Related papers: Minimal surfaces and CPE metric

We prove that in the three dimensional sphere with a bumpy metric or a metric with positive Ricci curvature, there exist at least four distinct embedded minimal two-spheres. This confirms a conjecture of S. T. Yau in 1982 for bumpy metrics…

Seiberg-Witten theory leads to a delicate interplay between Riemannian geometry and smooth topology in dimension four. In particular, the scalar curvature of any metric must satisfy certain non-trivial estimates if the manifold in question…

Entropy is a natural geometric quantity measuring the complexity of a surface embedded in $\mathbb{R}^3$. For dynamical reasons relating to mean curvature flow, Colding-Ilmanen-Minicozzi-White conjectured that the entropy of any closed…

This paper is a continuation of I, (same title), and is concerned with the existence, regularity and degeneration of metrics minimizing natural curvature functionals on the space of metrics on 3-manifolds. The functionals chosen are…

In this paper, we consider some rigidity results for the Einstein metrics as the critical points of some known quadratic curvature functionals on complete manifolds, characterized by some point-wise inequalities. Moreover, we also provide…

In this work we are interested in studying deformations of the $\sigma_2$-curvature and the volume. For closed manifolds, we relate critical points of the total $\sigma_2$-curvature functional to the $\sigma_2$-Einstein metrics and, as a…

In this paper we study warped product Einstein metrics over spaces with constant scalar curvature. We call such a manifold rigid if the universal cover of the base is Einstein or is isometric to a product of Einstein manifolds. When the…

Minimal surfaces and Einstein manifolds are among the most natural structures in differential geometry. Whilst minimal surfaces are well understood, Einstein manifolds remain far less so. This exposition synthesises together a set of…

In a remarkable article published in 1982, M. Gromov introduced the concept of minimal volume, namely, the minimal volume of a manifold $M^n$ is defined to be the greatest lower bound of the total volumes of $M^n$ with respect to complete…

Let $(M^{n+1},g)$ be a closed Riemannian manifold of dimension $3\le n+1\le 5$. We show that, if the metric $g$ is generic or if the metric $g$ has positive Ricci curvature, then $M$ contains infinitely many geometrically distinct constant…

For a compact 3-manifold $M$ which is a circle bundle over a compact Riemann surface $\Sigma$ with even Euler number $e(M)$, and with a Riemannian metric compatible with the bundle projection, there exists a compact minimal surface $S$ in…

An almost Einstein manifold satisfies equations which are a slight weakening of the Einstein equations; Einstein metrics, Poincare-Einstein metrics, and compactifications of certain Ricci-flat asymptotically locally Euclidean structures are…

We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In…

We use PDE methods as developed for the Liouville equation to study the existence of conformal metrics with prescribed singularities on surfaces with boundary, the boundary condition being constant geodesic curvature. Our first result shows…

A left invariant metric on a nilpotent Lie group is called minimal, if it minimizes the norm of the Ricci tensor among all left invariant metrics with the same scalar curvature. Such metrics are unique up to isometry and scaling and the…

This paper is devoted to the first systematic investigation of manifolds that are Einstein for a connection with skew symmetric torsion. We derive the Einstein equation from a variational principle and prove that, for parallel torsion, any…

In this paper, we prove some rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals on closed manifolds, characterized by some point-wise inequalities. Moreover, we also provide…

Minimal surfaces in the Riemannian product of surfaces of constant curvature have been considered recently, particularly as these products arise as spaces of oriented geodesics of 3-dimensional space-forms. This papers considers more…

The main point of this paper is that, under suitable conditions on the mean curvature and the Ricci curvature of the ambient space, we can extend Choi-Schoen's Compactness Theorem to compact embedded minimal surfaces to simple immersed…

We show that each of the topological 4-manifolds $CP^2#k\bar{CP^2}, for $k = 6, 7$ admits a smooth structure which has an Einstein metric of scalar curvature $s > 0$, a smooth structure which has an Einstein metric with $s < 0$ and…