Period problems for mean curvature one surfaces in $H^3$ (with applications to surfaces of low total curvature)

2019 ◽  
Author(s):  
Wayne Rossman ◽  
Masaaki Umehara ◽  
Kotaro Yamada
Keyword(s):  
Mean Curvature ◽  
Total Curvature

The focusing of radiation by a random surface when the source is at a finite distance

1960 ◽  
Vol 56 (1) ◽  
pp. 27-40 ◽  
Author(s):  
M. S. Longuet-Higgins
Keyword(s):  
Mean Curvature ◽  
Joint Distribution ◽  
Total Curvature ◽  
Statistical Distribution ◽  
Constant Parameter ◽  
Random Surface ◽  
Second Derivatives ◽  
Finite Distance ◽  
The Mean ◽  
Image Position

Imagine a nearly horizontal, statistically uniform, random surface ζ(x, y), Gaussian in the sense that the second derivatives , , have a normal joint distribution. The problem considered is the statistical distribution of the quantitywhere J and Ω denote the mean curvature and total curvature of the surface, respectively, and ν is a constant parameter.

scholarly journals Some examples of critical points for the total mean curvature functional

2000 ◽  
Vol 43 (3) ◽  
pp. 587-603 ◽  
Author(s):  
Josu Arroyo ◽  
Manuel Barros ◽  
Oscar J. Garay
Keyword(s):  
Critical Points ◽  
Mean Curvature ◽  
Space Form ◽  
Total Curvature ◽  
Constant Scalar Curvature ◽  
Real Space ◽  
Closed Curves ◽  
Three Sphere ◽  
Total Mean Curvature

AbstractWe study the following problem: establish existence and classification of closed curves which are critical points for the total curvature functional, defined on spaces of curves in a Riemannian manifold. This problem is completely solved in a real space form. Next, we give examples of critical points for this functional in a class of metrics with constant scalar curvature on the three sphere. Also, we obtain a rational one-parameter family of closed helices which are critical points for that functional in ℂℙ2 (4) when it is endowed with its usual Kaehlerian structure. Finally, we use the principle of symmetric criticality to get equivariant submanifolds, constructed on the above curves, which are critical points for the total mean curvature functional.

scholarly journals Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature. I

2004 ◽  
Vol 34 (1) ◽  
pp. 21-56 ◽  
Author(s):  
Wayne Rossman ◽  
Masaaki Umehara ◽  
Kotaro Yamada
Keyword(s):  
Mean Curvature ◽  
Total Curvature
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