We prove that the Teichmüller space of surfaces of genus $g$ with $p$ punctures contains balls which are not convex in the Teichmüller metric whenever its complex dimension $(3g −3+p)$ is greater than $1$.

In this paper we introduce a flow on the spectral data for symmetric CMC surfaces in the $3$-sphere. The flow is designed in such a way that it changes the topology but fixes the intrinsic (metric) and certain extrinsic (periods) closing conditions of the CMC surfaces. By construction the flow yields closed (possibly branched) CMC surfaces at rational times and immersed higher genus CMC surfaces at integer times. We prove the short time existence of this flow near the spectral data of (certain classes of) CMC tori and obtain thereby the existence of new families of closed (possibly branched) connected CMC surfaces of higher genus. Moreover, we prove that flowing the spectral data for the Clifford torus is equivalent to the flow of Plateau solutions by varying the angle of the fundamental piece in Lawson’s construction for the minimal surfaces $\xi_{g,1}$.

We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem. As a corollary we derive an elliptic analogue of the main result which solves a Calderón problem for connections on a cylinder.

For projective conifold transitions between Calabi–Yau threefolds $X$ and $Y$, with $X$ close to $Y$ in the moduli, we show that the combined information provided by the $A$ model (Gromov–Witten theory in all genera) and $B$ model (variation of Hodge structures) on $X$, linked along the vanishing cycles, determines the corresponding combined information on $Y$. Similar result holds in the reverse direction when linked with the exceptional curves.

Recently, Huang, Lutwak, Yang & Zhang discovered the duals of Federer’s curvature measures within the dual Brunn–Minkowski theory and stated the “Minkowski problem” associated with these new measures. As they showed, this dual Minkowski problem has as special cases the Aleksandrov problem (when the index is $0$) and the logarithmic Minkowski problem (when the index is the dimension of the ambient space)—two problems that were never imagined to be connected in any way. Huang, Lutwak, Yang & Zhang established sufficient conditions to guarantee existence of solution to the dual Minkowski problem in the even setting. In this work, existence of solution to the even dual Minkowski problem is established under new sufficiency conditions. It was recently shown by Böröczky, Henk & Pollehn that these new sufficiency conditions are also necessary.