An asymptotic curve on a regular surface 
is a curve 
on 
such that the normal curvature
is 0 in the direction 
for all 
in the domain of 
.
The differential equation for the parametric representation of an asymptotic curve
is
 |
(1)
|
where 
,
, and 
are coefficients of the second
fundamental form. The differential equation for asymptotic curves on a Monge
patch 
is
 |
(2)
|
and on a polar patch 
is
 |
(3)
|
The images below show asymptotic curves for the elliptic helicoid, funnel, hyperbolic paraboloid, and monkey saddle.
|
|
Banks et al. (2026) constructed a smooth embedded annulus in 
with negative Gaussian curvature in its interior
and two closed asymptotic curves. The strictly convex
boundary curves lie in parallel planes tangent to the surface.
This refutes the Nirenberg asymptotic
curve conjecture.
