ANALYSIS OF PDE & GENERAL RELATIVITY

1. June, 2000 Sergiu Klainerman ANALYSIS OF PDE & GENERAL RELATIVITY

2. SHORT HISTORY OF GENERAL RELATIVITY most mathematical of all physical theories • Formulation of the theory • Exact Solutions • Schwartzschild, Kerr • Cosmological solutions • Causality, Geodesics • Singularity Theorems (global hyperbolicity, trapped surface) • Initial Data Sets • Construction of Solutions to the Constraint Equations Positive Mass Theorem • Penrose Inequality • Space-time Foliation'sMaximal, Constant Mean Curvature • Problem of Evolution • Maximal Future Developments • Stability of Minkowski • Full Analysis of the Spherically Symmetric Einstein Equations coupled with a Scalar Field

3. INITIAL DATA SETS (S, g, k) Strong Energy condition Strong Asymptotic Flatness Flatness Center of Mass Frame E=m, P=0 • Positive Mass Theorem (Schoen-Yau, Witten) • M  0, M = 0 => (, g,k) is flat • Penrose Inequality (Huisken-Ilmanen, Bray) • Assume k=0, Rg0 => Where A is the total area of the outermost horizon H. (H not enclosed by a surface with less area)

4. PROBLEM OF EVOLUTION (in Vacuum) • Global Existence and Uniqueness (Bruhat-Geroch) • Any (Σ, g, k) has a unique, maximal, Cauchy development MVCD. • May not be complete • Global Stability of Minkowski (Christodoulou-Klainerman) • If (Σ, g, k) is SAF and verifies a global smallness assumption=> MVCD is complete. • Space-time becomes flat in all directions. • Foliated by maximal hypersurfaces • Theorem (Christodoulou-Klainerman-Nicolo) • For any (Σ, g, k) there exists a suitable domain such that the future of the complement of is null outgoing complete. • Foliated by a double null foliation {C(u)} and {C(u)} with complete outgoing leaves {C(u)}.

5. COSMIC CENSORSHIP CONJECTURS • Weak Cosmic Censorship Conjecture • Generic SAF Data have maximal, future, Cauchy developments which have a complete future null infinity. • Strong Cosmic Censorship Conjecture • Generic SAF initial data sets have a maximal future Cauchy development which is locally inextendible as a Lorentzian manifold. Curvature singularities • Spherically Symmetric-Scalar Field Model (D. Christodoulou) • Formation of trapped surfaces • Sharp smallness assumption (implies complete regular solutions). Scale invariant BV space • Examples of solutions with naked singularities • Rigorous proof of the weak and strong Cosmic Censorship

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7. MATHEMATICAL TECHNIQUES • Development of Elliptic-Parabolic Theory • A-priori Estimates • Method of Continuity, Bootstrap Arguments • Scaling Analysis • Development of Hyperbolic Theory • Energy Estimates, Sobolev Inequalities (Symmetric Hyperbolic Systems) • - Example 1: Proof of the Kay-Wald Result • - Example 2: Sketch of proof of local existence and uniqueness • Symmetries, Generalized Energy Estimates, Global Sobolev Inequalities Decay Estimates • Null Frames and Null Condition • Elaborate Bootstrap Strategies • Special role of 2 dimensions. Can have sup norm estimates along characteristics

8. MAIN LIMITATIONS • The current techniques require too many derivative conditions on the initial data. Would like a dimensionless smallness assumption for the stability of Minkowski space. • Certain type of naked singularities may not destroy causality. Need to work with rough solutions • The identification of a dimensionless quantity, whose smallness implies completeness, has played a fundamental role in Christodoulou’s work on spherically symmetric solutions. • The current techniques to prove decay estimates do not extend to the Schwarzschild and Kerr space-times, near the black hole. • Precise decay estimates have played a crucial role in the stability of Minkowski. We need similar estimates in the study of the stability of Schwarzschild and Kerr solutions. • The problem of linear decay is certainly solvable. Need a robust argument. • Have no results on formation of trapped surfaces in the non-spherically symmetric case.

9. STRONG STABILITY OF MINKOWSKI Conjecture There exists a scale invariant smallness conditions such that the maximal future extension is complete. Locally it has to involve the L2 norm of 3/2 derivatives of g and 1/2 derivatives of k. L2 is the only norm preserved by evolution. Leads to the question of Cauchy developments of initial data sets with low regularity. • L2 -Curvature Conjecture • The Bruhat-Geroch result can be extended to initial data sets (Σ, g, k) with R(g)L2 and kL2 .

10. WAVE MAPS u: R n+1 M • Classical WP (local well posedness) Theorem • There exists a unique solution u in [0,T] x Rn depending continuously on the data. • The size of T depends only on the size of • Is (u0,u1) = || Ds u0 || L2(Rn)+ || Ds-1 u1 || L2 (Rn); s > + 1. • Existence, uniqueness and continuous dependence on the data for u0 • Hs(Rn), u1Hs-1 (Rn). • Critical WP exponent sc = . The quantity Isc is dimensionless • Conjecture Local WP for s > sc. Global WP for s = sc and Isc small. Can prove the first of the Conjecture (K-Machedon, K-Selberg, Tataru). Account for cancellations of the quadratic term Q0.

12. WAVE MAPS