Scalar curvature rigidity of domains in a 3-dimensional warped product

TL;DR

This work proves a Llarull-type scalar curvature rigidity for domains within a 3-dimensional spherical warped product with a logarithmically concave warping function, establishing that a metric $g$ dominating the warped product background $ar{g}$ and satisfying $R_g\, geq\,R_{ar{g}}$ and appropriate boundary inequalities must coincide with $ar{g}$. The authors develop a robust framework based on stable capillary μ-bubbles, a variational functional, and a boundary angle analysis to derive rigidity from barrier constructions, both near cone points and in smooth regions. A key innovation is the tangent-cone analysis allowing barrier arguments even with non-isometric cones, supplemented by a capillary foliation with constant $H-ar{h}$ that propagates local rigidity to the global setting. The results generalize Llarull-type rigidity to three-dimensional domains in spherical warped products, including cases with antipodal conical singularities and Euclidean cone limits, and offer a systematic barrier-based approach alongside a foliation technique to obtain sharp rigidity conclusions with boundary data.

Abstract

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.

Figures (3)

• Figure 2.1: Notations.
• Figure 3.1: Construction of $\Sigma_{t,t^2u}$.
• Figure 4.1: Construction of $\Sigma_{\lambda,s}$ and $\Sigma_{s,u}$.

Theorems & Definitions (52)

• Theorem 1.1: llarull-sharp-1998
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof