Dissertation Defense: Yizhou Zeng

Defense titled "The H\"older Extension Problem for the first Heisenberg Group". The H\"older extension problem asks when a map defined on a subset of a metric space admits an extension with controlled H\"older regularity. For maps into the Heisenberg groups, this problem is complicated due to the fractal-type behavior of the equipped Carnot metric. In particular, the horizontal and vertical directions scale at different rates, and the Carnot metric exhibits square-root behavior in the vertical direction, which creates significant difficulties in controlling the regularity of an extension. Through technically intricate and explicit constructions, Lytchak, Wenger and Young proved that every Lipschitz map from $S^1$ into the first Heisenberg group $\mathbb{H}^1$ admits a H\"older extension to $D^2$ for exponents below 2/3. Later, Haj{\l}asz and Schikorra considered the H\"older extension problem in a broader two-metric framework, namely the extensions whose regularity is controlled simultaneously with respect to the Carnot metric and the Euclidean metric. As a corollary, they proved that the exponent 2/3 cannot be exceeded in general. Together with the result of Lytchak, Wenger, and Young, this shows the corollary of Hajłasz and Schikorra is sharp. However, the sharpness of the full two-metric theorem remained unresolved in the first Heisenberg group. In this thesis, we construct explicit extensions with simultaneously controlled Carnot and Euclidean H\"older regularity. Such extensions realize the full range of the H\"older exponents allowed by the theorem of Haj{\l}asz and Schikorra. Consequently, we prove that their two-metric theorem is sharp in $\mathbb{H}^1$. The proof relies on a refinement of the Lytchak–Wenger–Young construction, combined with an iterative extension procedure with controlled regularity under the two target metrics at the same time. This identifies the optimal tradeoff between H\"older regularity in the intrinsic Carnot geometry and in the