Analysis (multivariable calculus, some measure theory, function spaces).
Functional analysis (The Hahn-Banach theorem, Riesz representation theorem, Open mapping theorem. Ideally, the spectral theory of compact self-adjoint operators too, but we will recall the statement if not the proof)
Basics of Riemannian geometry (Metrics, Levi-Civita connection,
curvature, Geodesics, Normal coordinates, Riemannian Volume form), The
Laplace equation on compact manifolds (Existence, Uniqueness, Sobolev
spaces, Schauder estimates), Hodge theory, more
general elliptic equations (Fredholmness etc), Uniformization theorem.
Suggested books and references:
Do Carmo, Riemannian Geometry.
Griffiths and Harris, Principles of Algebraic Geometry.
S. Donaldson, Lecture Notes for TCC Course “Geometric Analysis”.
J. Kazdan, Applications of Partial Differential Equations To Problems in Geometry.
L. Nicolaescu, Lectures on the Geometry of Manifolds.
T. Aubin, Some nonlinear problems in geometry.
C. Evans, Partial differential equations.
Gilbarg and Trudinger, Elliptic partial differential equations of the second order.