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Near Horizon Extremal Geometries (NHEG), are geometries which may appear inthe near horizon region of the extremal black holes. These geometries have$SL(2,\mathbb{R})\!\times\!U(1)^n$ isometry, and constitute a family ofsolutions to the theory under consideration. In the first part of this report,their thermodynamic properties are reviewed, and their three universal laws arederived. In addition, at the end of the first part, the role of these laws inblack hole thermodynamics is presented.  In the second part of this thesis, we review building their classical phasespace in the Einstein-Hilbert theory. The elements in the NHEG phase spacemanifold are built by appropriately chosen coordinate transformations of theoriginal metric. These coordinate transformations are generated by some vectorfields, dubbed "symplectic symmetry generators." To fully specify the phasespace, we also need to identify the symplectic structure. In order to fix thesymplectic structure, we use the formulation of Covariant Phase Space method.The symplectic structure has two parts, the Lee-Wald term and a boundarycontribution. The latter is fixed requiring on-shell vanishing of thesymplectic current, which guarantees the conservation and integrability of thesymplectic structure, and leads to the new concept of "symplectic symmetry."Given the symplectic structure, we construct the corresponding conservedcharges, the "symplectic symmetry generators." We also specify the explicitexpression of the charges as a functional over the phase space. These symmetrygenerators constitute the "NHEG algebra," which is an infinite dimensionalalgebra (may be viewed as a generalized Virasoro), and admits a centralextension which is equal to the black hole entropy.

On Thermodynamics and Phase Space of Near Horizon Extremal Geometries