On boundary value problems for Einstein metrics

2006/12/31 by Michael T. Anderson, Michael T Anderson · 1 voice · 1 citation
Mathematics · #Boundary (topology) #Boundary value problem #Dirichlet distribution #Einstein #Einstein's constant #Geometric Analysis and Curvature Flows #Manifold (fluid mechanics) #Moduli space #Neumann boundary condition #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Space (punctuation) #math.DG

paper · pdf · doi:10.2140/gt.2008.12.2009

published as Geom. Topol. 12 (2008) 2009-2045 · 23pp, significant corrections

arxiv created 2008/03/13 · openalex publication_date 2008/07/23 · arxiv updated 2014/11/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

On any given compact manifold M nC1 with boundary @M , it is proved that the moduli space E of Einstein metrics on M , if non-empty, is a smooth, infinite dimensional Banach manifold, at least when 1 .M; @M / D 0. Thus, the Einstein moduli space is unobstructed. The usual Dirichlet and Neumann boundary maps to data on @M are smooth, but not Fredholm. Instead, one has natural mixed boundaryvalue problems which give Fredholm boundary maps.

Citations

• Multiple Integrals in the Calculus of Variations
• Elliptic Partial Differential Equations of Second Order
• The inverse function theorem of Nash and Moser
• Unique continuation results for Ricci curvature and applications
• Einstein Manifolds

Cited by

• On the Bartnik conjecture for the static vacuum Einstein equations

Discussions

• Basically, the h principle likes on open and ample and underdetermined relations. Einstein’s equation is a determined quasilinear system with gauge and after fixing gauge it’s hyperbolic in Lorentzian [stackexchange, 1 points]