We study various aspects related to boundary regularity of complete properly embedded Willmore surfaces in $\mathbb{H}^3$, particularly those related to assumptions on boundedness or smallness of a certain weighted version of the Willmore energy. We prove, in particular, that small energy controls $\mathcal{C}^1$ boundary regularity. We examine the possible lack of $\mathcal{C}^1$ convergence for sequences of surfaces with bounded Willmore energy and find that the mechanism responsible for this is a bubbling phenomenon, where energy escapes to infinity.

We show that there are vast families of contact 3-manifolds each member of which admits infinitely many Stein fillings with arbitrarily large Euler characteristics and arbitrarily small signature— disproving a conjecture of Stipsicz and Ozbagci. To produce our examples, we use a framework which generalizes the construction of Stein structures on allowable Lefschetz fibrations over the 2-disk to those over any orientable base surface, along with the construction of contact structures via open books on 3-manifolds to spinal open books introduced in On symplectic fillings of spinal open book decompositions by S. Lisi, J. Van Horn-Morris & C. Wendl [24].

We describe and partially solve a natural Yamabe-type problem on smooth metric measure spaces which interpolates between the Yamabe problem and the problem of finding minimizers for Perelman’s $\nu$-entropy. In Euclidean space, this problem reduces to the characterization of the minimizers of the family of Gagliardo–Nirenberg inequalities studied by Del Pino and Dolbeault. We show that minimizers always exist on a compact manifold provided the weighted Yamabe constant is strictly less than its value on Euclidean space. We also show that strict inequality holds for a large class of smooth metric measure spaces, but we also give an example which shows that minimizers of the weighted Yamabe constant do not always exist.

We introduce a diffeomorphism invariant of 4-manifolds, the $\mathrm{Pin}^{-} (2)$-monopole invariant, defined by using the $\mathrm{Pin}^{-} (2)$-monopole equations. We compute the invariants of several 4-manifolds, and prove gluing formulae. By using the invariants, we construct exotic smooth structures on the connected sum of an elliptic surface $E(n)$ with arbitrary number of the 4-manifolds of the form of $S^2 \times \Sigma$ or $S^1 \times Y$ where $\Sigma$ is a compact Riemann surface with positive genus and $Y$ is a closed 3-manifold. As another application, we give an estimate of the genus of surfaces embedded in a 4-manifold $X$ representing a class $\alpha \in H_2 (X; l)$, where $l$ is a local coefficient on $X$.