scholarly journals CR-HARMONIC MAPS

2019 ◽  
pp. 1-21
Author(s):  
GAUTIER DIETRICH
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Riemannian Manifold ◽  
Critical Points ◽  
Harmonic Maps ◽  
Hermitian Manifold ◽  
Cr Geometry ◽  
Paneitz Operator ◽  
Renormalized Energy ◽  
Invariant Functional

We develop the notion of renormalized energy in Cauchy–Riemann (CR) geometry for maps from a strictly pseudoconvex pseudo-Hermitian manifold to a Riemannian manifold. This energy is a CR invariant functional whose critical points, which we call CR-harmonic maps, satisfy a CR covariant partial differential equation. The corresponding operator coincides on functions with the CR Paneitz operator.

BUBBLING LOCATION FOR SEQUENCES OF APPROXIMATE f-HARMONIC MAPS FROM SURFACES

2010 ◽  
Vol 21 (04) ◽  
pp. 475-495 ◽  
Author(s):  
YUXIANG LI ◽  
YOUDE WANG
Keyword(s):  
Riemannian Manifold ◽  
Riemann Surface ◽  
Critical Points ◽  
Homotopy Class ◽  
Blow Up ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Identity ◽  
Minimal Point ◽  
Closed Riemann Surface

Let f be a positive smooth function on a closed Riemann surface (M, g). The f-energy of a map u from M to a Riemannian manifold (N, h) is defined as [Formula: see text] and its L2-gradient is: [Formula: see text] We will study the blow-up properties of some approximate f-harmonic map sequences in this paper. For a sequence uk : M → N with ‖τf(uk)‖L2 < C1 and Ef(uk) < C2, we will show that, if the sequence is not compact, then it must blow-up at some critical points of f or some concentrate points of |τf(uk)|2dVg. For a minimizing α-f-harmonic map sequence in some homotopy class of maps from M into N we show that, if the sequence is not compact, the blow-up points must be the minimal point of f and the energy identity holds true.

scholarly journals On Finite Energy Solutions of 4-harmonic and ES-4-harmonic Maps

2021 ◽  
Author(s):  
Volker Branding
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Euclidean Space ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Elliptic Partial Differential Equation ◽  
Variational Problems ◽  
Finite Energy ◽  
Qualitative Behavior ◽  
Nonlinear Elliptic

Abstract4-harmonic and ES-4-harmonic maps are two generalizations of the well-studied harmonic map equation which are both given by a nonlinear elliptic partial differential equation of order eight. Due to the large number of derivatives it is very difficult to find any difference in the qualitative behavior of these two variational problems. In this article we prove that finite energy solutions of both 4-harmonic and ES-4-harmonic maps from Euclidean space must be trivial. However, the energy that we require to be finite is different for 4-harmonic and ES-4-harmonic maps pointing out a first difference between these two variational problems.

The Local Möbius Equation and Decomposition Theorems in Riemannian Geometry

2002 ◽  
Vol 45 (3) ◽  
pp. 378-387
Author(s):  
Manuel Fernández-López ◽  
Eduardo García-Río ◽  
Demir N. Kupeli
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Riemannian Geometry ◽  
Warped Product ◽  
Product Structure ◽  
Twisted Product ◽  
Partial Differential ◽  
Decomposition Theorems

AbstractA partial differential equation, the local Möbius equation, is introduced in Riemannian geometry which completely characterizes the local twisted product structure of a Riemannian manifold. Also the characterizations of warped product and product structures of Riemannian manifolds are made by the local Möbius equation and an additional partial differential equation.

Harmonic G-structures

2009 ◽  
Vol 146 (2) ◽  
pp. 435-459 ◽  
Author(s):  
J. C. GONZÁLEZ–DÁVILA ◽  
F. MARTÍN CABRERA
Keyword(s):  
Riemannian Manifold ◽  
Critical Points ◽  
Compact Riemannian Manifold ◽  
Harmonic Maps ◽  
Energy Functional ◽  
Second Variation ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Intrinsic Torsion ◽  
First And Second Variation

AbstractFor closed and connected subgroups G of SO(n), we study the energy functional on the space of G-structures of a (compact) Riemannian manifold (M, 〈⋅, ⋅〉), where G-structures are considered as sections of the quotient bundle (M)/G. We deduce the corresponding first and second variation formulae and the characterising conditions for critical points by means of tools closely related to the study of G-structures. In this direction, we show the rôle in the energy functional played by the intrinsic torsion of the G-structure. Moreover, we analyse the particular case G=U(n) for 2n-dimensional manifolds. This leads to the study of harmonic almost Hermitian manifolds and harmonic maps from M into (M)/U(n).

Qxygen transfer in gravity flow sewers

2000 ◽  
Vol 42 (3-4) ◽  
pp. 417-422 ◽  
Author(s):  
T.Y. Pai ◽  
C.F. Ouyang ◽  
Y.C. Liao ◽  
H.G. Leu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Dissolved Oxygen ◽  
Flow Velocity ◽  
Molecular Diffusion ◽  
Eddy Diffusion ◽  
Partial Differential ◽  
Diffusion Term ◽  
Sewer Pipes ◽  
Gravity Sewer

Oxygen diffused to water in gravity sewer pipes was studied in a 21 m long, 0.15 m diameter model sewer. At first, the sodium sulfide was added into the clean water to deoxygenate, then the pump was started to recirculate the water and the deoxygenated water was reaerated. The dissolved oxygen microelectrode was installed to measure the dissolved oxygen concentrations varied with flow velocity, time and depth. The dissolved oxygen concentration profiles were constructed and observed. The partial differential equation diffusion model that considered Fick's law including the molecular diffusion term and eddy diffusion term were derived. The analytic solution of the partial differential equation was used to determine the diffusivities by the method of nonlinear regression. The diffusivity values for the oxygen transfer was found to be a function of molecular diffusion, eddy diffusion and flow velocity.

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