The <mean value property for harmonic functions> says that whenever $\overline{B_r(x_0)}\subset\Omega$,
$$
u(x_0)=\frac1{|B_r|}\int_{B_r(x_0)}u
=\frac1{|\partial B_r|}\int_{\partial B_r(x_0)}u.
$$
If $u$ attains its maximum $M$ at an interior point, the average of the nonnegative function $M-u$ over every sufficiently small centred ball is zero. Continuity makes $u=M$ on each such ball, and connectedness propagates this equality throughout the domain. Applying the same argument to $-u$ proves the .

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 107 / 1 / a / Solution

The forms $dz_I\wedge d\overline z_J$ are parallel for the flat metric. Consequently the Hodge Laplacian acts coefficientwise:
$$
\Delta\alpha=\sum_{I,J}(\Delta\alpha_{I,J}),dz_I\wedge d\overline z_J.
$$
If every coefficient is constant, this vanishes. Conversely, if $\Delta\alpha=0$, orthogonality of the constant frame gives $\Delta\alpha_{I,J}=0$ for every $I,J$. Each coefficient is a harmonic function on a compact connected manifold and hence is constant by the .

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 118 / 2 / b / Solution

{c}
{wiki=Maximum_principle}

If a harmonic function on a connected domain attains an interior maximum or minimum, it is constant. The mean value property forces equality throughout every sufficiently small ball around an interior extremum, and connectedness propagates the equality.


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 Incoming links: Strong maximum principle for harmonic functions