[Paper Review] The Plateau problem for marginally outer trapped surfaces

Michael Eichmair|arXiv (Cornell University)|Nov 27, 2007

Geometric Analysis and Curvature Flows32 references4 citations

TL;DR

This paper solves the Plateau problem for marginally outer trapped surfaces (MOTS) in general initial data sets of general relativity by applying the Perron method and geometric measure theory to force and control blow-up of Jang's equation. The key contribution is a direct, flexible technique that establishes existence and regularity of MOTS spanning a given boundary, revealing new low-order geometric properties analogous to minimal and constant mean curvature surfaces.

ABSTRACT

We solve the Plateau problem for marginally outer trapped surfaces in general Cauchy data sets. We employ the Perron method and tools from geometric measure theory to force and control a blow-up of Jang's equation. Substantial new geometric insights regarding the lower order properties of marginally outer trapped surfaces are gained in the process. The techniques developed in this paper are flexible and can be adapted to other non-variational existence problems.

Motivation & Objective

• To establish the existence of marginally outer trapped surfaces (MOTS) spanning a prescribed boundary in general initial data sets of general relativity.
• To overcome the lack of variational structure in the MOTS equation by developing a non-variational existence method.
• To introduce and apply low-order geometric measure theory tools to analyze the structure and regularity of MOTS.
• To provide a robust, adaptable framework for solving non-variational existence problems in geometric analysis.
• To offer a new, direct alternative to the delicate surgery and stability-based methods used in prior existence proofs of apparent horizons.

Proposed method

• The Perron method is employed to construct subsolutions and supersolutions for a regularization of Jang’s equation.
• A blow-up of Jang’s equation is forced and controlled using barrier constructions and boundary gradient estimates.
• Geometric measure theory tools, including the concept of C-almost minimizing currents, are introduced to analyze the singular set of the limiting current.
• Allard’s regularity theorem and dimension reduction arguments are applied to show that the singular set of the limiting current has Hausdorff dimension at most n−7.
• The method relies on uniform bounds on mean curvature and weak compactness in the space of integral currents.
• The construction is carried out in the ambient space Mⁿ×ℝ, where solutions to Jang’s equation correspond to graphs of functions over Mⁿ.

Experimental results

Research questions

• RQ1 Can the Plateau problem for marginally outer trapped surfaces be solved without relying on variational or stability-based methods?
• RQ2 What geometric properties do marginally outer trapped surfaces possess at the level of low-order regularity and measure-theoretic structure?
• RQ3 Can the blow-up of Jang’s equation be systematically controlled to yield embedded MOTS with prescribed boundary?
• RQ4 How do the singular sets of MOTS solutions behave, and what is their dimension in general initial data sets?
• RQ5 To what extent can geometric measure theory techniques be adapted to non-variational problems in Lorentzian geometry?

Key findings

• The paper establishes the existence of a marginally outer trapped surface spanning any given boundary in a general initial data set, solving the Plateau problem for MOTS.
• The singular set of the limiting current has Hausdorff dimension at most n−7, with isolated singularities when n=7.
• The method provides a direct alternative to the surgery-based approach in [AM07], avoiding the need for delicate injectivity radius estimates.
• The outermost MOTS is shown to be 3|p|_C(Ω)-almost minimizing in a neighborhood of the horizon, yielding a new area estimate.
• The results extend to all dimensions n≤7 and recover the regularity and existence of the apparent horizon without relying on stability or Pogorelov-type curvature estimates.
• The framework is flexible and can be adapted to other non-variational existence problems in geometric analysis.