Existence and Compactness of Conformal Metrics on the Plane with Unbounded and Sign-Changing Gaussian Curvature

2021 ◽  
Author(s):  
Chiara Bernardini
Keyword(s):  
Gaussian Curvature ◽  
Conformal Metrics

scholarly journals Scalar curvatures of conformal metrics on Sn

1995 ◽  
Vol 140 ◽  
pp. 151-166
Author(s):  
Shigeo Kawai
Keyword(s):  
Scalar Curvature ◽  
Unit Sphere ◽  
Gaussian Curvature ◽  
Smooth Function ◽  
Conformal Metrics ◽  
Conformal Metric ◽  
Canonical Metric ◽  
Dimensional Unit ◽  
Dimension 2

In this paper we consider the following problem: Given a smooth function K on the n-dimensional unit sphere Sn(n ≥ 3) with its canonical metric g0, is it possible to find a pointwise conformal metric which has K as its scalar curvature? This problem was presented by J. L. Kazdan and F. W. Warner. The associated problem for Gaussian curvature in dimension 2 had been presented by L. Nirenberg several years before.

scholarly journals A note on the harmonic quasiconformal diffeomorphisms of the unit disc

Filomat ◽  
2015 ◽  
Vol 29 (2) ◽  
pp. 335-341
Author(s):  
Miljan Knezevic
Keyword(s):  
Gaussian Curvature ◽  
Unit Disc ◽  
Quasiconformal Mappings ◽  
Conformal Metrics

We analyze the properties of harmonic quasiconformal mappings and by comparing some suitably chosen conformal metrics defined in the unit disc we obtain some geometrically motivated inequalities for those mappings (see for instance [15, 17, 20]). In particular, we obtain the answers to many questions concerning these classes of functions which are related to the determination of different properties that are of essential importance for validity of the results such as those that generalize famous inequalities of the Schwarz-Pick type. The approach used is geometrical in nature, via analyzing the properties of the Gaussian curvature of the conformal metrics we are dealing with. As a consequence of this approach we give a note to the co-Lipschicity of harmonic quasiconformal self mappings of the unit disc at the origin.

scholarly journals Universally bistable shells with nonzero Gaussian curvature for two-way transition waves

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Nikolaos Vasios ◽  
Bolei Deng ◽  
Benjamin Gorissen ◽  
Katia Bertoldi
Keyword(s):  
Gaussian Curvature ◽  
Energy Landscape ◽  
Propagation Distance ◽  
Energy Landscapes ◽  
Bending Energy ◽  
Nonlinear Dynamic Response ◽  
One Dimensional ◽  
Doubly Curved Shells ◽  
Doubly Curved ◽  
The Waves

AbstractMulti-welled energy landscapes arising in shells with nonzero Gaussian curvature typically fade away as their thickness becomes larger because of the increased bending energy required for inversion. Motivated by this limitation, we propose a strategy to realize doubly curved shells that are bistable for any thickness. We then study the nonlinear dynamic response of one-dimensional (1D) arrays of our universally bistable shells when coupled by compressible fluid cavities. We find that the system supports the propagation of bidirectional transition waves whose characteristics can be tuned by varying both geometric parameters as well as the amount of energy supplied to initiate the waves. However, since our bistable shells have equal energy minima, the distance traveled by such waves is limited by dissipation. To overcome this limitation, we identify a strategy to realize thick bistable shells with tunable energy landscape and show that their strategic placement within the 1D array can extend the propagation distance of the supported bidirectional transition waves.

scholarly journals Comment on ,,On the integrability of 2D Hamiltonian systems with variable Gaussian curvature” by A. A. Elmandouh

2021 ◽  
Author(s):  
Wojciech Szumiński ◽  
Andrzej J. Maciejewski
Keyword(s):  
Hamiltonian Systems ◽  
Gaussian Curvature ◽  
Necessary Conditions ◽  
Constant Gaussian Curvature ◽  
Local Coordinates ◽  
Hamiltonian Functions

AbstractIn the paper [1], the author formulates in Theorem 2 necessary conditions for integrability of a certain class of Hamiltonian systems with non-constant Gaussian curvature, which depends on local coordinates. We give a counterexample to show that this theorem is not correct in general. This contradiction is explained in some extent. However, the main result of this note is our theorem that gives new simple and easy to check necessary conditions to integrability of the system considered in [1]. We present several examples, which show that the obtained conditions are effective. Moreover, we justify that our criterion can be extended to wider class of systems, which are given by non-meromorphic Hamiltonian functions.

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