Yamabe’s problem consists in finding metrics with constant scalar curvature in a given conformal class. Analytically, this reduces to finding a positive solution to a certain semilinear elliptic PDE with critical exponent. The case of a smooth closed manifold is well understood: the combined work of many authors implies that the aforementioned equation is always solvable (with a solution of minimum type). The same problem can also be formulated for less regular spaces, such as closed orbifolds or manifolds with conical singularities. However, in these settings, there are examples where the equation admits no solutions at all.
In the seminar, we will prove the existence of a positive solution to the Yamabe equation on manifolds with conical singularities, under a generic condition on the behavior of the metric near the conical points. In particular, we will derive an analogue of Aubin’s classical result. Interestingly, the singular nature of the metric determines a different condition on the dimension, compared to the regular case. If time permits, we will discuss possible extensions of this result to other singular manifolds.
This is based on a joint work with Andrea Malchiodi.
