Recently discovered exoplanets on close-in orbits should have surface temperatures of hundreds to thousands of Kelvin. They are likely tidally locked and synchronously rotating around their parent stars and, if an atmosphere is absent, have surface temperature contrasts of many hundreds to thousands of Kelvin between permanent day and night sides. We investigated the effect of elevated surface temperature and strong surface temperature contrasts for Earth-mass planets on the (1) pattern of mantle convection, (2) tectonic regime, and (3) rate and distribution of partial melting, using numerical simulations of mantle convection with a composite viscous/pseudo-plastic rheology. Our simulations indicate that if a close-in rocky exoplanet lacks an atmosphere to redistribute heat, a 400 K surface temperature contrast can maintain an asymmetric degree 1 pattern of mantle convection in which the surface of the planet moves preferentially toward subduction zones on the cold night side. The planetary surface features a hemispheric dichotomy, with plate-like tectonics on the night side and a continuously evolving mobile lid on the day side with diffuse surface deformation and vigorous volcanism. If volcanic outgassing establishes an atmosphere and redistributes heat, plate tectonics is globally replaced by diffuse surface deformation and volcanism accelerates and becomes distributed more uniformly across the planetary surface.