scholarly journals A Pinching Theorem for Compact Minimal Submanifolds in Warped Products I×fSm(c)

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1445
Author(s):  
Xin Zhan ◽  
Zhonghua Hou
Keyword(s):  
Minimal Submanifolds ◽  
Positive Function ◽  
Open Interval ◽  
Type I ◽  
Warped Products ◽  
Euclidean Sphere ◽  
Smooth Positive Function

Let Sm(c) be a Euclidean sphere of curvature c>0 and R be a Euclidean line. We prove a pinching theorem for compact minimal submanifolds immersed in Riemannian warped products of the type I×fSm(c), where f:I→R+ is a smooth positive function on an open interval I of R. This allows us to generalize Chen-Cui’s pinching theorem from Riemannian products Sm(c)×R to Riemannian warped products I×fSm(c).

On the Simple Inductive Limits of Splitting Interval Algebras with Dimension Drops

2012 ◽  
Vol 64 (3) ◽  
pp. 544-572 ◽  
Author(s):  
Zhiqiang Li
Keyword(s):  
Open Interval ◽  
Finite Union ◽  
Type I ◽  
Inductive Limits ◽  
Direct Sums ◽  
Theoretic Classification

Abstract A K-theoretic classification is given of the simple inductive limits of finite direct sums of the type I C*-algebras known as splitting interval algebras with di- mension drops. (These are the subhomogeneous C*-algebras, each having spectrum a finite union of points and an open interval, and torsion K1-group.)

scholarly journals Non-degeneracy of least-energy solutions for an elliptic problem with nearly critical nonlinearity

2010 ◽  
Vol 140 (1) ◽  
pp. 203-222
Author(s):  
Futoshi Takahashi
Keyword(s):  
Recent Result ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Positive Function ◽  
Coefficient Function ◽  
Critical Nonlinearity ◽  
Smooth Bounded Domain ◽  
Least Energy Solutions ◽  
Least Energy ◽  
Smooth Positive Function

We consider the problem −Δu = c0K(x)upε, u > 0 in Ω, u = 0 on δΩ, where Ω is a smooth, bounded domain in ℝN, N ≥ 3, c0 = N(N − 2), pε = (N + 2)/(N − 2) − ε and K is a smooth, positive function on . We prove that least-energy solutions of the above problem are non-degenerate for small ε > 0 under some assumptions on the coefficient function K. This is a generalization of the recent result by Grossi for K ≡ 1, and needs precise estimates and a new argument.

scholarly journals Deforming convex bodies in Minkowski geometry

2021 ◽  
Author(s):  
V. Rovenski ◽  
P. Walczak
Keyword(s):  
Recent Work ◽  
Finsler Geometry ◽  
Convex Bodies ◽  
Positive Function ◽  
Euclidean Norm ◽  
Minkowski Geometry ◽  
Linearly Independent ◽  
Minkowski Norms ◽  
Smooth Positive Function ◽  
Axis Passing

We introduce and study certain deformation of Minkowski norms in [Formula: see text] determined by a set of [Formula: see text] linearly independent 1-forms and a smooth positive function of [Formula: see text] variables. In particular, the deformation of a Euclidean norm [Formula: see text] produces a Minkowski norm defined in our recent work; its indicatrix is a rotation hypersurface with a [Formula: see text]-dimensional axis passing through the origin. For [Formula: see text], our deformation generalizes the construction of [Formula: see text]-norms which form a rich class of “computable” Minkowski norms and play an important role in Finsler geometry. We characterize such pairs of a Minkowski norm and its image that Cartan torsions of the two norms either coincide or differ by a [Formula: see text]-reducible term. We conjecture that for [Formula: see text] any Minkowski norm can be approximated by images of a Euclidean norm.

