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We consider ferromagnetic instabilities of two-dimensional helical Diracfermions hosted on the surface of three-dimensional topological insulators. Weinvestigate ways to increase the role of interactions by means of modifying thebulk properties which in turn changes both the surface Dirac theory and thescreening of interactions. We discuss both the long-ranged part of the Coulombinteractions controlled by the dimensionless coupling constant $\alpha =e^{2}/(\hbar \epsilon v_{F}^{\mathrm{surf}})$ as well as the effects of localinteractions parametrized by the ratio $U_{\mathrm{surf}}/D_{\mathrm{surf}}$ ofa local interaction on the surface, $U_{\mathrm{surf}}$, and the surfacebandwidth, $D_{\mathrm{surf}}$. If large compared to 1, both mechanisms caninduce spontaneously surface ferromagnetism, thereby gapping the surface Diracmetal and inducing an anomalous quantum Hall effect. We investigate twomechanisms which can naturally lead to small Fermi velocities$v_{F}^{\mathrm{surf}}$ and a corresponding small bandwidth $D_{\mathrm{surf}}$at the surface when the bulk band gap is reduced. The same mechanisms can,however, also lead to an enhanced screening of surface interactions. While inall considered cases the long-ranged part of the Coulomb interaction isscreened efficiently, $\alpha \lesssim 1$, we discuss situations, where$U_{\mathrm{surf}}/D_{\mathrm{surf}}$ becomes \emph{parametrically}\ largecompared to 1, thus inducing surface magnetism.

Interaction effects on almost flat surface bands in topological insulators