= Solution
On a bosonic configuration, the target-space <supersymmetry transformation> changes the fermionic coordinate by $\delta_\epsilon\Theta=\epsilon$, where $\epsilon$ is a background <Killing spinor>. A <kappa symmetry> convention is
$$
\delta_\kappa\Theta=(1+\Gamma_\kappa)\kappa,\qquad\Gamma_\kappa^2=1.
$$
The configuration preserves a supersymmetry if this change can be cancelled by a local <kappa symmetry> transformation. Multiplying $\epsilon+(1+\Gamma_\kappa)\kappa=0$ by $1-\Gamma_\kappa$ proves necessity of
$$
\boxed{\Gamma_\kappa\epsilon=\epsilon.}
$$
Conversely, if this holds, choosing $\kappa=-\epsilon/2$ cancels the variation. Reversing the brane orientation reverses the corresponding <kappa symmetry projector>.
The preserved parameters must solve this equation everywhere on the brane and satisfy the background <Killing spinor> equations. Thus the fraction of all thirty-two eleven-dimensional supersymmetries is
$$
\boxed{\frac1{32}\dim_{\mathbb R}\{\epsilon:\epsilon\text{ is a background Killing spinor and }
\Gamma_\kappa(\sigma)\epsilon(X(\sigma))=\epsilon(X(\sigma))\text{ for all }\sigma\}.}
$$
One traceless constant <kappa symmetry projector> in flat space preserves sixteen parameters. Further independent commuting projectors often halve that number again, but a position-dependent projector requires a common global solution; its pointwise rank alone does not determine the answer. To quote a fraction of the background supersymmetry, divide instead by the number of its <Killing spinors>.
For the <supermembrane>, let $Z^{\mathcal M}=(X^M,\Theta^\alpha)$ be the embedding in eleven-dimensional <superspace>. Define the pulled-back <supervielbein>, the <induced worldvolume metric>, and induced <gamma matrices> by
$$
\Pi_i^a=\partial_iZ^{\mathcal M}E_{\mathcal M}{}^a(Z),\qquad
h_{ij}=\Pi_i^a\Pi_j^b\eta_{ab},\qquad \gamma_i=\Pi_i^a\Gamma_a.
$$
For a chosen orientation,
$$
\boxed{\Gamma_\kappa=\frac{\varepsilon^{ijk}}{3!\sqrt{-\det h}}\gamma_{ijk},\qquad
\gamma_{ijk}=\gamma_{[i}\gamma_j\gamma_{k]}.}
$$
The <Clifford algebra> shows that $\Gamma_\kappa^2=1$; in an orthonormal membrane frame this reduces to $(\Gamma_0\Gamma_1\Gamma_2)^2=1$. Its trace is zero, so its two eigenspaces each have real dimension sixteen.
The full <supermembrane action> in an on-shell <eleven-dimensional supergravity> superspace is
$$
\boxed{S=-T_2\int_{\Sigma_3}d^3\sigma\,\sqrt{-\det h}
+qT_2\int_{\Sigma_3}Z^*\mathcal A_3,\qquad q=\pm1.}
$$
Here $\mathcal A_3$ is the super-three-form and $Z^*$ denotes the <pullback of a differential form>. Its bosonic restriction is the ordinary supergravity potential, but its fermionic components are retained in this action. The magnitude of its <Wess-Zumino brane coupling> equals the membrane tension; the sign $q$ fixes the orientation and the matching sign of $\Gamma_\kappa$. In flat superspace one may take $\Pi^a=dX^a-i\overline\Theta\Gamma^a d\Theta$, with the super-three-form chosen to obey the standard supergravity superspace constraints. Thus this is a fermionic, <kappa symmetry>-invariant action, rather than merely its bosonic truncation.
To construct an ordinary <calibration>, first take a flux-free static background $ds^2=-dt^2+g_{mn}dx^m dx^n$ with a unit covariantly constant spinor $\epsilon$. In a compatible orientation and mostly-plus <Clifford algebra>, the membrane <spinor calibration form> is
$$
\boxed{\varphi=\frac12\epsilon^\dagger\Gamma_0\Gamma_{ab}\epsilon\;e^a\wedge e^b.}
$$
For an oriented orthonormal spatial pair $u,v$, $\varphi(u,v)=\epsilon^\dagger\Gamma_0\gamma(u)\gamma(v)\epsilon$. The matrix on the right is the static membrane $\Gamma_\kappa$, a Hermitian involution, so this value is at most one, with equality precisely when $\Gamma_\kappa\epsilon=\epsilon$. Parallel transport of the spinor and <gamma matrices> gives $\nabla\varphi=0$, hence $d\varphi=0$.
A <calibration> is a closed <differential form> whose value on each oriented unit tangent plane is at most one; its <comass> is at most one. A <calibrated submanifold> $\Sigma$ saturates the bound, $\varphi|_\Sigma=\operatorname{vol}_\Sigma$. For any homologous competitor $\Sigma'$ with the same boundary, <Stokes theorem> gives
$$
\operatorname{Vol}(\Sigma)=\int_\Sigma\varphi
=\int_{\Sigma'}\varphi\leq\operatorname{Vol}(\Sigma').
$$
This proves that <calibration implies volume minimization>, and in particular that the membrane's spatial surface is a <minimal surface>. The closure condition is indispensable. A general flux-coupled <Killing spinor> need not produce a closed ordinary <calibration>; the corresponding <generalized calibration> bounds the full brane energy including its potential coupling. The flux-free construction above is the ordinary volume-minimizing case.
For the flux-coupled membrane, the standard spinor bilinears give a <Killing vector> $K$ and a two-form $\Omega$ satisfying $d\Omega=\iota_KF$, in matching supergravity conventions. In a stationary gauge $\mathcal L_KA=0$, <Cartan's magic formula> gives
$$
d(\Omega+\iota_KA)=\iota_KF+\mathcal L_KA-\iota_KdA=0.
$$
This closed charge form provides the membrane <generalized calibration>; its potential term is what changes the volume bound into an energy bound.
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