TITLE: What is a minimal surface of finite topology?
QUESTION [5 upvotes]: For example this paper uses that term: https://arxiv.org/pdf/1506.07793v1.pdf.

REPLY [5 votes]: A (necessarily non-compact) surface is said to have finite topology if it is homeomorphic to a compact surface (i.e., has finite genus) with finitely many points removed (i.e., has finitely many ends).
For minimal surfaces, complete examples embedded in $\mathbf{R}^{3}$ include:

Genus zero, one end: A plane (finite total curvature) or helicoid (infinite total curvature).
Genus zero, two ends: A catenoid.
Genus one, three ends: Costa's surface. Generalizations of arbitrary finite genus and with three or more ends are known to exist.

Non-examples include:

Doubly-periodic: Scherk surfaces.
Triply-periodic Schwarz surfaces.