scholarly journals Variational problems of total mean curvature of submanifolds in a sphere

2016 ◽  
Vol 144 (8) ◽  
pp. 3563-3568 ◽  
Author(s):  
Zhen Guo ◽  
Bangchao Yin
Keyword(s):  
Mean Curvature ◽  
Variational Problems ◽  
Total Mean Curvature

On the total mean curvature of a compact space-like submanifold in Lorentz–Minkowski spacetime

2017 ◽  
Vol 148 (1) ◽  
pp. 199-210 ◽  
Author(s):  
Oscar Palmas ◽  
Francisco J. Palomo ◽  
Alfonso Romero
Keyword(s):  
Euclidean Space ◽  
Compact Space ◽  
Upper Bound ◽  
Mean Curvature ◽  
Light Cone ◽  
Minkowski Spacetime ◽  
Totally Umbilical ◽  
Total Mean Curvature ◽  
Lower Estimation ◽  
Compact Submanifold

By means of several counterexamples, the impossibility to obtain an analogue of the Chen lower estimation for the total mean curvature of any compact submanifold in Euclidean space for the case of compact space-like submanifolds in Lorentz–Minkowski spacetime is shown. However, a lower estimation for the total mean curvature of a four-dimensional compact space-like submanifold that factors through the light cone of six-dimensional Lorentz–Minkowski spacetime is proved by using a technique completely different from Chen's original one. Moreover, the equality characterizes the totally umbilical four-dimensional round spheres in Lorentz–Minkowski spacetime. Finally, three applications are given. Among them, an extrinsic upper bound for the first non-trivial eigenvalue of the Laplacian of the induced metric on a four-dimensional compact space-like submanifold that factors through the light cone is proved.

The equivalence of some variational problems for surfaces of prescribed mean curvature

1979 ◽  
Vol 20 (1) ◽  
pp. 87-104 ◽  
Author(s):  
Graham H. Williams
Keyword(s):  
Boundary Conditions ◽  
Minimization Problem ◽  
Mean Curvature ◽  
Variational Problems ◽  
Generalized Solution ◽  
Smooth Functions ◽  
Prescribed Mean Curvature ◽  
Class Of Functions ◽  
General Conditions ◽  
Non Parametric

One method of finding non-parametric hypersurfaces of prescribed mean curvature which span a given curve in Rn is to find a function which minimizes a particular integral amongst all smooth functions satisfying certain boundary conditions. A new problem can be considered by changing the integral slightly and then minimizing over a larger class of functions. It is possible to show that a solution to this new problem exists under very general conditions and it is usually known as the generalized solution. In this paper we show that the two problems are equivalent in the sense that the least value for the original minimization problem and the generalized problem are the same even though no solution may exist. The case where the surfaces are constrained to lie above an obstacle is also considered.

scholarly journals An integral formula for the total mean curvature matrix

2015 ◽  
Vol 68 ◽  
pp. 1-17 ◽  
Author(s):  
Chunna Zeng ◽  
Wenxue Xu ◽  
Jiazu Zhou ◽  
Lei Ma
Keyword(s):  
Mean Curvature ◽  
Integral Formula ◽  
Total Mean Curvature

scholarly journals On the Minimization of Total Mean Curvature

2015 ◽  
Vol 26 (4) ◽  
pp. 2729-2750 ◽  
Author(s):  
J. Dalphin ◽  
A. Henrot ◽  
S. Masnou ◽  
T. Takahashi
Keyword(s):  
Mean Curvature ◽  
Total Mean Curvature

scholarly journals On the Kinematic Formula of the Total Mean Curvature Matrix

2017 ◽  
Vol 21 (1) ◽  
pp. 43-54
Author(s):  
Chunna Zeng ◽  
Lei Ma ◽  
Yin Tong
Keyword(s):  
Mean Curvature ◽  
Kinematic Formula ◽  
Total Mean Curvature

scholarly journals On the total mean curvature of a nonrigid surface

2009 ◽  
Vol 50 (5) ◽  
pp. 757-759 ◽  
Author(s):  
Victor A. Alexandrov
Keyword(s):  
Mean Curvature ◽  
Total Mean Curvature

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.

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