Mean curvature

of a surface $ \Phi ^ {2} $ in $ 3 $-dimensional Euclidean space $ \mathbf R ^ {3} $

Half of the sum of the principal curvatures (cf. Principal curvature) $ k _ {1} $ and $ k _ {2} $, calculated at a point $ A $ of this surface:

$$ H( A) = \frac{k _ {1} + k _ {2} }{2} . $$

For a hypersurface $ \Phi ^ {n} $ in the Euclidean space $ \mathbf R ^ {n+1} $, this formula is generalized in the following way:

$$ H( A) = \frac{k _ {1} + \dots + k _ {n} }{n} , $$

where $ k _ {i} $, $ i = 1 \dots n $, are the principal curvatures of the hypersurface, calculated at a point $ A \in \Phi ^ {n} $.

The mean curvature of a surface in $ \mathbf R ^ {3} $ can be expressed by means of the coefficients of the first and second fundamental forms of this surface:

$$ H( A) = \frac{1}{2} \frac{LG - 2MF + NE }{EG - F ^ { 2 } } , $$

where $ E, F, G $ are the coefficients of the first fundamental form, and $ L, M, N $ are the coefficients of the second fundamental form, calculated at a point $ A \in \Phi ^ {2} $. In the particular case where the surface is defined by an equation $ z = f( x, y) $, the mean curvature is calculated using the formula:

$$ H ( A) = $$

$$ = \ \frac{\left ( 1 + \left ( \frac{\partial f }{\partial y } \right ) ^ {2} \right ) \frac{\partial ^ {2} f }{\partial x ^ {2} } - 2 \frac{\partial f }{\partial x } \frac{\partial f }{\partial y } \frac{\partial ^ {2} f }{\partial x \partial y } + \left ( 1 + \left ( \frac{\partial f }{ \partial x } \right ) ^ {2} \right ) \frac{\partial ^ {2} f }{\partial y ^ {2} } }{\left ( 1 + \left ( \frac{\partial f }{\partial x } \right ) ^ {2} + \left ( \frac{\partial f }{\partial y } \right ) ^ {2} \right ) ^ {3/2} } , $$

which is generalized for a hypersurface $ \Phi ^ {n} $ in $ \mathbf R ^ {n+1} $, defined by the equation $ x _ {n+1} = f( x _ {1} \dots x _ {n} ) $, as follows:

$$ H ( A) = \frac{\sum _ { i= 1} ^ { n } \left ( 1 + p ^ {2} - \left ( \frac{\partial f }{\partial x _ {i} } \right ) ^ {2} \right ) \frac{\partial ^ {2} f }{\partial x _ {i} ^ {2} } - \sum _ { i,j= 1} ^ { n } \frac{\partial f }{\partial x _ {i} } \frac{\partial f }{\partial x _ {j} } \frac{ \partial ^ {2} f }{\partial x _ {i} \partial x _ {j} } }{( 1 + p ^ {2} ) ^ {3/2} } , $$

where

$$ p ^ {2} = | \mathop{\rm grad} f | ^ {2} = \ \left ( \frac{\partial f }{\partial x _ {1} } \right ) ^ {2} + \dots + \left ( \frac{\partial f }{\partial x _ {n} } \right ) ^ {2} . $$

Comments

For an $ m $-dimensional submanifold $ M $ of an $ n $-dimensional Euclidean space of codimension $ n - m > 1 $, the mean curvature generalizes to the notion of the mean curvature normal

$$ \nu _ {p} = \frac{1}{m} \sum _ { j= 1} ^ { n- m} [ \mathop{\rm Tr} A ( e _ {j} ) ] e _ {j} , $$

where $ e _ {1} \dots e _ {n-m} $ is an orthonormal frame of the normal space (cf. Normal space (to a surface)) of $ M $ at $ p $ and $ A ( e _ {j} ) : T _ {p} M \rightarrow T _ {p} M $ ($ T _ {p} M $ denotes the tangent space to $ M $ at $ p $) is the shape operator of $ M $ at $ p $ in the direction $ e _ {j} $, which is related to the second fundamental tensor $ V $ of $ M $ at $ p $ by $ \langle A ( e _ {j} ) ( X) , Y \rangle = \langle V ( X , Y ) , e _ {j} \rangle $.

References

[a1]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2]	M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976)
[a3]	W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)
[a4]	B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973)