scholarly journals Laplacian of the distance function on the cut locus on a Riemannian manifold

Nonlinearity ◽  
2020 ◽  
Vol 33 (8) ◽  
pp. 3928-3939
Author(s):  
François Générau
Keyword(s):  
Riemannian Manifold ◽  
Distance Function ◽  
Cut Locus

scholarly journals Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry

2021 ◽  
Author(s):  
Piermarco Cannarsa ◽  
Wei Cheng ◽  
Albert Fathi
Keyword(s):  
Riemannian Manifold ◽  
Distance Function ◽  
Riemannian Geometry ◽  
Jacobi Equation ◽  
Compact Manifold ◽  
Complete Riemannian Manifold ◽  
Time Dependent ◽  
Jacobi Equations ◽  
Uniformly Continuous ◽  
Hamilton Jacobi Equation

AbstractIf $U:[0,+\infty [\times M$ U : [ 0 , + ∞ [ × M is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$ ∂ t U + H ( x , ∂ x U ) = 0 , where $M$ M is a not necessarily compact manifold, and $H$ H is a Tonelli Hamiltonian, we prove the set $\Sigma (U)$ Σ ( U ) , of points in $]0,+\infty [\times M$ ] 0 , + ∞ [ × M where $U$ U is not differentiable, is locally contractible. Moreover, we study the homotopy type of $\Sigma (U)$ Σ ( U ) . We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.

scholarly journals Growth of the Eigensolutions of Laplacians on Riemannian Manifolds I: Construction of Energy Function

2018 ◽  
Vol 2020 (9) ◽  
pp. 2561-2587 ◽  
Author(s):  
Wencai Liu
Keyword(s):  
Riemannian Manifold ◽  
Distance Function ◽  
Riemannian Manifolds ◽  
Energy Function ◽  
Complete Riemannian Manifold ◽  
Essential Spectra ◽  
Radial Curvature

Abstract In this paper, we consider the eigensolutions of $-\Delta u+ Vu=\lambda u$, where $\Delta $ is the Laplacian on a non-compact complete Riemannian manifold. We develop Kato’s methods on manifold and establish the growth of the eigensolutions as r goes to infinity based on the asymptotical behaviors of $\Delta r$ and V (x), where r = r(x) is the distance function on the manifold. As applications, we prove several criteria of absence of eigenvalues of Laplacian, including a new proof of the absence of eigenvalues embedded into the essential spectra of free Laplacian if the radial curvature of the manifold satisfies $ K_{\textrm{rad}}(r)= -1+\frac{o(1)}{r}$.

scholarly journals Scalar Curvature via Local Extent

2018 ◽  
Vol 6 (1) ◽  
pp. 146-164 ◽  
Author(s):  
Giona Veronelli
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Distance Function ◽  
Metric Space ◽  
Integral Curvature ◽  
Alexandrov Spaces ◽  
Metric Characterization ◽  
Maximal Distance

AbstractWe give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between (n + 1) points in infinitesimally small neighborhoods of a point. Since this characterization is purely in terms of the distance function, it could be used to approach the problem of defining the scalar curvature on a non-smooth metric space. In the second part we will discuss this issue, focusing in particular on Alexandrov spaces and surfaces with bounded integral curvature.

scholarly journals Cut locus of a separating fractal set in a Riemannian manifold

1998 ◽  
Vol 50 (4) ◽  
pp. 455-467
Author(s):  
Hyeong In Choi ◽  
Doo Seok Lee ◽  
Joung-Hahn Yoon
Keyword(s):  
Riemannian Manifold ◽  
Cut Locus ◽  
Fractal Set

scholarly journals An integral formula for hypersurfaces in space forms

1995 ◽  
Vol 37 (3) ◽  
pp. 337-341 ◽  
Author(s):  
Theodoros Vlachos
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Distance Function ◽  
Integral Formula ◽  
Constant Sectional Curvature ◽  
Position Vector ◽  
Space Forms ◽  
Image Position ◽  
Simply Connected

Let be an n+ 1-dimensional, complete simply connected Riemannian manifold of constant sectional curvature c and We consider the function r(·) = d(·, P0) where d stands for the distance function in and we denote by grad r the gradient of The position vector (see [1]) with origin P0 is defined as where ϕ(r)equalsr, if c = 0, c< 0 or c <0 respectively.

scholarly journals A link between harmonicity of 2-distance functions and incompressibility of canonical vector fields

2018 ◽  
Vol 49 (4) ◽  
pp. 339-347
Author(s):  
Bang-Yen Chen
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Distance Function ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Tangential Component ◽  
Ambient Space ◽  
Distance Functions ◽  
Necessary And Sufficient ◽  
Canonical Vector

Let $M$ be a Riemannian submanifold of a Riemannian manifold $\tilde M$ equipped with a concurrent vector field $\tilde Z$. Let $Z$ denote the restriction of $\tilde Z$ along $M$ and let $Z^T$ be the tangential component of $Z$ on $M$, called the canonical vector field of $M$. The 2-distance function $\delta^2_Z$ of $M$ (associated with $Z$) is defined by $\delta^2_Z=\$. In this article, we initiate the study of submanifolds $M$ of $\tilde M$ with incompressible canonical vector field $Z^T$ arisen from a concurrent vector field $\tilde Z$ on the ambient space $\tilde M$. First, we derive some necessary and sufficient conditions for such canonical vector fields to be incompressible. In particular, we prove that the 2-distance function $\delta^2_Z$ is harmonic if and only if the canonical vector field $Z^T$ on $M$ is an incompressible vector field. Then we provide some applications of our main results.

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