Heat localization through reduced dimensionality

We present a model to show that heat propagation away from a local source depends strongly on dimensionality, leading to dramatic localization in low-dimensional systems. An example of such a system is a carbon nanotube array. We further show that this localization is amplified due to a runaway mechanism if thermal conductivity declines rapidly with temperature. Extremely high temperatures of thousands of Kelvins and gradients of hundreds of K/{\mu}m may thus be obtained in a conductor using a modest local power source such as a laser pointer. This is of fundamental importance for high-efficiency energy conversion through thermoelectric and thermionic mechanisms, as well as various other applications.