scholarly journals A characterization related to Schrödinger equations on Riemannian manifolds

2019 ◽  
Vol 21 (08) ◽  
pp. 1850060
Author(s):  
Francesca Faraci ◽  
Csaba Farkas
Keyword(s):  
Riemannian Manifold ◽  
Variational Methods ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Compact Riemannian Manifold ◽  
Existence Of Solutions ◽  
Bounded Function ◽  
Real Parameter ◽  
Schrödinger Equations ◽  
Characterization Result

In this paper, we consider the following problem: [Formula: see text] where [Formula: see text] is a [Formula: see text]-dimensional ([Formula: see text], non-compact Riemannian manifold with asymptotically non-negative Ricci curvature, [Formula: see text] is a real parameter, [Formula: see text] is a positive coercive potential, [Formula: see text] is a bounded function and [Formula: see text] is a suitable nonlinearity. By using variational methods, we prove a characterization result for existence of solutions for [Formula: see text].

scholarly journals Torsional rigidity on compact Riemannian manifolds with lower Ricci curvature bounds

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Najoua Gamara ◽  
Abdelhalim Hasnaoui ◽  
Akrem Makni
Keyword(s):  
Riemannian Manifold ◽  
Dirichlet Problem ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Compact Riemannian Manifold ◽  
Torsional Rigidity ◽  
Hölder Inequality ◽  
Reverse Hölder Inequality ◽  
Curvature Bounds ◽  
Reverse Hölder

AbstractIn this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains

scholarly journals Clustered solutions for supercritical elliptic equations on Riemannian manifolds

2018 ◽  
Vol 8 (1) ◽  
pp. 1213-1226 ◽  
Author(s):  
Wenjing Chen
Keyword(s):  
Riemannian Manifold ◽  
Positive Integer ◽  
Riemannian Manifolds ◽  
Elliptic Problem ◽  
Elliptic Equations ◽  
Compact Riemannian Manifold ◽  
Blow Up ◽  
Beltrami Operator ◽  
Real Parameter ◽  
Reduction Procedure

Abstract Let {(M,g)} be a smooth compact Riemannian manifold of dimension {n\geq 5} . We are concerned with the following elliptic problem: -\Delta_{g}u+a(x)u=u^{\frac{n+2}{n-2}+\varepsilon},\quad u>0\text{ in }M, where {\Delta_{g}=\mathrm{div}_{g}(\nabla)} is the Laplace–Beltrami operator on M, {a(x)} is a {C^{2}} function on M such that the operator {-\Delta_{g}+a} is coercive, and {\varepsilon>0} is a small real parameter. Using the Lyapunov–Schmidt reduction procedure, we obtain that the problem under consideration has a k-peaks solution for positive integer {k\geq 2} , which blow up and concentrate at one point in M.

scholarly journals On the structure of Riemannian manifolds of almost nonnegative Ricci curvature

2004 ◽  
Vol 2004 (39) ◽  
pp. 2085-2090
Author(s):  
Gabjin Yun
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Betti Number ◽  
Compact Riemannian Manifold ◽  
Nonnegative Real Number ◽  
Sphere Bundle ◽  
Finite Cover ◽  
Nonnegative Ricci Curvature ◽  
Bounded Curvature

We study the structure of manifolds with almost nonnegative Ricci curvature. We prove a compact Riemannian manifold with bounded curvature, diameter bounded from above, and Ricci curvature bounded from below by an almost nonnegative real number such that the first Betti number havingcodimension two is an infranilmanifold or a finite cover is a sphere bundle over a torus. Furthermore, if we assume the Ricci curvature is bounded and volume is bounded from below, then the manifold must be an infranilmanifold.

scholarly journals Gradient Estimates on Dirichlet and Neumann Eigenfunctions

2018 ◽  
Vol 2020 (20) ◽  
pp. 7279-7305 ◽  
Author(s):  
Marc Arnaudon ◽  
Anton Thalmaier ◽  
Feng-Yu Wang
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Stochastic Analysis ◽  
Compact Riemannian Manifold ◽  
Gradient Estimates ◽  
Nonnegative Ricci Curvature

