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 A volume estimate for strong subharmonicity and maximum principle on complete Riemannian manifolds

1998 ◽  
Vol 151 ◽  
pp. 25-36 ◽  
Author(s):  
Kensho Takegoshi
Keyword(s):  
Riemannian Manifold ◽  
Maximum Principle ◽  
Growth Condition ◽  
Riemannian Manifolds ◽  
Volume Growth ◽  
Complete Riemannian Manifold ◽  
Volume Estimate ◽  
Complete Riemannian Manifolds ◽  
Generalized Maximum Principle

Abstract.A generalized maximum principle on a complete Riemannian manifold (M, g) is shown under a certain volume growth condition of (M, g) and its geometric applications are given.

scholarly journals Solutions of a Certain Nonlinear Elliptic Equation on Riemannian Manifolds

2001 ◽  
Vol 162 ◽  
pp. 149-167
Author(s):  
Yong Hah Lee
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Nonlinear Elliptic Equation ◽  
Complete Riemannian Manifold ◽  
Doubling Condition ◽  
Finite Covering ◽  
Nonlinear Elliptic ◽  
Finite Dimensional ◽  
Volume Doubling

In this paper, we prove that if a complete Riemannian manifold M has finitely many ends, each of which is a Harnack end, then the set of all energy finite bounded A-harmonic functions on M is one to one corresponding to Rl, where A is a nonlinear elliptic operator of type p on M and l is the number of p-nonparabolic ends of M. We also prove that if a complete Riemannian manifold M is roughly isometric to a complete Riemannian manifold with finitely many ends, each of which satisfies the volume doubling condition, the Poincaré inequality and the finite covering condition near infinity, then the set of all energy finite bounded A-harmonic functions on M is finite dimensional. This result generalizes those of Yau, of Donnelly, of Grigor’yan, of Li and Tam, of Holopainen, and of Kim and the present author, but with a barrier argument at infinity that the peculiarity of nonlinearity demands.

Long time behavior of quasi-convex and pseudo-convex gradient systems on Riemannian manifolds

Filomat ◽  
2017 ◽  
Vol 31 (14) ◽  
pp. 4571-4578 ◽  
Author(s):  
P. Ahmadi ◽  
H. Khatibzadeh
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Gradient Flow ◽  
Complete Riemannian Manifold ◽  
Discrete Version ◽  
Gradient System ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Pseudo Convex

In this paper, we study the following gradient system on a complete Riemannian manifold M, {-x?(t) = grad'(x(t)) x(0) = x0, where ? : M ? R is a C1 function with Argmin ? ? ?. We prove that the gradient flow x(t) converges to a critical point of ? if ? is pseudo-convex, or if ? is quasi-convex and M is Hadamard. As an application to minimization, we consider a discrete version of the system to approximate a minimum point of a given pseudo-convex function ?.

A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions

2019 ◽  
Vol 19 (3) ◽  
pp. 291-296 ◽  
Author(s):  
Sergey Stepanov ◽  
Irina Tsyganok
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Type Theorem ◽  
Complete Riemannian Manifold ◽  
Warped Products ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Totally Umbilical

Abstract We prove a Liouville-type theorem for two orthogonal complementary totally umbilical distributions on a complete Riemannian manifold with non-positive mixed scalar curvature. This is applied to some special types of complete doubly twisted and warped products of Riemannian manifolds.

scholarly journals EIGENVALUES OF THE LAPLACIAN ON RIEMANNIAN MANIFOLDS

2012 ◽  
Vol 23 (07) ◽  
pp. 1250067
Author(s):  
QING-MING CHENG ◽  
XUERONG QI
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Orthonormal Basis ◽  
Smooth Boundary ◽  
Complete Riemannian Manifold ◽  
Piecewise Smooth Boundary ◽  
Dirichlet Eigenvalue ◽  
Eigenvalues Of The Laplacian ◽  
Dirichlet Eigenvalue Problem ◽  
The Laplace Operator

For a bounded domain Ω with a piecewise smooth boundary in a complete Riemannian manifold M, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of L2(Ω) in place of the Rayleigh–Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, for lower order eigenvalues, our results extend the results of Chen and Cheng [D. Chen and Q.-M. Cheng, Extrinsic estimates for eigenvalues of the Laplace operator, J. Math. Soc. Japan 60 (2008) 325–339].

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.

ON THE REGULARITY OF SETS IN RIEMANNIAN MANIFOLDS

2020 ◽  
pp. 1-20 ◽  
Author(s):  
A. SEPAHVAND ◽  
A. BARANI
Keyword(s):  
Riemannian Manifold ◽  
Distance Function ◽  
Riemannian Manifolds ◽  
Closed Subset ◽  
Normal Cones ◽  
Closed Set ◽  
Directional Regularity ◽  
Tangential Regularity

This paper is devoted to the study of the normal (tangential) regularity of a closed set and the subdifferential (directional) regularity of its distance function in the context of Riemannian manifolds. The Clarke, Fréchet and proximal subdifferentials of the distance function from a closed subset in a Riemannian manifold are represented by corresponding normal cones of the set.

scholarly journals Distance difference functions on nonconvex boundaries of Riemannian manifolds

2021 ◽  
Vol 33 (1) ◽  
pp. 57-64
Author(s):  
S. Ivanov
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Complete Riemannian Manifold ◽  
Manifold With Boundary ◽  
Distance Difference

It is shown that a complete Riemannian manifold with boundary is uniquely determined, up to isometry, by its distance difference representation on the boundary. Unlike previously known results, no restrictions on the boundary are imposed.

Essential Self-adjointness of Differential Operators on Riemannian Manifolds

2021 ◽  
Author(s):  
Hany Atia ◽  
Hassan Abu Donia ◽  
Hala Emam
Keyword(s):  
Riemannian Manifold ◽  
Vector Bundle ◽  
Differential Operator ◽  
Riemannian Manifolds ◽  
Differential Operators ◽  
Positive Function ◽  
Complete Riemannian Manifold ◽  
Sufficient Condition ◽  
Hermitian Vector Bundle

Abstract In this paper we have studied the essential self-adjointness for the differential operator of the form: T=Δ⁸+V, on sections of a Hermitian vector bundle over a complete Riemannian manifold, with the potential V satisfying a bound from below by a non-positive function depending on the distance from a point. We give sufficient condition for the essential self-adjointness of such differential operator on Riemannian Manifolds.

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