The stability of capillary hypersurfaces in the Euclidean ball

Capillary hypersurfaces in an ambient manifold with boundary are the ones having constant mean curvature and constant contact angle with the boundary of the ambient manifold. In this talk I will first introduce this object and its stability from the variational viewpoint. Then I will talk about our result (joint with Prof. Haizhong Li), which shows that the Delaunay type capillary hypersurfaces with some symmetry in the Euclidean ball are unstable. The proof uses the natural conformal transformations on the Euclidean ball. An open and interesting question is whether the hyperplanes and spherical caps, known being stable, are the only stable capillary hypersurfaces in the Euclidean ball.