For surfaces embedded in $\mathbb R^3$ with principal curvatures $ \kappa\_1, \kappa\_2 $ we know bending/isometric mappings conserve $ K= \kappa\_1 \kappa\_2 $ and CMC DeLaunay type minimal surfaces conserve $ 2 H =\kappa\_1 +\kappa\_2 $ by a physical differential pressure.


What type of deformations are conserved with $ (1/\kappa\_1 +1/\kappa\_2 )?$ Is there any physical or mathematical description for this class of surfaces? 



Constant Harmonic Mean surfaces
Surfaces with constant $(\tfrac{1}{\kappa\_1}+\tfrac{1}{\kappa\_2})$ are contained in the class of linear Weingarten surfaces, i.e., surfaces such that 
$$2aH+bK=c$$
for some $a,b,c\in\mathbb R.$ 
These surfaces are well-studied. They are critical points
for an energy functional which contains area, enclosed volume and total mean curvature terms.
You can find many papers on this topic by a short search, for example 
"Linear Weingarten Surfaces in R3" by Galvez, Martinez and Milan, and the references therein.


