Mean curvature flow in a Riemannian manifold endowed with a Killing vector field

2020 ◽  
Vol 308 (2) ◽  
pp. 435-472
Author(s):  
Liangjun Weng
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Killing Vector ◽  
Curvature Flow ◽  
Killing Vector Field

scholarly journals Twistor Bundle of a Neutral Kähler Surface

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 43
Author(s):  
Włodzimierz Jelonek
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Complex Structure ◽  
Killing Vector ◽  
Kähler Manifold ◽  
Killing Vector Field ◽  
Kahler Manifold ◽  
Kähler Surface ◽  
Kähler Surfaces ◽  
Twistor Bundle

In this paper, we characterize neutral Kähler surfaces in terms of their positive twistor bundle. We prove that an O+,+(2,2)-oriented four-dimensional neutral semi-Riemannian manifold (M,g) admits a complex structure J with ΩJ∈⋀−M, such that (M,g,J) is a neutral-Kähler manifold if and only if the twistor bundle (Z1(M),gc) admits a vertical Killing vector field.

Notes on Hypersurfaces in a Riemannian Manifold

1967 ◽  
Vol 19 ◽  
pp. 439-446 ◽  
Author(s):  
Kentaro Yano
Keyword(s):  
Vector Field ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Killing Vector ◽  
Einstein Space ◽  
Inner Product ◽  
Killing Vector Field ◽  
Parameter Group ◽  
Fixed Sign ◽  
Orientable Hypersurface

H. Liebmann (3) and W. Süss (7) provedTheorem A. The only convex closed hypersurface with constant mean curvature in a Euclidean space is a sphere.Y. Katsurada (1; 2) gave the following generalization.Theorem B. Let M be an orientable Einstein space which admits a proper conformai Killing vector field, that is, a vector field generating a local one-parameter group of conformai transformations which is not that of isometries, and S a closed orientable hypersurface in M whose first mean curvature is constant. If the inner product of the conformai Killing vector field and the normal to the hypersurface has fixed sign on S, then every point of S is umbilical.

scholarly journals On Jacobi-Type Vector Fields on Riemannian Manifolds

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1139 ◽  
Author(s):  
Bang-Yen Chen ◽  
Sharief Deshmukh ◽  
Amira A. Ishan
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Compact Riemannian Manifold ◽  
Killing Vector ◽  
Sufficient Conditions ◽  
Killing Vector Field ◽  
Natural Question ◽  
Necessary And Sufficient

In this article, we study Jacobi-type vector fields on Riemannian manifolds. A Killing vector field is a Jacobi-type vector field while the converse is not true, leading to a natural question of finding conditions under which a Jacobi-type vector field is Killing. In this article, we first prove that every Jacobi-type vector field on a compact Riemannian manifold is Killing. Then, we find several necessary and sufficient conditions for a Jacobi-type vector field to be a Killing vector field on non-compact Riemannian manifolds. Further, we derive some characterizations of Euclidean spaces in terms of Jacobi-type vector fields. Finally, we provide examples of Jacobi-type vector fields on non-compact Riemannian manifolds, which are non-Killing.

scholarly journals On Affine Transformations of a Riemannian Manifold

1955 ◽  
Vol 9 ◽  
pp. 99-109 ◽  
Author(s):  
Jun-Ichi Hano
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Affine Transformation ◽  
Integral Formula ◽  
Killing Vector ◽  
Decomposition Theorem ◽  
Complete Riemannian Manifold ◽  
Killing Vector Field ◽  
Affine Transformations ◽  
Infinitesimal Affine Transformation

In this paper we establish some theorems about the group of affine transformations on a Riemannian manifold. First we prove a decomposition theorem (Theorem 1) of the largest connected group of affine transformations on a simply connected complete Riemannian manifold, which corresponds to the decomposition theorem of de Rham [4] for the manifold. In the case of the largest group of isometries, a theorem of the same type is found in de Rham’s paper [4] in a weaker form. Using Theorem 1 we obtain a sufficient condition for an infinitesimal affine transformation to be a Killing vector field (Theorem 2). This result includes K. Yano’s theorem [13] which states that on a compact Riemannian manifold an infinitesimal affine transformation is always a Killing vector field. His proof of the theorem depends on an integral formula which is valid only for a compact manifold. Our method is quite different and is based on a result [11] of K. Nomizu.

scholarly journals Fixed Points of Isometries

1958 ◽  
Vol 13 ◽  
pp. 63-68 ◽  
Author(s):  
Shoshichi Kobayashi
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Fixed Points ◽  
Killing Vector ◽  
Connected Components ◽  
Killing Vector Field ◽  
Set Of Points ◽  
Infinitesimal Isometry

The purpose of this paper is to prove the followingTheorem. Let M be a Riemannian manifold of dimension n and let ξ be a Killing vector field (i.e., infinitesimal isometry) of M. Let F be the set of points x of M where ξ vanishes and let F = ∪ Vi, where the Vi’s are the connected components of F. Then (assuming F to be non-empty)

scholarly journals Local Well-Posedness of Skew Mean Curvature Flow for Small Data in $$d\ge 4$$ Dimensions

2022 ◽  
Author(s):  
Jiaxi Huang ◽  
Daniel Tataru
Keyword(s):  
Riemannian Manifold ◽  
Evolution Equation ◽  
Sobolev Spaces ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Small Data ◽  
Well Posedness ◽  
The Mean ◽  
Schrödinger Map

AbstractThe skew mean curvature flow is an evolution equation for d dimensional manifolds embedded in $${{\mathbb {R}}}^{d+2}$$ R d + 2 (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension $$d\ge 4$$ d ≥ 4 .

scholarly journals A Note on Killing Calculus on Riemannian Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 307
Author(s):  
Sharief Deshmukh ◽  
Amira Ishan ◽  
Suha B. Al-Shaikh ◽  
Cihan Özgür
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Laplace Operator ◽  
Compact Riemannian Manifold ◽  
Killing Vector ◽  
Adjoint Operator ◽  
Dimensional Sphere ◽  
Killing Vector Field ◽  
Smooth Functions ◽  
The Laplace Operator

In this article, it has been observed that a unit Killing vector field ξ on an n-dimensional Riemannian manifold (M,g), influences its algebra of smooth functions C∞(M). For instance, if h is an eigenfunction of the Laplace operator Δ with eigenvalue λ, then ξ(h) is also eigenfunction with same eigenvalue. Additionally, it has been observed that the Hessian Hh(ξ,ξ) of a smooth function h∈C∞(M) defines a self adjoint operator ⊡ξ and has properties similar to most of properties of the Laplace operator on a compact Riemannian manifold (M,g). We study several properties of functions associated to the unit Killing vector field ξ. Finally, we find characterizations of the odd dimensional sphere using properties of the operator ⊡ξ and the nontrivial solution of Fischer–Marsden differential equation, respectively.

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