Is there a complete classification of compact surface $S\subset \mathbb{R}^3$ for which $\Delta \kappa \geq 0$ where $\kappa $ is the Gaussian curvature of $S$.


Does every (compact) $2$ dimensional manifold admit a Riemannian metric with this property?


Is there a name for this property in $2$ (or higher dimensions)?



Compact surfaces whose Gaussian curvature is a subharmonic function
The Laplacian of any function $f \colon S \to \mathbb{R}$ is $\Delta f$ defined by $\Delta f \, dA = -d(\*df)$, so is exact, and hence has integral zero if $S$ has empty boundary. So if $\Delta f \ge 0$ then $0=\int \Delta f$ forces $\Delta f=0$ everywhere, and so (if $S$ is connected) $f$ is constant.


