Analysis of weighted p-harmonic forms and applications

2019 ◽  
Vol 30 (11) ◽  
pp. 1950058
Author(s):  
Nguyen Thac Dung ◽  
Chiung-Jue Anna Sung
Keyword(s):  
Harmonic Functions ◽  
Measure Space ◽  
Harmonic Maps ◽  
Curvature Operator ◽  
Vanishing Theorems ◽  
Liouville Property ◽  
Harmonic Forms ◽  
Smooth Metric Measure Space ◽  
Weighted Poincaré Inequality ◽  
Bounded Below

In this paper, we study weighted [Formula: see text]-harmonic forms on smooth metric measure space [Formula: see text] with a weighted Sobolev or a weighted Poincaré inequality. When [Formula: see text] is constant, we derive a splitting theorem for Kähler manifolds with maximal bottom spectrum for the [Formula: see text]-Laplacian. For general [Formula: see text] we also obtain various splitting and vanishing theorems when the weighted curvature operator of [Formula: see text] is bounded below. As applications, we conclude Liouville property for weighted [Formula: see text]-harmonic functions and [Formula: see text]-harmonic maps.

HARMONIC MAPS FROM SMOOTH METRIC MEASURE SPACES

2012 ◽  
Vol 23 (09) ◽  
pp. 1250095 ◽  
Author(s):  
GUOFANG WANG ◽  
DELIANG XU
Keyword(s):  
Riemannian Manifold ◽  
Ricci Tensor ◽  
Measure Space ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Metric Measure Spaces ◽  
Rigidity Results ◽  
Smooth Metric Measure Space ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces

In this paper, we study a generalized harmonic map, ϕ-harmonic map, from a smooth metric measure space (M, g, e-ϕ dv) into a Riemannian manifold. We proved various rigidity results for the ϕ-harmonic maps under conditions in terms of the Bakry–Émery Ricci tensor.

scholarly journals L 2 harmonic forms and finiteness of ends

2013 ◽  
Vol 85 (2) ◽  
pp. 457-471 ◽  
Author(s):  
PENG ZHU
Keyword(s):  
Riemannian Manifolds ◽  
Poincaré Inequality ◽  
Vanishing Theorems ◽  
Poincare Inequality ◽  
Harmonic Forms ◽  
Minimal Hypersurfaces ◽  
Weighted Poincaré Inequality ◽  
Complete Riemannian Manifolds

In this paper, we obtain vanishing theorems and finitely many ends theorems of complete Riemannian manifolds with weighted Poincaré inequality, applying them to minimal hypersurfaces.

Harmonic functions on ℝ-covered foliations

2009 ◽  
Vol 29 (4) ◽  
pp. 1141-1161
Author(s):  
S. FENLEY ◽  
R. FERES ◽  
K. PARWANI
Keyword(s):  
Harmonic Functions ◽  
Riemannian Metrics ◽  
Related Result ◽  
Continuous Functions ◽  
Property A ◽  
Liouville Property ◽  
Foliated Manifold ◽  
Curved Manifolds ◽  
Codimension One ◽  
Negatively Curved

AbstractLet (M,ℱ) be a compact codimension-one foliated manifold whose leaves are endowed with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of ℱ. If every such function is constant on leaves, we say that (M,ℱ) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. A related result for ℝ-covered foliations is also established.

Harmonic Spinors on Hyperbolic Space

1993 ◽  
Vol 36 (3) ◽  
pp. 257-262 ◽  
Author(s):  
Pierre-Yves Gaillard
Keyword(s):  
Differential Equations ◽  
Symmetric Space ◽  
Representation Theory ◽  
Hyperbolic Space ◽  
Harmonic Functions ◽  
Short Note ◽  
Harmonic Forms ◽  
Semisimple Symmetric Space ◽  
Invariant Differential ◽  
Harmonic Spinors

AbstractThe purpose for this short note is to describe the space of harmonic spinors on hyperbolicn-spaceHn. This is a natural continuation of the study of harmonic functions onHnin [Minemura] and [Helgason]—these results were generalized in the form of Helgason's conjecture, proved in [KKMOOT],—and of [Gaillard 1, 2], where harmonic forms onHnwere considered. The connection between invariant differential equations on a Riemannian semisimple symmetric spaceG/Kand homological aspects of the representation theory ofG, as exemplified in (8) below, does not seem to have been previously mentioned. This note is divided into three main parts respectively dedicated to the statement of the results, some reminders, and the proofs. I thank the referee for having suggested various improvements.

scholarly journals Eigenvalues of the drifted Laplacian on complete metric measure spaces

2016 ◽  
Vol 19 (01) ◽  
pp. 1650001 ◽  
Author(s):  
Xu Cheng ◽  
Detang Zhou
Keyword(s):  
Lower Bound ◽  
Measure Space ◽  
Logarithmic Sobolev Inequality ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Dimensional Manifold ◽  
Metric Measure Spaces ◽  
Smooth Metric Measure Space ◽  
Measure Spaces ◽  
Multiplicity Formula

In this paper, first we study a complete smooth metric measure space [Formula: see text] with the ([Formula: see text])-Bakry–Émery Ricci curvature [Formula: see text] for some positive constant [Formula: see text]. It is known that the spectrum of the drifted Laplacian [Formula: see text] for [Formula: see text] is discrete and the first nonzero eigenvalue of [Formula: see text] has lower bound [Formula: see text]. We prove that if the lower bound [Formula: see text] is achieved with multiplicity [Formula: see text], then [Formula: see text], [Formula: see text] is isometric to [Formula: see text] for some complete [Formula: see text]-dimensional manifold [Formula: see text] and by passing an isometry, [Formula: see text] must split off a gradient shrinking Ricci soliton [Formula: see text], [Formula: see text]. This result can be applied to gradient shrinking Ricci solitons. Secondly, we study the drifted Laplacian [Formula: see text] for properly immersed self-shrinkers in the Euclidean space [Formula: see text], [Formula: see text] and show the discreteness of the spectrum of [Formula: see text] and a logarithmic Sobolev inequality.

scholarly journals Brownian motion on stationary random manifolds

2016 ◽  
Vol 16 (02) ◽  
pp. 1660001
Author(s):  
Pablo Lessa
Keyword(s):  
Brownian Motion ◽  
Harmonic Functions ◽  
Dimensional Space ◽  
Volume Growth ◽  
Upper And Lower Bounds ◽  
Entropy Theory ◽  
Liouville Property ◽  
Average Volume ◽  
Infinite Dimensional ◽  
Bounded Harmonic Functions

We introduce the notion of a stationary random manifold and develop the basic entropy theory for it. Examples include manifolds admitting a compact quotient under isometries and generic leaves of a compact foliation. We prove that the entropy of an ergodic stationary random manifold is zero if and only if the manifold satisfies the Liouville property almost surely, and is positive if and only if it admits an infinite dimensional space of bounded harmonic functions almost surely. Upper and lower bounds for the entropy are provided in terms of the linear drift of Brownian motion and average volume growth of the manifold. Other almost sure properties of these random manifolds are also studied.

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