[Paper Review] Minkowski formulae and Alexandrov theorems in spacetime

Mu‐Tao Wang, Ye-Kai Wang+1 more|arXiv (Cornell University)|Sep 8, 2014

Geometric Analysis and Curvature Flows9 references4 citations

TL;DR

This paper extends the classical Minkowski formula to spacelike codimension-two submanifolds in spacetimes with conformal Killing-Yano two-forms, introducing new integral identities that generalize curvature invariants. The key result establishes an Alexandrov-type theorem: a spacelike submanifold with constant normalized null expansion in a static spherically symmetric spacetime must lie in a shear-free (umbilical) null hypersurface, generalizing the Euclidean CMC hypersurface classification to Lorentzian geometry.

ABSTRACT

The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimension-two submanifold with constant normalized null expansion (null mean curvature) must lie in a shear-free (umbilical) null hypersurface. These results are generalized for higher order curvature invariants. In particular, the notion of mixed higher order mean curvature is introduced to highlight the special null geometry of the submanifold. Finally, Alexandrov type theorems are established for spacelike submanifolds with constant mixed higher order mean curvature, which are generalizations of hypersurfaces of constant Weingarten curvature in the Euclidean space.

Motivation & Objective

• To generalize the classical Minkowski formula to spacelike codimension-two submanifolds in Lorentzian manifolds with hidden conformal symmetry via conformal Killing-Yano two-forms.
• To establish an Alexandrov-type theorem characterizing spacelike submanifolds with constant normalized null expansion in static spherically symmetric spacetimes.
• To introduce the notion of mixed higher-order mean curvature to capture the null geometry of submanifolds and generalize Weingarten curvature conditions to spacetime.
• To prove that such submanifolds must lie in shear-free (umbilical) null hypersurfaces, extending results from Euclidean to Lorentzian geometry.

Proposed method

• Derives a new Minkowski-type integral formula using the conformal Killing-Yano two-form $ Q = r\,dr \wedge dt $ in the Schwarzschild spacetime.
• Introduces the connection one-form $ \zeta_L $ associated with a null normal field $ L $, and defines torsion-freeness via $ \zeta_L = 0 $, which simplifies the Minkowski formula.
• Applies the divergence theorem on the submanifold $ \Sigma $, leveraging the closedness and symmetry of the ambient spacetime to relate curvature invariants to geometric quantities like mean curvature vector $ \vec{H} $.
• Uses the curvature tensor expression in terms of $ Q $, showing that $ \bar{R}_{\alpha\beta\gamma\delta} $ can be written as a combination of $ \bar{g} $, $ Q $, and $ Q^2 $, enabling curvature contraction in the formula.
• Introduces the concept of mixed higher-order mean curvature to generalize the notion of constant Weingarten curvature to codimension-two submanifolds in spacetime.
• Establishes the equivalence between constant mixed higher-order mean curvature and lying in a shear-free null hypersurface under appropriate geometric conditions.

Experimental results

Research questions

• RQ1 Can the classical Minkowski formula be extended to spacelike codimension-two submanifolds in Lorentzian spacetimes with hidden conformal symmetry?
• RQ2 What geometric characterization arises when a spacelike submanifold has constant normalized null expansion in a static spherically symmetric spacetime?
• RQ3 How can higher-order curvature invariants be generalized in spacetime to capture the null geometry of submanifolds?
• RQ4 Under what conditions does a spacelike submanifold with constant mixed higher-order mean curvature lie in a shear-free null hypersurface?
• RQ5 Is there a spacetime analogue of the Alexandrov theorem for hypersurfaces of constant mean curvature in Euclidean space?

Key findings

• A new Minkowski-type formula is derived for closed spacelike codimension-two submanifolds in the Schwarzschild spacetime, involving the conformal Killing-Yano two-form $ Q = r\,dr \wedge dt $, the mean curvature vector $ \vec{H} $, and the normal connection of a null frame.
• When the submanifold is torsion-free with respect to a null frame $ L, \underline{L} $, the Minkowski formula simplifies to $ -(n-1)\int_\Sigma \langle \partial_t, \underline{L} \rangle \, d\mu - \frac{1}{2}\int_\Sigma \langle \vec{H}, \underline{L} \rangle Q(L, \underline{L}) \, d\mu = 0 $, linking geometric and causal structure.
• An Alexandrov-type theorem is proven: a closed, spacelike codimension-two submanifold with constant normalized null expansion lies in a shear-free (umbilical) null hypersurface in a static spherically symmetric spacetime.
• The notion of mixed higher-order mean curvature is introduced to generalize the concept of constant Weingarten curvature to spacetime submanifolds, capturing their intrinsic null geometry.
• The curvature tensor of the Schwarzschild spacetime is explicitly expressed in terms of $ Q $, showing that $ \bar{R}_{\alpha\beta\gamma\delta} $ is a linear combination of $ \bar{g} $, $ b(Q) $, and $ \bar{g} \circ Q^2 $, which underpins the derivation of the Minkowski formula.
• The existence of conformal Killing-Yano two-forms is generalized to warped product spacetimes, with $ Q = r \sqrt{ |\det(g_{tt}, g_{tr}; g_{rt}, g_{rr})| } \, dr \wedge dt $ being a CKY two-form under mild conditions on the metric coefficients.