Mean Curvature

Let $\kappa_1$ and $\kappa_2$ be the Principal Curvatures, then their Mean

$$H={\textstyle{1\over 2}}(\kappa_1+\kappa_2)$$ (1)

is called the mean curvature. Let $R_1$ and $R_2$ be the radii corresponding to the Principal Curvatures, then the multiplicative inverse of the mean curvature $H$ is given by the multiplicative inverse of the Harmonic Mean,

$$H\equiv {1\over 2}\left({{1\over R_1}+{1\over R_2}}\right)= {R_1+R_2\over 2R_1R_2}.$$ (2)

In terms of the Gaussian Curvature $K$,

$$H = {\textstyle{1\over 2}}(R_1+R_2)K.$$ (3)

The mean curvature of a Regular Surface in $\Bbb{R}^3$ at a point p is formally defined as

$$H({\bf p})={\textstyle{1\over 2}}\mathop{\rm tr}(S({\rm p})),$$ (4)

where $S$ is the Shape Operator and $\mathop{\rm tr}(S)$ denotes the Trace. For a Monge Patch with $z=h(x,y)$,

$$H={(1+{h_v}^2)h_{uu}-2h_u h_v h_{uv}+(1+{h_u}^2)h_{vv}\over (1+{h_u}^2+{h_v}^2)^{3/2}}$$ (5)

(Gray 1993, p. 307).

If ${\bf x}:U\to\Bbb{R}^3$ is a Regular Patch, then the mean curvature is given by

$$H={eG-2fF+gE\over 2(EG-F^2)},$$ (6)

where $E$, $F$, and $G$ are coefficients of the first Fundamental Form and $e$, $f$, and $g$ are coefficients of the second Fundamental Form (Gray 1993, p. 282). It can also be written

$$H={\mathop{\rm det}({\bf x}_{uu} {\bf x}_u {\bf x}_v)\vert{\... ...rt^2 \vert{\bf x}_v\vert^2-({\bf x}_u\cdot{\bf x}_v)^2]^{3/2}}$$ (7)

Gray (1993, p. 285).

The Gaussian and mean curvature satisfy

$$H^2\geq K,$$ (8)

with equality only at Umbilic Points, since

$$H^2-K={\textstyle{1\over 4}}(\kappa_1-\kappa_2)^2.$$ (9)

If p is a point on a Regular Surface $M\subset\Bbb{R}^3$ and ${\bf v}_{\bf p}$ and ${\bf w}_{\bf p}$ are tangent vectors to $M$ at p, then the mean curvature of $M$ at p is related to the Shape Operator $S$ by

$$S({\bf v}_{\bf p})\times{\bf w}_{\bf p}+{\bf v}_{\bf p}\time... ...w}_{\bf p}) =2H({\bf p}){\bf v}_{\bf p}\times{\bf w}_{\bf p}.$$ (10)

Let Z be a nonvanishing Vector Field on $M$ which is everywhere Perpendicular to $M$, and let $V$ and $W$ be Vector Fields tangent to $M$ such that $V\times W={\bf Z}$, then

$$H=-{{\bf Z}\cdot(D_V{\bf Z}\times W+V\times D_W{\bf Z})\over 2\vert{\bf Z}\vert^3}$$ (11)

(Gray 1993, pp. 291-292).

Wente (1985, 1986, 1987) found a nonspherical finite surface with constant mean curvature, consisting of a self-intersecting three-lobed toroidal surface. A family of such surfaces exists.

See also Gaussian Curvature, Principal Curvatures, Shape Operator

References

Gray, A. ``The Gaussian and Mean Curvatures.'' §14.5 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 279-285, 1993.

Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, p. 108, 1992.

Peterson, I. The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W. H. Freeman, pp. 69-70, 1988.

Wente, H. C. ``A Counterexample in 3-Space to a Conjecture of H. Hopf.'' In Workshop Bonn 1984, Proceedings of the 25th Mathematical Workshop Held at the Max-Planck Institut für Mathematik, Bonn, June 15-22, 1984 (Ed. F. Hirzebruch, J. Schwermer, and S. Suter). New York: Springer-Verlag, pp. 421-429, 1985.

Wente, H. C. ``Counterexample to a Conjecture of H. Hopf.'' Pac. J. Math. 121, 193-243, 1986.

Wente, H. C. ``Immersed Tori of Constant Mean Curvature in $\Bbb{R}^3$.'' In Variational Methods for Free Surface Interfaces, Proceedings of a Conference Held in Menlo Park, CA, Sept. 7-12, 1985 (Ed. P. Concus and R. Finn). New York: Springer-Verlag, pp. 13-24, 1987.