Mean curvature and minimal submanifolds

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. A positive-dimensional Riemannian immersion $f:M^m\to \bar{M}$ is called minimal when its averaged mean curvature vector vanishes identically:

${\mathbf{H}}_f\equiv 0.$

For boundaryless $M$, the general compact-support clause of First variation of volume for a normal variation then makes the first variation of volume zero for every compactly supported variation. If a boundary is allowed, the proposition still gives this stationarity for the normal variations covered there; no boundary-moving assertion is implicit.

Minimality is weaker than total geodesicity. A concrete witness is the Clifford torus in the unit round three-sphere,

$f(u,v)=2^{-1\mathrm{/}2}(\cos u,\sin u,\cos v,\sin v)\in S^3\subset {\mathbb{R}}^4\mathrm{.}$

With $e_1=(-\sin u,\cos u,0,0)$, $e_2=(0,0,-\sin v,\cos v)$, and

$\nu =2^{-1\mathrm{/}2}(\cos u,\sin u,-\cos v,-\sin v),$

the four vectors $f,e_1,e_2,\nu$ are orthonormal. Differentiation in the unit directions gives $D_{e_1}\nu =e_1$ and $D_{e_2}\nu =-e_2$. In Cartesian coordinates the Euclidean metric coefficients are constant, so Christoffel formula for the levi civita connection and Connection laws in directional form identify $D$ with the Euclidean Levi–Civita connection. These two derivatives are tangent to $S^3$, and The induced connection is Levi–Civita therefore makes them the corresponding round-sphere covariant derivatives. Hence the shape operator satisfies

$S_{\nu }{e}_1=-e_1,S_{\nu }{e}_2=e_2\mathrm{.}$

Its averaged trace, and therefore ${\mathbf{H}}_f$, is zero, but $S_{\nu }$ and ${\mathrm{I}\mathrm{I}}_f$ are not zero. Thus this minimal immersion is not totally geodesic.

Stationarity is only a first-order condition and need not mean local volume minimization. For $m\ge 1$, the equator $S^m\times \{0\}\subset S^{m+1}$ has constant unit normal in its last coordinate, so its shape operator and mean-curvature vector vanish. But the normal latitude variation

$F_t(x)=(\cos tx,\sin t)$

has pullback metric $g_t={\cos }^{2}t{g}_0$ and volume density ${\mu }_{g_t}=\mid \cos t{\mid }^m{\mu }_{g_0}$. Consequently, for $0<\mid t\mid <\pi \mathrm{/}2$,

$\mathrm{Vol}(F_t(S^m))=(\cos t{)}^m\mathrm{Vol}(S^m)<\mathrm{Vol}(S^m)\mathrm{.}$

The equator is therefore stationary but not a local minimizer among nearby immersions. Calegari's Example 2.3 and warning make precisely this distinction; Lee likewise identifies zero mean curvature with the variational equation.

These statements are Euler–Lagrange facts only. Regularity, existence, stability, second variation, singular minimal varieties, and the wider theory of minimal surfaces are outside this page. The assumption ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited through the mean-curvature and first-variation constructions; both displayed finite calculations add no choice. The empty positive-dimensional immersion is vacuously minimal, dimension one is included, dimension zero is excluded by the averaged convention, and degenerate induced metrics are excluded by immersivity.