  We study the geometry of the smooth projective surfaces that are defined by
Frobenius forms, a class of homogenous polynomials in prime characteristic
recently shown to have minimal possible F-pure threshold among forms of the
same degree. We call these surfaces $\textit{extremal surfaces}$, and show that
their geometry is reminiscent of the geometry of smooth cubic surfaces,
especially non-Frobenius split cubic surfaces of characteristic two, which are
examples of extremal surfaces. For example, we show that an extremal surface
$X$ contains $d^2(d^2-3d+3)$ lines where $d$ is the degree, which is notable
since the number of lines on a complex surface is bounded above by a quadratic
function in $d$. Whenever two of those lines meet, they determine a $d$-tangent
plane to $X$ which consists of a union of $d$ lines meeting in one point; we
count the precise number of such "star points" on $X$, showing that it is
quintic in the degree, which recovers the fact that there are exactly 45
Eckardt points on an extremal cubic surface.
  Finally, we generalize the classical notion of a double six for cubic
surfaces to a double $2d$ on an extremal surface of degree $d$. We show that,
asymptotically in $d$, smooth extremal surfaces have at least
$\frac{1}{16}d^{14}$ double $2d$'s. A key element of the proofs is using the
large automorphism group of extremal surfaces which we show acts transitively
on many sets, such as the set of (triples of skew) lines on the extremal
surface. Extremal surfaces are closely related to finite Hermitian geometries,
which we recover as the $\mathbb F_{q^2}$-rational points of special extremal
surfaces defined by Hermitian forms over $\mathbb F_{q^2}$.
