The extrusion of polymer melts is often rate limited by the onset of an elastic surface instability known as sharkskin. Appearance of these surface distortions is generally unacceptable for commercial applications. The desire to forestall the onset of sharkskin to higher output rates has motivated a considerable amount of research to characterize the nature of the instability. In this manuscript, we will present a series of detailed experiments using a custom fabricated extruder and die. By incorporating thermal breaks and precise localized temperature control of the die and barrel, predetermined temperature gradients could be induced across the extrudate. Polymers are typically very poor thermal conductors, and therefore the effects of heating or cooling from a boundary can be designed to only affect the properties of the extrudate very close to the die wall. We will present data correlating the amplitude and frequency of the sharkskin instability to the bulk and die surface temperature as well as the shear rates. The result is a quantitative processing map that characterizes the instability and demonstrates that by modifying the rheology of the polymeric fluid very near the die exit corner, it is possible to suppress or control the sharkskin instability through isolated die heating or cooling. By reformulating our data into Weissenberg and Deborah numbers using the relaxation time evaluated at the wall temperature, we demonstrate that the sharkskin surface instability is dependent only on flow kinematics and viscometric properties of the fluid very near the die wall, a result of the stress singularity present at the die exit, and independent of bulk fluid properties. This technique could conceivably increase the profitability of extrusion processes and be extended to develop precisely-controlled sharkskin for designing specific functionality into extruded surfaces.