Volume comparison for C 1 , 1 -metrics

Area of Science:

• Differential Geometry
• General Relativity
• Geometric Analysis

Background:

• Volume comparison theorems (e.g., Bishop-Gromov) are fundamental in Riemannian geometry.
• Existing theorems typically require smooth (C-infinity) metrics.
• Generalizing these results to lower regularity metrics is crucial for broader applications.

Purpose of the Study:

• To extend volume comparison theorems to $C^{1,1}$ metrics.
• To establish a volume monotonicity result for compact subsets of hypersurfaces in globally hyperbolic spacetimes.
• To demonstrate applications of these generalized theorems to singularity theorems.

Main Methods:

• Approximation methods are employed to handle $C^{1,1}$ metrics.
• Analysis of the cut locus for hypersurfaces with $C^{1,1}$ regularity.
• Direct proofs of singularity theorems using the established volume comparison results.

Main Results:

• A volume monotonicity result is established for compact subsets of spacelike, acausal, future causally complete hypersurfaces.
• The cut locus of such hypersurfaces is shown to have measure zero, generalizing a known result for smooth metrics.
• The generalized theorems are applied to prove Myers' theorem and singularity theorems for globally hyperbolic spacetimes.

Conclusions:

• Volume comparison theorems can be successfully generalized to $C^{1,1}$ metrics.
• This generalization has significant implications for understanding spacetime geometry and singularities.
• The study provides a unified approach to proving fundamental theorems in general relativity and differential geometry.