Abstract By methods of stochastic analysis on Riemannian manifolds, we derive explicit constants $c_1(D)$ and $c_2(D)$ for a $d$-dimensional compact Riemannian manifold $D$ with boundary such that $c_1(D)\sqrt \lambda \|\phi \|_\infty \leqslant \|\nabla \phi \|_\infty \leqslant c_2(D)\sqrt \lambda \|\phi \|_\infty $ holds for any Dirichlet eigenfunction $\phi $ of $-\Delta $ with eigenvalue $\lambda $. In particular, when $D$ is convex with nonnegative Ricci curvature, the estimate holds for $c_1(D)= 1/{d\mathrm{e}}$ and $c_2(D)=\sqrt{\mathrm{e}}\left (\frac{\sqrt{2}}{\sqrt{\pi }}+\frac{\sqrt{\pi }}{4\sqrt{2}}\right ).$ Corresponding two-sided gradient estimates for Neumann eigenfunctions are derived in the second part of the paper.

scholarly journals On Killing Vector Fields on Riemannian Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 259
Author(s):  
Sharief Deshmukh ◽  
Olga Belova
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Laplace Operator ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Smooth Function ◽  
Compact Riemannian Manifold ◽  
Killing Vector ◽  
Killing Vector Field ◽  
The Laplace Operator

We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field w on a connected Riemannian manifold (M,g) we show that for each non-constant smooth function f∈C∞(M) there exists a non-zero vector field wf associated with f. In particular, we show that for an eigenfunction f of the Laplace operator on an n-dimensional compact Riemannian manifold (M,g) with an appropriate lower bound on the integral of the Ricci curvature S(wf,wf) gives a characterization of the odd-dimensional unit sphere S2m+1. Also, we show on an n-dimensional compact Riemannian manifold (M,g) that if there exists a positive constant c and non-constant smooth function f that is eigenfunction of the Laplace operator with eigenvalue nc and the unit Killing vector field w satisfying ∇w2≤(n−1)c and Ricci curvature in the direction of the vector field ∇f−w is bounded below by n−1c is necessary and sufficient for (M,g) to be isometric to the sphere S2m+1(c). Finally, we show that the presence of a unit Killing vector field w on an n-dimensional Riemannian manifold (M,g) with sectional curvatures of plane sections containing w equal to 1 forces dimension n to be odd and that the Riemannian manifold (M,g) becomes a K-contact manifold. We also show that if in addition (M,g) is complete and the Ricci operator satisfies Codazzi-type equation, then (M,g) is an Einstein Sasakian manifold.

scholarly journals Gaussian heat kernel estimates: From functions to forms

2020 ◽  
Vol 2020 (761) ◽  
pp. 25-79
Author(s):  
Thierry Coulhon ◽  
Baptiste Devyver ◽  
Adam Sikora
Keyword(s):  
Riemannian Manifold ◽  
Heat Kernel ◽  
Ricci Curvature ◽  
Compact Riemannian Manifold ◽  
Kernel Estimates ◽  
Doubling Property ◽  
Negative Part ◽  
Heat Kernel Estimates ◽  
Volume Doubling ◽  
Upper Estimate

AbstractOn a complete non-compact Riemannian manifold satisfying the volume doubling property, we give conditions on the negative part of the Ricci curvature that ensure that, unless there are harmonic 1-forms, the Gaussian heat kernel upper estimate on functions transfers to one-forms. These conditions do no entail any constraint on the size of the Ricci curvature, only on its decay at infinity.

scholarly journals TRANSVERSAL INFINITESIMAL AUTOMORPHISMS ON KÄHLER FOLIATIONS

2012 ◽  
Vol 86 (3) ◽  
pp. 405-415 ◽  
Author(s):  
SEOUNG DAL JUNG ◽  
HUILI LIU
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Ricci Curvature ◽  
Compact Riemannian Manifold ◽  
Killing Field ◽  
Nonzero Constant ◽  
Conformal Field ◽  
Infinitesimal Automorphisms

