scholarly journals Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds

2020 ◽  
Vol 2020 (764) ◽  
pp. 71-109 ◽  
Author(s):  
Paul Bryan ◽  
Mohammad N. Ivaki ◽  
Julian Scheuer
Keyword(s):  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Mean Curvature Flow ◽  
Einstein Manifolds ◽  
De Sitter ◽  
Harnack Estimate ◽  
Nonnegative Sectional Curvature ◽  
Harnack Inequalities ◽  
Harnack Estimates ◽  
New Type

AbstractWe obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant nonnegative sectional curvature as well as in the Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack estimate with a bonus term for mean curvature flow in locally symmetric Riemannian Einstein manifolds of nonnegative sectional curvature. Using a concept of “duality” for strictly convex hypersurfaces, we also obtain a new type of inequality, so-called “pseudo”-Harnack inequality, for expanding flows in the sphere and in the hyperbolic space.

scholarly journals CHARACTERIZATIONS OF LINEAR WEINGARTEN SPACELIKE HYPERSURFACES IN EINSTEIN SPACETIMES

2013 ◽  
Vol 55 (3) ◽  
pp. 567-579 ◽  
Author(s):  
HENRIQUE F. DE LIMA ◽  
JOSEÍLSON R. DE LIMA
Keyword(s):  
Maximum Principle ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Sufficient Conditions ◽  
Analytical Tool ◽  
Isoparametric Hypersurface ◽  
De Sitter ◽  
Principal Curvatures ◽  
Spacelike Hypersurfaces ◽  
Generalized Maximum Principle

AbstractOur purpose is to study the geometry of linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Einstein spacetime, whose sectional curvature is supposed to obey some standard restrictions. In this setting, by using as main analytical tool a generalized maximum principle for complete non-compact Riemannian manifolds, we establish sufficient conditions to guarantee that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. Applications to the de Sitter space are given.

scholarly journals Differential Harnack estimates for conjugate heat equation under the Ricci flow

2015 ◽  
Vol 08 (04) ◽  
pp. 1550063
Author(s):  
Abimbola Abolarinwa
Keyword(s):  
Heat Equation ◽  
Ricci Flow ◽  
Beltrami Operator ◽  
Nonlinear Heat Equation ◽  
Curvature Operator ◽  
Harnack Estimate ◽  
Diffusion Operator ◽  
Harnack Inequalities ◽  
Harnack Estimates ◽  
Conjugate Heat Equation

We prove (local and global) differential Harnack inequalities for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace–Beltrami operator is replaced with “[Formula: see text]”, where [Formula: see text] is the scalar curvature of the Ricci flow, which is well generalized to the case of nonlinear heat equation with potential. Our estimates improve on some well known results by weakening the curvature constraints. As a by-product, we obtain some Li–Yau-type differential Harnack estimate. The localized version of our estimate is very useful in extending the results obtained to noncompact case.

Stein manifolds of nonnegative curvature

2018 ◽  
Vol 18 (3) ◽  
pp. 285-287
Author(s):  
Xiaoyang Chen
Keyword(s):  
Fixed Point ◽  
Sectional Curvature ◽  
Normal Bundle ◽  
Stein Manifold ◽  
Riemannian Metric ◽  
Nonnegative Curvature ◽  
Fixed Point Set ◽  
Nonnegative Sectional Curvature ◽  
Point Set ◽  
Nonempty Compact

AbstractLet X bea Stein manifold with an anti-holomorphic involution τ and nonempty compact fixed point set Xτ. We show that X is diffeomorphic to the normal bundle of Xτ provided that X admits a complete Riemannian metric g of nonnegative sectional curvature such that τ*g = g.

Boundary conditions for Maxwell fields in Kerr-AdS spacetimes

2016 ◽  
Vol 25 (09) ◽  
pp. 1641011 ◽  
Author(s):  
Mengjie Wang
Keyword(s):  
Black Holes ◽  
Angular Momentum ◽  
Boundary Conditions ◽  
Energy Flux ◽  
Momentum Flux ◽  
De Sitter ◽  
Perturbative Methods ◽  
Flux Boundary Conditions ◽  
Maxwell Fields ◽  
New Type

Perturbative methods are useful to study the interaction between black holes and test fields. The equation for a perturbation itself, however, is not complete to study such a composed system if we do not assign physically relevant boundary conditions. Recently we have proposed a new type of boundary conditions for Maxwell fields in Kerr-anti-de Sitter (Kerr-AdS) spacetimes, from the viewpoint that the AdS boundary may be regarded as a perfectly reflecting mirror, in the sense that energy flux vanishes asymptotically. In this paper, we prove explicitly that a vanishing energy flux leads to a vanishing angular momentum flux. Thus, these boundary conditions may be dubbed as vanishing flux boundary conditions.

scholarly journals On the Study of Jacobi Fields in Riemannian Manifolds

1970 ◽  
Vol 43 (4) ◽  
pp. 521-528
Author(s):  
Khondokar M Ahmed
Keyword(s):  
Field Equation ◽  
Key Words ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Jacobi Equation ◽  
Standard Form ◽  
Jacobi Field ◽  
Jacobi Fields ◽  
New Approach

A new approach of finding a Jacobi field equation with the relation between curvature and geodesics of a Riemanian manifold M has been derived. Using this derivation we have made an attempt to find a standard form of this equation involving sectional curvature K and other related objects. Key words: Riemanign curvature, Sectional curvature, Jacobi equation, Jacobifield.    doi: 10.3329/bjsir.v43i4.2242 Bangladesh J. Sci. Ind. Res. 43(4), 521-528, 2008

Jacobi fields and geodesic spheres

1979 ◽  
Vol 82 (3-4) ◽  
pp. 233-240 ◽  
Author(s):  
L. Vanhecke ◽  
T. J. Willmore
Keyword(s):  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Holomorphic Sectional Curvature ◽  
Principal Curvatures ◽  
Jacobi Fields ◽  
Geodesic Spheres ◽  
Jacobi Vector ◽  
Nearly Kähler ◽  
Nearly Kähler Manifolds

SynopsisThis is a contribution to the general problem of determining the extent to which the geometry of a riemannian manifold is determined by properties of its geodesic spheres. In particular we show that total umbilicity of geodesic spheres determines riemannian manifolds of constant sectional curvature; quasi-umbilicity of geodesic spheres determines Kähler and nearly-Kähler manifolds of constant holomorphic sectional curvature; and the condition that geodesic spheres have only two different principal curvatures, one having multiplicity 3, determines manifolds locally isometric to the quaternionic projective spaces. The use of Jacobi vector fields leads to a unified treatment of these different cases.

Sign in / Sign up

Export Citation Format

Share Document