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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)?

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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.

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    $\begingroup$ This answer proves that $\kappa$ must be constant. To answer the original question it remains to say that yes, on every compact surface, there is a metric of constant curvature. The sign of this constant depends only on the topology of the surface: + for the sphere, 0 for tori, and - for every other surface. $\endgroup$
    – Alexandre Eremenko
    Commented Dec 25, 2017 at 18:53

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