4 problems

Let $E_\varepsilon$ be the energy functional, and let $I_0$ be an interface that may pass through the conical singularity. Consider sequences of saddle points of $E_\varepsilon$ wh…

Nonvanishing energy integral conjecture. The quantity $\int_{\mathbb{S}^2}\varepsilon|\Phi|^2\,ds$ vanishes if and only if the contrast $\kappa_{\varepsilon}$ coincides with a boun…

Donaldson–Witten conjecture. The nodal singularities along circles may be replaced by isolated conical singularities with this homogeneous space as the link. The conjecture propose…

The linear conical-defect Gauss–Bonnet conjecture. There should be no higher-order terms in $1-\alpha$, and