scholarly journals Compactness issues and bubbling phenomena for the prescribed Gaussian curvature equation on the torus

2015 ◽  
Vol 54 (3) ◽  
pp. 2483-2501 ◽  
Author(s):  
Luca Galimberti
Keyword(s):  
Gaussian Curvature ◽  
Curvature Equation ◽  
Prescribed Gaussian Curvature ◽  
Bubbling Phenomena

Boundary trace of the solutions of the prescribed Gaussian curvature equation

2000 ◽  
Vol 130 (3) ◽  
pp. 527-560 ◽  
Author(s):  
Michele Grillot ◽  
Laurent Véron
Keyword(s):  
Gaussian Curvature ◽  
Radon Measure ◽  
Borel Measure ◽  
Sufficient Conditions ◽  
Open Ball ◽  
Curvature Equation ◽  
Borel Measures ◽  
Boundary Trace ◽  
Regular Borel Measure ◽  
Prescribed Gaussian Curvature

We study the existence of a boundary trace for minorized solutions of the equation Δu + K (x) e2u = 0 in the unit open ball B2 of R2. We prove that this trace is an outer regular Borel measure on ∂B2, not necessarily a Radon measure. We give sufficient conditions on Borel measures on ∂B2 so that they are the boundary trace of a solution of the above equation. We also give boundary removability results in terms of generalized Bessel capacities.

scholarly journals The behaviour of solutions of the Gaussian curvature equation near an isolated boundary point

2008 ◽  
Vol 145 (3) ◽  
pp. 643-667 ◽  
Author(s):  
DANIELA KRAUS ◽  
OLIVER ROTH
Keyword(s):  
Continuous Function ◽  
Liouville Equation ◽  
Gaussian Curvature ◽  
Boundary Point ◽  
Classical Result ◽  
Riemannian Metrics ◽  
Higher Order ◽  
Isolated Singularities ◽  
Hölder Continuous ◽  
Curvature Equation

AbstractA classical result of Nitsche [22] about the behaviour of the solutions to the Liouville equation Δu= 4e2unear isolated singularities is generalized to solutions of the Gaussian curvature equation Δu= −κ(z)e2uwhere κ is a negative Hölder continuous function. As an application a higher–order version of the Yau–Ahlfors–Schwarz lemma for complete conformal Riemannian metrics is obtained.

Timelike surfaces in the Lorentz–Minkowski space with prescribed Gaussian curvature and Gauss map

2006 ◽  
Vol 56 (8) ◽  
pp. 1357-1369 ◽  
Author(s):  
Juan A. Aledo ◽  
José M. Espinar ◽  
José A. Gálvez
Keyword(s):  
Minkowski Space ◽  
Gaussian Curvature ◽  
Gauss Map ◽  
Prescribed Gaussian Curvature

HELICOIDAL SURFACES WITH PRESCRIBED CURVATURES IN Nil3

2013 ◽  
Vol 24 (14) ◽  
pp. 1350107 ◽  
Author(s):  
DAE WON YOON ◽  
DONG-SOO KIM ◽  
YOUNG HO KIM ◽  
JAE WON LEE
Keyword(s):  
Heisenberg Group ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Smooth Functions ◽  
Constant Gaussian Curvature ◽  
3 Dimensional ◽  
Helicoidal Surfaces ◽  
Prescribed Gaussian Curvature

In the present paper, we study helicoidal surfaces in the 3-dimensional Heisenberg group Nil3. Also, we construct helicoidal surfaces in Nil3 with prescribed Gaussian curvature or mean curvature given by smooth functions. As the results, we classify helicoidal surfaces with constant Gaussian curvature or constant mean curvature.

On the structure of the conformal Gaussian curvature equation on R2. II

1991 ◽  
Vol 290 (1) ◽  
pp. 671-680 ◽  
Author(s):  
Kuo-Shung Cheng ◽  
Wei-Ming Ni
Keyword(s):  
Gaussian Curvature ◽  
Curvature Equation
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