Total Mean Curvature and Submanifolds of Finite Type

10.1142/9237 ◽  
2014 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Finite Type ◽  
Mean Curvature ◽  
Total Mean Curvature ◽  
Submanifolds Of Finite Type

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.

scholarly journals A Note on Marcinkiewicz Integrals along Submanifolds of Finite Type

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Daiqing Zhang
Keyword(s):  
Finite Type ◽  
Unit Sphere ◽  
Radial Direction ◽  
Rough Kernels ◽  
Marcinkiewicz Integrals ◽  
Submanifolds Of Finite Type

We study the parametric Marcinkiewicz integrals along submanifolds of finite type with rough kernels. The kernels of our operators are allowed to be very rough both on the unit sphere and in the radial direction. Under the rather weakened size conditions on the integral kernels, the Lp bounds will be established for such operators. As applications, the corresponding results for parametric Marcinkiewicz integrals related to area integrals and Littlewood-Paley gλ⁎ functions are also given.

Hyperbolic submanifolds with finite type hyperbolic Gauss map

2015 ◽  
Vol 26 (02) ◽  
pp. 1550014 ◽  
Author(s):  
Uğur Dursun ◽  
Rüya Yeğin
Keyword(s):  
Scalar Curvature ◽  
Finite Type ◽  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Constant Scalar Curvature ◽  
Gauss Map ◽  
Hyperbolic Spaces ◽  
Nonzero Constant ◽  
Hyperbolic Gauss Map

We study submanifolds of hyperbolic spaces with finite type hyperbolic Gauss map. First, we classify the hyperbolic submanifolds with 1-type hyperbolic Gauss map. Then we prove that a non-totally umbilical hypersurface Mn with nonzero constant mean curvature in a hyperbolic space [Formula: see text] has 2-type hyperbolic Gauss map if and only if M has constant scalar curvature. We also classify surfaces with constant mean curvature in the hyperbolic space [Formula: see text] having 2-type hyperbolic Gauss map. Moreover we show that a horohypersphere in [Formula: see text] has biharmonic hyperbolic Gauss map.

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
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