AbstractLet ℱ be a Kähler foliation on a compact Riemannian manifold M. If the transversal scalar curvature of ℱ is nonzero constant, then any transversal conformal field is a transversal Killing field; and if the transversal Ricci curvature is nonnegative and positive at some point, then there are no transversally holomorphic fields.

scholarly journals THE BEST-CONSTANT PROBLEM FOR A FAMILY OF GAGLIARDO–NIRENBERG INEQUALITIES ON A COMPACT RIEMANNIAN MANIFOLD

2003 ◽  
Vol 46 (1) ◽  
pp. 117-146 ◽  
Author(s):  
Christophe Brouttelande
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Compact Riemannian Manifold ◽  
Logarithmic Sobolev Inequality ◽  
Sobolev Inequalities ◽  
Best Constant ◽  
Nash Inequality ◽  
Primary 58J05 ◽  
Compact Case ◽  
Limiting Case

AbstractThe best-constant problem for Nash and Sobolev inequalities on Riemannian manifolds has been intensively studied in the last few decades, especially in the compact case. We treat this problem here for a more general family of Gagliardo–Nirenberg inequalities including the Nash inequality and the limiting case of a particular logarithmic Sobolev inequality. From the latter, we deduce a sharp heat-kernel upper bound.AMS 2000 Mathematics subject classification: Primary 58J05

scholarly journals F-biharmonic maps into general Riemannian manifolds

2019 ◽  
Vol 17 (1) ◽  
pp. 1249-1259
Author(s):  
Rong Mi
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Compact Riemannian Manifold ◽  
Biharmonic Maps ◽  
Type Result ◽  
Integral Conditions ◽  
Liouville Type ◽  
Nonexistence Result ◽  
Biharmonic Map ◽  
Euclidean Type

Abstract Let ψ:(M, g) → (N, h) be a map between Riemannian manifolds (M, g) and (N, h). We introduce the notion of the F-bienergy functional $$\begin{array}{} \displaystyle E_{F,2}(\psi)=\int\limits_{M}F\left(\frac{|\tau(\psi)|^{2}}{2}\right)\text{d}V_{g}, \end{array}$$ where F : [0, ∞) → [0, ∞) be C3 function such that F′ > 0 on (0, ∞), τ(ψ) is the tension field of ψ. Critical points of τF,2 are called F-biharmonic maps. In this paper, we prove a nonexistence result for F-biharmonic maps from a complete non-compact Riemannian manifold of dimension m = dimM ≥ 3 with infinite volume that admit an Euclidean type Sobolev inequality into general Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the Lp-norm (p > 1) of the tension field is bounded and the m-energy of the maps is sufficiently small, then every F-biharmonic map must be harmonic. We also get a Liouville-type result under proper integral conditions which generalize the result of [Branding V., Luo Y., A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds, 2018, arXiv: 1806.11441v2].

scholarly journals Weighted Hardy inequality on Riemannian manifolds

2016 ◽  
Vol 18 (06) ◽  
pp. 1550072 ◽  
Author(s):  
El Hadji Abdoulaye Thiam
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Hardy Inequality ◽  
Compact Riemannian Manifold ◽  
Sufficient Conditions ◽  
Extremal Functions ◽  
Dimension Formula ◽  
Existence Of Minimizers ◽  
Necessary And Sufficient ◽  
Funct Anal

Let [Formula: see text] be a smooth compact Riemannian manifold of dimension [Formula: see text] and let [Formula: see text] to be a closed submanifold of dimension [Formula: see text]. In this paper, we study existence and non-existence of minimizers of Hardy inequality with weight function singular on [Formula: see text] within the framework of Brezis–Marcus–Shafrir [Extremal functions for Hardy’s inequality with weight, J. Funct. Anal. 171 (2000) 177–191]. In particular, we provide necessary and sufficient conditions for existence of minimizers.

Sign in / Sign up

Export Citation Format

Share Document