Note on a Weighted Geometric Inequality in Hyperbolic Space

Area of Science:

• Differential Geometry
• Geometric Analysis
• Hyperbolic Geometry

Background:

• Locally constrained flows in hyperbolic space are crucial for understanding geometric evolution.
• Previous studies on these flows often required initial hypersurfaces to be star-shaped.

Purpose of the Study:

• To establish the long-time existence and exponential convergence of solutions to a locally constrained flow in hyperbolic space.
• To relax the restrictive star-shaped condition for initial hypersurfaces.
• To refine existing geometric inequalities.

Main Methods:

• Analysis of a locally constrained flow in hyperbolic space.
• Utilizing techniques for proving long-time existence and exponential convergence.
• Developing methods to handle strictly horo-convex initial data without the star-shaped assumption.

Main Results:

• The solution to the locally constrained flow exists for a long time.
• The solution converges exponentially fast to a geodesic sphere centered at the origin.
• The requirement of star-shaped initial hypersurfaces is removed.

Conclusions:

• The findings extend the applicability of locally constrained flows in hyperbolic geometry.
• A refined Alexandrov-Fenchel type weighted geometric inequality is presented, featuring isometry-invariant sides.