Globale Regularitaet des kritischen Einstein-aequivarianten Wave Map Systems
Zusammenfassung der Projektergebnisse
Given a time-dependent physical system, a natural question for a mathematican to ask is: Suppose data at a given instance is known, can one characterize the system at a later instance? This question is referred to as the initial value problem or the Cauchy problem of the system. In the context, of the Einstein’s equations for general relativity, due to the complexity of the equations, global results of the Cauchy problem are intricate in general. If we consider Einstein’s equations with a translational symmetry, in view of the fact that it is an energy critical problem, the global existence follows from non-concentration of energy and global existence for small data. In the fellowship period, extending my PhD work on non-concentration for equivariant 2+1 Einstein-wave maps, the small data result was proved (jointly with L. Andersson and J. Szeftel). A long-standing open problem in mathematical general relativity is the stability of the famous Kerr black hole family. The Kerr family of black holes is a solution of Einstein’s equations with rotational symmetry. In the case of Einstein’s equations with rotational symmetry, although the local structure of the equations is identical to the translational symmetry case, the system is not energy-critical. Therefore, a direct consideration of nonlinear problem analogous to the aforementioned two step approach is infeasible. Traditionally, a natural first step in the super-critical stability problems is to consider the linear pertubation theory, with the aim of controlling the nonlinear terms at a subsequent stage. An important obstacle in the Kerr stability problem is that the conserved energy of even the linear waves is not necessarily positive-definite, due to the ergo-region that always surrounds a rotating Kerr black hole. The lack of a positive-definite and conserved energy limits the immediate application of the PDE techniques that establish decay − an essential notion for stability. In the fellowship period, using the Hamiltonian methods, a positivedefinite, gauge-invariant energy functional is contructed for the rotationally symmetric Maxwell and linear perturbative theory of sub-extremal Kerr black holes (jointly with V. Moncrief). Furthermore, explicit positive-mass theorems were proved for asymptotically de Sitter metrics and a positive energy functional was constructed also for Maxwell perturbations of Kerr-de Sitter. In view of these results, it is expected that the resolution of the black hole stability problem is within reach, for the special, but geometrically and physically important, rotationally symmetric case.
Projektbezogene Publikationen (Auswahl)
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A Positive definite energy functional for axially symmetric Maxwell’s equations on Kerr-de Sitter black hole spacetimes
Nishanth Gudapati
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Global regularity for 2+1 dimensional equivariant Einstein-wave map system, Ann. PDE, 3(13), 2017
Andersson, Lars; Gudapati, Nishanth & Szeftel, Jérémie
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On 3+1 Lorentzian Einstein Manifolds with one Rotational Isometry
Nishanth Gudapati
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On Scattering for small data of 2+1 dimensional equivariant Einstein wave map system. Ann. Henri Poincaré. 18(9):3097-3142, 2017
Dodson, Benjamin & Gudapati, Nishanth
