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In this paper, we study mixed-type solutions of $${SU(3)}$$ Chern-Simons system (see (1.4) below) on a two dimensional flat torus. Nolasco and Tarantello (Commun Math Phys 213:599-639, ), among other things, Nolasco and Tarantello obtained solutions of (1.4) as minimizers of several functionals closely related to (1.4), and showed that if $${N_1+N_2=1}$$ , then one of those minimizers turns out to be a mixed-type solution, that is, one component tends to $${\ln\frac{1}{2}}$$ pointwise a.e. and the other component converges to a solution of a mean field equation. We call these kinds of solutions mixed-type (I) solutions. In this paper, we prove two main results: (i) the asymptotic analysis of mixed-type (I) solutions with arbitrary configuration of vortex points, and (ii) the existence of mixed-type (I) solutions under a non-degenerate condition. This non-degenerate condition also ensures some uniqueness result. In particular, our results imply that when $${N_1+N_2=1}$$ , there are only two mixed-type (I) solutions of (1.4). [ABSTRACT FROM AUTHOR]

In this paper, we first derive the CR analogue of matrix Li-Yau-Hamilton inequality for the positive solution to the CR heat equation in a closed pseudohermitian $$(2n+1)$$ -manifold with nonnegative bisectional curvature and bitorsional tensor. We then obtain the CR Li-Yau gradient estimate in the Heisenberg group. We apply this CR gradient estimate and extend the CR matrix Li-Yau-Hamilton inequality to the case of the Heisenberg group. As a consequence, we derive the Hessian comparison property for the Heisenberg group. [ABSTRACT FROM AUTHOR]