Special multiply Einstein warped products with an affine connection

2018 ◽  
Vol 15 (07) ◽  
pp. 1850107
Author(s):  
Dan Dumitru
Keyword(s):  
Warped Product ◽  
Affine Connection ◽  
Open Interval ◽  
Einstein Space ◽  
Warped Products ◽  
Ricci Flat ◽  
Metric Connection

The aim of this paper is to study special multiply Einstein warped products having an affine connection. Let [Formula: see text] be a multiply warped product such that [Formula: see text] is an open interval, [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] for every [Formula: see text] [Formula: see text] [Formula: see text] and [Formula: see text] an affine connection on [Formula: see text] We compute the warping functions that make [Formula: see text] an Einstein space in the following cases: (a) [Formula: see text] is a semi-symmetric metric/non-metric connection and all the fibers are Ricci flat. (b) [Formula: see text] is a quarter-symmetric metric/non-metric connection and all the fibers are Ricci flat.

scholarly journals The determination of caloric morphisms on Euclidean domains

2000 ◽  
Vol 158 ◽  
pp. 133-166 ◽  
Author(s):  
Katsunori Shimomura
Keyword(s):  
Heat Equation ◽  
Positive Function ◽  
Smooth Mapping ◽  
Direct Sum ◽  
Smooth Positive Function ◽  
Euclidean Domains

AbstractLetDbe a domain in ℝm+1andEbe a domain in ℝn+1. A pair of a smooth mappingf:D → Eand a smooth positive function ϕ onDis called a caloric morphism if ϕ˙uofis a solution of the heat equation inDwheneveruis a solution of the heat equation inE. We give the characterization of caloric morphisms, and then give the determination of caloric morphisms. In the case ofm < n, there are no caloric morphisms. In the case ofm = n, caloric morphisms are generated by the dilation, the rotation, the translation and the Appell transformation. In the case ofm > n, under some assumption onf, every caloric morphism is obtained by composing a projection with a direct sum of caloric morphisms of ℝn+1.

scholarly journals On the principal Ricci curvatures of a Riemannian 3-manifold

2019 ◽  
Vol 19 (2) ◽  
pp. 251-262
Author(s):  
Amir Babak Aazami ◽  
Charles M. Melby-Thompson
Keyword(s):  
Vector Field ◽  
Positive Constant ◽  
Ricci Tensor ◽  
Unit Length ◽  
Positive Function ◽  
Universal Cover ◽  
Riemannian Metric ◽  
Ricci Solitons ◽  
Topological Obstruction ◽  
Smooth Positive Function

Abstract We study global obstructions to the eigenvalues of the Ricci tensor on a Riemannian 3-manifold. As a topological obstruction, we first show that if the 3-manifold is closed, then certain choices of the eigenvalues are prohibited: in particular, there is no Riemannian metric whose corresponding Ricci eigenvalues take the form (−μ, f, f), where μ is a positive constant and f is a smooth positive function. We then concentrate on the case when one of the eigenvalues is zero. Here we show that if the manifold is complete and its Ricci eigenvalues take the form (0, λ, λ), where λ is a positive constant, then its universal cover must split isometrically. If the manifold is closed, scalar-flat, and its zero eigenspace contains a unit length vector field that is geodesic and divergence-free, then the manifold must be flat. Our techniques also apply to the study of Ricci solitons in dimension three.

On the geometry of submanifolds in certain warped products

2021 ◽  
pp. 2150044
Author(s):  
Jogli G. Araújo ◽  
Henrique F. de Lima ◽  
Eraldo A. Lima ◽  
Márcio S. Santos
Keyword(s):  
Warped Product ◽  
Minimal Submanifolds ◽  
Maximum Principles ◽  
Warping Function ◽  
Warped Products ◽  
Rigidity Results ◽  
Type Formula

In this paper, we deal with [Formula: see text]-dimensional submanifolds immersed in a slab of a warped product of the type [Formula: see text]. Under suitable constraints on the warping function [Formula: see text] and assuming that such a submanifold [Formula: see text] is either complete or stochastically complete, we apply some maximum principles in order to show that [Formula: see text] must be contained in a slice of [Formula: see text]. In particular, from our results we guarantee the nonexistence of [Formula: see text]-dimensional closed minimal submanifolds immersed in [Formula: see text]. Furthermore, we construct a nontrivial duo-graph in [Formula: see text] which illustrates the importance of our rigidity results.

