A warped product with a spherical factor and a logarithmically concave warping function satisfies a scalar curvature rigidity of the Llarull type. We develop a scalar curvature rigidity of the Llarull type for a general class of domains in a three dimensional spherical warped product. In the presence of rotational symmetry, we identify this class of domains as those satisfying a boundary condition analogous to the logarithmic concavity of the warping function.
The paper establishes Llarull-type scalar-curvature rigidity for domains in three-dimensional spherical warped products, including models with boundary and singular endpoint geometry.
Domain rigidity
For a domain whose side boundary is convex in the conformally related product metric, the inequalities $g\geq\bar g$, $R_g\geq R_{\bar g}$, and $H_g\geq H_{\bar g}$ force $g=\bar g$ under the stated endpoint hypotheses.
Endpoint geometry
The rigidity theorem covers smooth poles, Euclidean or metric tangent cones, and truncations by end disks with matching mean-curvature and angle inequalities.
Conical Llarull theorem
For warped three-spheres with $\psi(t_\pm)=a_\pm|t-t_\pm|+o(|t-t_\pm|)$ and $0
Prescribed-mean-curvature capillary $\mu$-bubbles play the role of the minimizing hypersurfaces in Llarull’s theorem. The stability inequality and a Gauss–Bonnet comparison make a minimizing bubble infinitesimally rigid, and a local foliation propagates equality through the domain. Near a vanishing warping factor or a conical endpoint, local barriers and tangent-cone analysis replace a formal singular perturbation and ensure that the minimizing problem cannot escape through the pole.
My research interests include Geometric Analysis and Partial Differential Equations.