scholarly journals On n-superharmonic functions and some geometric applications

2021 ◽  
Vol 60 (6) ◽  
Author(s):  
Shiguang Ma ◽  
Jie Qing
Keyword(s):  
Conformal Geometry ◽  
Type Inequality ◽  
Conformally Flat ◽  
Nonlinear Potential Theory ◽  
Conformally Flat Manifolds ◽  
Superharmonic Functions ◽  
Laplace Equations ◽  
Nonlinear Potential ◽  
Wolff Potential ◽  
Flat Manifolds

AbstractIn this paper we study asymptotic behaviors of n-superharmonic functions at singularity using the Wolff potential and capacity estimates in nonlinear potential theory. Our results are inspired by and extend [6] of Arsove–Huber and [63] of Taliaferro in 2 dimensions. To study n-superharmonic functions we use a new notion of thinness in terms of n-capacity motivated by a type of Wiener criterion in [6]. To extend [63], we employ the Adams–Moser–Trudinger’s type inequality for the Wolff potential, which is inspired by the inequality used in [15] of Brezis–Merle. For geometric applications, we study the asymptotic end behaviors of complete conformally flat manifolds as well as complete properly embedded hypersurfaces in hyperbolic space. These geometric applications seem to elevate the importance of n-Laplace equations and make a closer tie to the classic analysis developed in conformal geometry in general dimensions.

COMPACTNESS AND GLOBAL ESTIMATES FOR A FOURTH ORDER EQUATION OF CRITICAL SOBOLEV GROWTH ARISING FROM CONFORMAL GEOMETRY

2006 ◽  
Vol 08 (01) ◽  
pp. 9-65 ◽  
Author(s):  
EMMANUEL HEBEY ◽  
FRÉDÉRIC ROBERT ◽  
YULIANG WEN
Keyword(s):  
Fourth Order ◽  
Compact Riemannian Manifold ◽  
Conformal Geometry ◽  
Conformally Flat ◽  
Conformally Flat Manifolds ◽  
Locally Conformally Flat ◽  
Constant Coefficients ◽  
Global Estimates ◽  
Definition Of ◽  
Flat Manifolds

Given (M,g) a smooth compact Riemannian manifold of dimension n ≥ 5, we investigate compactness for fourth order critical equations like Pgu = u2♯-1, where [Formula: see text] is a Paneitz–Branson operator with constant coefficients b and c, u is required to be positive, and [Formula: see text] is critical from the Sobolev viewpoint. We prove that such equations are compact on locally conformally flat manifolds, unless b lies in some closed interval associated to the spectrum of the smooth symmetric (2,0)-tensor field involved in the definition of the geometric Paneitz–Branson operator.

scholarly journals The duality of conformally flat manifolds

2011 ◽  
Vol 42 (1) ◽  
pp. 131-152 ◽  
Author(s):  
Huili Liu ◽  
Masaaki Umehara ◽  
Kotaro Yamada
Keyword(s):  
Conformally Flat ◽  
Conformally Flat Manifolds ◽  
Flat Manifolds

scholarly journals Twistor spinors on conformally flat manifolds

1997 ◽  
Vol 41 (3) ◽  
pp. 495-503 ◽  
Author(s):  
Wolfgang Kühnel ◽  
Hans–Bert Rademacher
Keyword(s):  
Conformally Flat ◽  
Conformally Flat Manifolds ◽  
Flat Manifolds

scholarly journals Prescribed Schouten Tensor in Locally Conformally Flat Manifolds

2019 ◽  
Vol 74 (4) ◽  
Author(s):  
Marcos Tulio Carvalho ◽  
Mauricio Pieterzack ◽  
Romildo Pina
Keyword(s):  
Sufficient Conditions ◽  
Conformally Flat ◽  
Symmetric Tensors ◽  
Conformally Flat Manifolds ◽  
Dimensional Group ◽  
Schouten Tensor ◽  
Locally Conformally Flat ◽  
Flat Manifolds ◽  
Necessary And Sufficient ◽  
Complete Metrics

Abstract We consider the pseudo-Euclidean space $$({\mathbb {R}}^n,g)$$(Rn,g), with $$n \ge 3$$n≥3 and $$g_{ij} = \delta _{ij} \varepsilon _{i}$$gij=δijεi, where $$\varepsilon _{i} = \pm 1$$εi=±1, with at least one positive $$\varepsilon _{i}$$εi and non-diagonal symmetric tensors $$T = \sum \nolimits _{i,j}f_{ij}(x) dx_i \otimes dx_{j} $$T=∑i,jfij(x)dxi⊗dxj. Assuming that the solutions are invariant by the action of a translation $$(n-1)$$(n-1)- dimensional group, we find the necessary and sufficient conditions for the existence of a metric $$\bar{g}$$g¯ conformal to g, such that the Schouten tensor $$\bar{g}$$g¯, is equal to T. From the obtained results, we show that for certain functions h, defined in $$\mathbb {R}^{n}$$Rn, there exist complete metrics $$\bar{g}$$g¯, conformal to the Euclidean metric g, whose curvature $$\sigma _{2}(\bar{g}) = h$$σ2(g¯)=h.

Sign in / Sign up

Export Citation Format

Share Document