We prove the existence of complete minimal surfaces, in Euclidean 3-space, of arbitrary positive genus and least total absolute curvature that have precisely two ends: a catenoidal end and an Enneper-type end. This talk is based on a joint work with Rivu Bardhan, Indranil Biswas, and Pradip Kumar. https://arxiv.org/abs/2509.03925

In this talk, I will describe a classification result for translating solitons of the mean curvature flow under two natural quantitative/topological assumptions: finite genus and entropy equal to 2. Roughly speaking, these hypotheses place the translator at the borderline between the simplest nontrivial singularity models and genuinely higher–complexity behavior: finite genus controls the global topology, while the entropy bound rigidifies the possible asymptotic geometry and rules out many exotic configurations. I will explain how one combines geometric measure theory and PDE tools for the translator equation to extract strong structural information from the entropy constraint, leading to a precise description of all such examples. Along the way, I will discuss the role of blow-down and compactness arguments, how the entropy pinching interacts with curvature and topology, and what the result says about the landscape of low-entropy translators. This is joint work with E. S. Gama and N. M. Møller.

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