Fernando C. Marques

Deformations of the hemisphere that increase scalar curvature

Consider a compact Riemannian manifold M of dimension n whose boundary ∂M is totally geodesic and is isometric to the standard sphere Sn−1. A natural conjecture of Min-Oo asserts that if the scalar curvature of M is at least n(n−1), then M is isometric to the hemisphere

Existence of infinitely many minimal hypersurfaces in positive Ricci curvature

S_{+}^{n}

Rigidity of min-max minimal spheres in three-manifolds

equipped with its standard metric. This conjecture is inspired by the positive mass theorem in general relativity, and has been verified in many special cases.In this paper, we construct counterexamples to Min-Oo’s Conjecture in dimension n≥3.

Compactness and non-compactness for Yamabe-type problems

In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min–max theory for the area functional to prove this conjecture in the positive Ricci curvature setting. More precisely, we show that every compact Riemannian manifold with positive Ricci curvature and dimension at most seven contains infinitely many smooth, closed, embedded minimal hypersurfaces. In the last section we mention some open problems related with the geometry of these minimal hypersurfaces.

Scalar Curvature, Conformal Geometry, and the Ricci Flow with Surgery

In this paper we consider min-max minimal surfaces in three-manifolds and prove some rigidity results. For instance, we prove that any metric on a 3-sphere which has scalar curvature greater than or equal to 6 and is not round must have an embedded minimal sphere of area strictly smaller than

Min-Max theory and the Willmore conjecture

4\pi

A compactness theorem for the Yamabe problem

and index at most one. If the Ricci curvature is positive we also prove sharp estimates for the width.

Blow-up phenomena for the Yamabe equation II

The Yamabe equation is one of the most natural and well-studied second-order semilinear elliptic equations arising in geometric variational problems. The existence theory is classical, while it follows from the works (Brendle, J Am Math Soc 21:951–979, 2008; Brendle and Marques, J Differ Geom 81:225–250, 2009) and (Khuri et al., J Differ Geom 81:143–196, 2009) that n = 25 is a critical dimension for compactness/noncompactness issues (or a priori estimates). In this survey article we review these results and discuss more recent related work on Yamabe-type problems.

A priori estimates for the Yamabe problem in the non-locally conformally flat case

In this note we will review recent results concerning two geometric problems associated to the scalar curvature. In the first part we will review the solution to Schoen’s conjecture about the compactness of the set of solutions to the Yamabe problem. It has been discovered, in a series of three papers, that the conjecture is true if and only if the dimension is less than or equal to 24. In the second part we will discuss the connectedness of the moduli space of metrics with positive scalar curvature in dimension three. In two dimensions this was proved by Weyl in 1916. This is a geometric application of the Ricci flow with surgery and Perelman’s work on Hamilton’s Ricci flow.