Maximum Circular Subarray sum
Let $A$ be your array, let $B$ a circular subarray of $A$ that maximizes $\sum_{b \in B} b$, and let $C$ be the subarray that contains all elements of $A$ that are not in $B$. Notice that $C$ is also a circular subarray.

You have $\sum_{a \in A} a = \sum_{b \in B} b + \sum_{c\in C} c$, which implies that $\sum_{c \in C} c = \sum_{a \in A} a - \sum_{b \in B} b $. Since $\sum_{a \in A} a $ is a constant (w.r.t. $B$) and $B$ maximizes $\sum_{b \in B} b$, $C$ minimizes $\sum_{c \in C} c$.

Moreover, at least one of $B$ and $C$ is not only circular but also contiguous and can therefore be found in $O(n)$ time.