= Solution
Use the <Monge representation> $\mathbf r(x,y)=(x,y,h(x,y))$ for a single-valued surface without overhangs. Its <area> element is $dA=\sqrt{1+|\nabla h|^2}\,dx\,dy$; at small slopes, $dA\simeq[1+|\nabla h|^2/2]dx\,dy$ and twice the <mean curvature> is $2\mathcal H\simeq\nabla^2h$, up to orientation sign. For zero spontaneous curvature, the quadratic excess <Helfrich energy> is
$$
\boxed{F=\frac12\int dx\,dy\,[\kappa(\nabla^2h)^2+\sigma|\nabla h|^2].}
$$
The <fluid membrane shear response> has no equilibrium static in-plane <shear modulus>: constituents can rearrange to relax shear. Bending and isotropic tension still cost <energy>; viscous <shear stresses> can matter during motion. A <Gaussian curvature> term is omitted under the stated approximation; for fixed closed-surface <topology> its integral is constant, but boundaries and <topology> changes require care. The small-slope Monge expression describes weak deformations, not an entire fully wrapped sphere.<Elastic Properties of Lipid Bilayers: Theory and Possible Experiments by Wolfgang Helfrich>[section II.C, equation (12), p. 695: curvature-elastic energy; only the generic continuum mechanics is used here]{ref}
For <spherical-particle wrapping by a fluid membrane>, integrate the spherical <area> element from polar angle $0$ to $\alpha$:
$$
A=\int_0^{2\pi}d\varphi\int_0^\alpha r_0^2\sin\theta\,d\theta
=2\pi r_0^2(1-\cos\alpha)=2\pi r_0^2\xi.
$$
Its footprint <radius> is $r_\parallel=r_0\sin\alpha$, so
$$
A_\parallel=\pi r_0^2(2\xi-\xi^2),\qquad
\boxed{\Delta A=\pi r_0^2\xi^2.}
$$
For a spherical surface $2\mathcal H=2/r_0$, making the bending cost $\kappa(2/r_0)^2A/2=4\pi\kappa\xi$. <adhesion> contributes $-2\pi wr_0^2\xi$, and tension contributes $\sigma\Delta A$. Therefore
$$
\boxed{\frac{E}{\pi\kappa}=(4-2\bar w)\xi+\bar\sigma\xi^2,\qquad
\bar w=\frac{wr_0^2}{\kappa},\quad\bar\sigma=\frac{\sigma r_0^2}{\kappa}.}
$$
All three contributions are energies; $\bar w$ and $\bar\sigma$ are dimensionless. This is the specified bound-patch model: deformation of the free membrane and the neck is omitted.
The allowed interval $0\leq\xi\leq2$ corresponds to no <adhesion>, partial wrapping and full wrapping. At $\xi=2$, the cap covers a whole sphere and its footprint shrinks to zero. This is a closed neck in the geometric limit. It does not by itself prove membrane scission: creating a detached enclosure changes <topology> and involves neck physics absent from this <energy>.
For positive <membrane tension>, $E''/(\pi\kappa)=2\bar\sigma>0$. Projecting the unconstrained <minimum> $(\bar w-2)/\bar\sigma$ onto the allowed interval gives
$$
\boxed{\xi_{\min}=\begin{cases}
0,&\bar w\leq2,\\
(\bar w-2)/\bar\sigma,&2<\bar w<2+2\bar\sigma,\\
2,&\bar w\geq2+2\bar\sigma.
\end{cases}}
$$
The <tension-dependent wrapping phase diagram> boundaries are $\bar w=2$ and $\bar w=2+2\bar\sigma$. At zero <membrane tension>, the <energy> is linear: the <minimum> jumps from $0$ to $2$ as $\bar w$ crosses $2$, while all indentation values are degenerate at $\bar w=2$. Negative <membrane tension> is outside the stable-flat-membrane interpretation used here. Increasing <membrane tension> makes the excess <area> required for wrapping more expensive, broadening the partially wrapped region.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2013/paper-1-wrapping-phases.png]
{title=Original phase diagram of the stipulated bound-patch energy, with finite-tension partial wrapping}
TODO: \b[user-deferred biological application]. The final request for biological relevance is intentionally left unsolved; all preceding generic geometry and membrane mechanics are included.
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