sufficient conditions to guarantee the existence of degenerate singularities in M, (in the sense of Segre varieties) and as a consequence we prove that \(\mathcal{F}\) can be defined by a global closed meromorphic 1-form.     

Local Hardy Spaces of Differential Forms on Riemannian Manifolds

We define local Hardy spaces of differential forms $h^{p}_{\mathcal{D}}(\wedge T^{*}M)$ for all p∈[1,∞] that are adapted to a class of first-order differential operators $\mathcal{D}$ on a complete Riemannian manifold M with at most exponential volume growth. In particular, if D is the Hodge–Dirac operator on M and Δ=D ² is the Hodge–Laplacian, then the local geometric Riesz transform D(Δ+aI)^?1/2 has a bounded extension to $h^{p}_{D}$ for all p∈[1,∞], provided that a>0 is large enough compared to the exponential growth of M. A characterization of $h^{1}_{\mathcal{D}}$ in terms of local molecules is also obtained. These results can be viewed as the localization of those for the Hardy spaces of differential forms $H^{p}_{D}(\wedge T^{*}M)$ introduced by Auscher, McIntosh, and Russ.     

Algebraic Levi-Flat Hypervarieties in Complex Projective Space

We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We define the rank of an algebraic Levi-flat hypersurface and study the connections between rank, degree, and the type and size of the singularity. In particular, we study degenerate singularities of algebraic Levi-flat hypersurfaces. We then give necessary and sufficient conditions for a Levi-flat hypersurface to be a pullback of a real-analytic curve in ℂ via a meromorphic function. Among other examples, we construct a nonalgebraic semianalytic Levi-flat hypersurface with compact leaves that is a perturbation of an algebraic Levi-flat variety.     

On Levi-Flat Hypersurfaces with Generic Real Singular Set

In this paper, we consider the holomorphic extension problem for the Levi foliation of a Levi-flat real-analytic hypersurface whose singular set is a generic real submanifold.     

应用测试元素判定Hadamard流形上等距群的离散性

我们应用给定斜驶元或抛物元作为测试元素,对Hadamard流形上的等距群Isom(X)的子群的离散性进行了研究,得到了几个离散判别准则.     

On the Local Behavior of Sobolev Classes on Two-Dimensional Riemannian Manifolds

Ukrainian Mathematical Journal - We study open discrete maps of two-dimensional Riemannian manifolds from the Sobolev class. For these mappings, we establish the lower estimates of distortions of...     

A Local Property of Large Polyhedral Maps on Compact 2-Dimensional Manifolds

We prove that each polyhedral map G on a compact 2-manifold, which has large enough vertices, contains a k-path, a path on k vertices, such that each vertex of it has, in G, degree at most 6k; this bound being best possible for k even. Moreover, if G has large enough vertices of degree >6k, than it contains a k-path such that each its vertex has degree, in G, at most 5k; this bound is best possible for any k. Received: December 8, 1997 Revised: April 27, 1998     

Local Hamilton type Gradient Estimates and Harnack Inequalities for Nonlinear Parabolic Equations on Riemannian Manifolds

In this paper,we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation u_t(x,t)=Δu(x,t)+au(x,t) ln u(x,t)+bu^α(x,t),on M×(-∞,∞) with α∈R,where a and b are constants.As application,the Harnack inequalities are derived.     

Steiner Ratio for Manifolds

The Steiner ratio characterizes the greatest possible deviation of the length of a minimal spanning tree from the length of the minimal Steiner tree. In this paper, estimates of the Steiner ratio on Riemannian manifolds are obtained. As a corollary, the Steiner ratio for flat tori, flat Klein bottles, and projective plane of constant positive curvature are computed.     

Local Stable and Unstable Manifolds and Their Control in Nonautonomous Finite-Time Flows

It is well known that stable and unstable manifolds strongly influence fluid motion in unsteady flows. These emanate from hyperbolic trajectories, with the structures moving nonautonomously in time. The local directions of emanation at each instance in time is the focus of this article. Within a nearly autonomous setting, it is shown that these time-varying directions can be characterised through the accumulated effect of velocity shear. Connections to Oseledets spaces and projection operators in exponential dichotomies are established. Availability of data for both infinite- and finite-time intervals is considered. With microfluidic flow control in mind, a methodology for manipulating these directions in any prescribed time-varying fashion by applying a local velocity shear is developed. The results are verified for both smoothly and discontinuously time-varying directions using finite-time Lyapunov exponent fields, and excellent agreement is obtained.     

On Local Metric Characteristics of Level Sets of CH1-Mappings of Carnot Manifolds

Siberian Mathematical Journal - Considering the level surfaces of the mappings of class C1 which are defined on Carnot manifolds and take values in Carnot—Carathéodory spaces, we...     

Local Nonautonomous Schrödinger Flows on Kähler Manifolds

In this paper, we prove that the nonautonomous Schrödinger flow from a compact Riemannian manifold into a Kähler manifold admits a local solution