= Solution
For uniform unconstrained <boundary conditions>, the positive <gradient> coefficient favors a spatially constant minimizer. Put $u=|\mathbf m|^2\ge0$. The remaining free-energy difference is $a(T-T_c)u+bu^2$. Its minimum is
$$
\boxed{\begin{cases}
\mathbf m=0,&T\ge T_c,\\
\mathbf m=m_0\hat{\mathbf n},\quad
m_0=\sqrt{\dfrac{a(T_c-T)}{2b}},\quad |\hat{\mathbf n}|=1,&T<T_c.
\end{cases}}
$$
Below $T_c$ the amplitude is fixed but its direction is arbitrary; the set of uniform minima is a two-sphere. The free-energy minimum is $f_0-a^2(T_c-T)^2/(4b)$ below $T_c$ and $f_0$ above. The <order parameter> vanishes continuously, so this is a continuous, second-order transition within <Landau theory>.
The <free energy> has internal $O(3)$ spin-rotation/reflection symmetry; a chosen nonzero vector leaves an $O(2)$ subgroup invariant. Restricting to proper rotations, this is \b[$SO(3)\to SO(2)$ <spontaneous symmetry breaking>]. There are two transverse orientation directions and a massive amplitude direction below the transition. Spatially varying directions cost <gradient> energy, and imposed boundaries or defects can prevent a uniform minimizer. In strictly two-dimensional equilibrium with short-range interactions, the fluctuation caveat of part (d) still applies: minimizing this polynomial functional at mean-field level does not overturn the absence of finite-temperature spontaneous <magnetization>.
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