Consider embeddings $\Sigma_g \subset \mathbb{R}^3$ of oriented surface of genus $g$ which all have area $1$. Call the curvature at a point $x \in \Sigma_g$ of the surface $\kappa(x)$, then can we compute:
$$C_g = \text{min}_{\Sigma_g} \text{max}_{x \in \Sigma_g} (|\kappa(x)|)$$
The loose 'intuition' behind this quantity is that if we bound the absolute value of the curvature of the surface, then we have a 'nice' embedding of the surface in $\mathbb{R}^3$. Using gauss-bonnet and some argument (as discussed in comments), we can deduce that $C_g \geq 4 \pi (1+g)$, with the bound saturated for a sphere, $C_0 = 4 \pi$.
Likely this result is in some book/paper, I am just not sure what the right search-terms are, "min max curvature" doesn't make a very good google search query