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We study two minimization questions: the nature of curves $\gamma \subset\mathbb{H}$ which minimize the Loewner energy among all curves from 0 to afixed $z_0 \in \mathbb{H}$, and the nature of $\gamma$ which minimize theLoewner energy among all curves that weld a given pair $x<0 <y$. The formerquestion was partially studied by Yilin Wang, who used SLE techniques tocalculate the minimal energy and show it is uniquely attained. We revisit thequestion using a purely deterministic methodology, and re-derive the energyformula and also obtain further results, such as an explicit computation of thedriving function. Our approach also yields existence and uniqueness ofminimizers for the welding question, as well as an explicit energy formula andexplicit driving function. In addition, we show both families have a"universality" property; for the welding minimizers this means that there is asingle, explicit algebraic curve $\Gamma$ such that truncations of $\Gamma$ orits reflection $-\overline{\Gamma}$ in the imaginary axis generate all weldingminimizers up to scaling. While Wang noted her minimizer is SLE$_0(-8)$, weshow the welding minimizers are SLE$_0(-4,-4)$. Our results also show sharpnessof a case of the driver-curve regularity theorem of Carto Wong.

A deterministic approach to Loewner-energy minimizers