Overview
In a newly released mathematical study, researchers establish the non-spin hyperbolic scalar--mean rigidity theorem for dimensions $2 \leq n \leq 5$. This result resolves the low-dimensional case of an open problem posed by Mikhail Gromov concerning scalar and mean curvature rigidity.
Approach
To establish the rigidity theorem without spin assumptions, the proof integrates two differential geometric techniques:
- The classical Jang deformation.
- Dimension reduction via capillary $\mu$-bubbles.
Findings
The paper establishes several concrete geometric results:
- The non-spin hyperbolic scalar--mean rigidity theorem holds in dimensions $2 \leq n \leq 5$, settling Gromov's problem within these dimensions.
- Weighted scalar--mean rigidity holds for Euclidean ball--torus products in total dimensions up to seven.
- In contrast to the Euclidean case, the hyperbolic analogue of the weighted scalar--mean rigidity for ball--torus products is shown to fail.