SPECTRAL STUDY OF A COUPLED COMPACT-NONCOMPACT PROBLEM IN A NON-SELF-ADJOINT CASE

1994 ◽  
Vol 04 (05) ◽  
pp. 607-624
Author(s):  
A. BENKADDOUR ◽  
J. SANCHEZ-HUBERT ◽  
A. RIDHA
Keyword(s):  
Porous Medium ◽  
Essential Spectrum ◽  
Neumann Boundary ◽  
Positive Function ◽  
Acoustic Vibration ◽  
Resolvent Set ◽  
First Case ◽  
Free Air ◽  
Impedance Condition ◽  
Smooth Positive Function

We consider a coupled problem of acoustic vibration of air in a porous medium Ω p that is in contact, by a plane interface Γ (x1=0), with free air in some region Ω f . The porous medium is made of infinitely narrow thin channels parallel to the x1-axis. In Refs. 1–3 and 5, problems of this type were considered with homogeneous Neumann boundary conditions on the exterior boundary of the whole domain Ω, Ω= Ω f ∪Γ∪Ω p . In this paper, we study the problem in the mentioned configuration but with the impedance condition ∂u/∂n=u/Z (where Z is a given complex number) on a part of the boundary of the porous medium Ω p defined by x1=–ℓ(x2, x3), where ℓ is a smooth positive function. Let us denote by A the operator associated with the coupled eigenvalue problem –Au=ω2u, and by Ap(x2, x3) the corresponding problem in a channel x2= const , x3= const . As in Ref. 1, two cases appear according to the values of ω2. In the first case ω2 is not an eigenvalue of Ap(x2, x3), and then we show that ω2 is either a point of the resolvent set or an isolated eigenvalue with finite multiplicity of A. In the second case ω2 is an eigenvalue of Ap(a2, a3) for some value (a2, a3) of (x2, x3), and then ω2 is a point of the essential spectrum of A. The novelty of this paper is the study of the spectrum of the operator A which is non self-adjoint and has a noncompact resolvent. We find a complex essential spectrum of A and we study its topological structure.

Heat-Etching with a Bullivant Type II Simple Freeze-Cleave Device

1970 ◽  
Vol 28 ◽  
pp. 106-107 ◽  
Author(s):  
Ronald S. Weinstein ◽  
N. Scott McNutt
Keyword(s):  
New Zealand ◽  
General Hospital ◽  
Massachusetts General Hospital ◽  
Type I ◽  
Type Ii ◽  
Collaborative Effort ◽  
Temperature And Pressure ◽  
The University

The Type I simple cold block device was described by Bullivant and Ames in 1966 and represented the product of the first successful effort to simplify the equipment required to do sophisticated freeze-cleave techniques. Bullivant, Weinstein and Someda described the Type II device which is a modification of the Type I device and was developed as a collaborative effort at the Massachusetts General Hospital and the University of Auckland, New Zealand. The modifications reduced specimen contamination and provided controlled specimen warming for heat-etching of fracture faces. We have now tested the Mass. General Hospital version of the Type II device (called the “Type II-MGH device”) on a wide variety of biological specimens and have established temperature and pressure curves for routine heat-etching with the device.

Two Distinct Types of Microfilaments in the Cytoplasm of Human Adenohypophysial Cells

1973 ◽  
Vol 31 ◽  
pp. 668-669
Author(s):  
E. Horvath ◽  
K. Kovacs ◽  
I. E. Stratmann ◽  
C. Ezrin
Keyword(s):  
Pituitary Tumors ◽  
Average Diameter ◽  
Anterior Lobe ◽  
Uranyl Acetate ◽  
Type I ◽  
Breast Cancer Patients ◽  
Human Pituitary ◽  
Severe Diabetes ◽  
Pituitary Glands ◽  
Membrane Bound

Surgically removed human pituitary glands as well as pituitary tumors fixed in glutaraldehyde, postfixed in osmium tetroxide, embedded in epon resin, stained with uranyl acetate and lead citrate have been investigated by electron microscopy in order to correlate ultrastructure with functional activity. In the course of this study two distinct types of microfilaments have been identified in the cytoplasm of adenohypophysiocytes.Type I microfilaments (Fig. 1) were found in the cytoplasm of anterior lobe cells of five female subjects with disseminated mammary cancer and two patients with severe diabetes mellitus. The breast cancer patients were treated pre-operatively for various periods of time with different doses of oxysteroids. The microfilaments had an average diameter of JO A, formed parallel bundles, were scattered irregularly in the cytoplasm and were frequently located in the perikaryon. They were not membrane-bound and failed to show any periodicity.

Sign in / Sign up

Export Citation Format

Share Document