Suppose $M$ is a rotationally symmetric convex set in $\mathbb{R}^3$ with two different metric $g$ and $\delta$ (the standard Euclidean metric). Then for some special $M$ with the condition $g_{\partial M}\ge \delta_{\partial M},H_g\ge H_\delta, R_g\ge R_\delta$, then we can show that $g=\delta$.
This paper treats scalar-curvature rigidity for compact three-manifolds whose reference boundary is rotationally symmetric and weakly convex, including smooth, conical, and truncated models.
Euclidean rigidity
If $R_g\geq0$, the induced boundary metric and mean curvature dominate those of the Euclidean reference, and the nonsmooth dihedral angles are no larger, then the manifold is flat under the stated pole hypotheses.
Endpoint geometries
The theorem covers boundaries cut off by planar disks, boundaries with conical tips, and boundaries with smooth spherical poles. Standard balls and right circular cones are important special cases.
Extensions
The same capillary framework yields corresponding rigidity statements in hyperbolic space and in nonpositively curved spaces with an $\mathbb S^1$ symmetry.
The argument minimizes a capillary functional whose prescribed contact angle varies along the rotational boundary. Stability and Gauss–Bonnet force the minimizing disk to be infinitesimally rigid, after which a foliation propagates the equality. The main difficulty is ensuring that a nontrivial minimizer exists near a conical or spherical pole; local constant-mean-curvature barrier foliations reduce those cases to the planar-end problem.
My research interests include Geometric Analysis and Partial Differential Equations.