Uniqueness of immersed spheres in three-manifolds

José A. Gálvez, Pablo Mira
J. Diff. Geom., to appear.

In this paper we prove an extremely general uniqueness result for immersed spheres that belong to classes of surfaces modeled by elliptic PDEs in three-manifolds. The most famous result in this respect is Hopf’s theorem (constant mean curvature spheres immersed in Euclidean three-space are round spheres). In our result we substitute: (1) the Euclidean ambient space by an arbitrary oriented three-manifold, (2) the constant mean curvature condition by an arbitrary field of elliptic PDEs on the ambient space, and (3) the role of round spheres by the existence of a family of candidate examples that are solutions the prescribed field of elliptic PDEs, and whose distributions of tangent planes foliate the unit tangent bundle of the ambient manifold. In these very general conditions, we prove that any immersed sphere of the class of surfaces modeled by the PDE field is a candidate sphere.

This result unifies and greatly generalizes essentially all previously known uniqueness theorems for immersed spheres that satisfy elliptic PDEs in three-manifolds.

As an application, we settle in the affirmative a 1956 conjecture by A.D. Alexandrov, which may be summarized as follows: Let S be an ovaloid in Euclidean three-space whose principal curvatures k1,k2 and unit normal N are linked by some C^1 elliptic equation F(k1,k2,N)=0. Then any other immersed sphere satisfying this condition is a translation of S.

A corollary of our solution to the Alexandrov conjecture also settles a classical open problem: round spheres are the only elliptic Weingarten spheres immersed in Euclidean three-space.

Constant mean curvature spheres in homogeneous three-spheres

William H. Meeks III, Pablo Mira, Joaquín Pérez, Antonio Ros
Preprint, 2013

In 1951 Hopf proved his famous theorem that any constant mean curvature sphere immersed in Euclidean three-space is a round sphere. His proof also worked when the ambient space is the hyperbolic three-space or the three-dimensional sphere, but failed to work in spaces of non-constant curvature. Much later, in 2004, Abresch and Rosenberg gave an unexpected extension of Hopf’s theorem: any constant mean curvature sphere immersed in a rotationally symmetric homogeneous three-manifold is a rotational sphere. The Abresch-Rosenberg proof failed to work, however, for general homogeneous three-manifolds.

In this paper we fully classified constant mean curvature spheres in compact simply connected homogeneous three-manifolds. We prove that, in any such homogeneous three-sphere, for every value H there exists a unique (up to ambient isometry) constant mean curvature sphere of value H, and we describe its most basic properties in what regards embeddedness, stability or Gauss map convexity. The proof uses many different ideas: a complex representation via the Gauss map, a study of the zeros of complex quadratic differentials, a priori estimates for curvature and area, limit examples of complete constant mean curvature surfaces, stability, Alexandrov embeddedness arguments, and others.

A classification of isolated singularities of elliptic Monge-Ampere equations in dimension two

José A. Gálvez, Asun Jiménez, Pablo Mira
Comm. Pure Appl. Math. 68 (2015), 2086-2107

 

Understanding isolated singularities is a central problem of PDE theory. A key feature of elliptic Monge-Ampère equations is that sometimes they present non-removable isolated singularities at which the solution extends continuously but not smoothly. This type of singularities is not covered by the usual theories of generalized solutions of elliptic PDEs, so some new ideas are needed to fully explain this situation.

In this paper we classified isolated singularities of any real analytic elliptic Monge-Ampere equation in two variables, by constructing an explicit correspondence between the moduli space of such singularities and the space of strictly convex, real analytic Jordan curves in the plane. In the non-analytic case, we still give a description of the asymptotic behavior of the solution at the singularity. The proof is a blend of ideas and techniques from surface theory, complex analysis and PDE theory. The problem of classifying isolated singularities of elliptic Monge-Ampère equations was first considered by K. Jörgens in 1956, in his study of solutions to the Hessian one equation.

Existence and uniqueness of constant mean curvature spheres in Sol_3

Benoit Daniel, Pablo Mira
J. Reine Angew Math. 685 (2013), 1-32.

The homogeneous three-manifold Sol is the least symmetric of the eight 3-dimensional Thurston geometries; the unique of them that is not rotationally invariant. This lack of rotational symmetry, and in particular of rotational constant mean curvature spheres, made the Hopf uniqueness problem in Sol (i.e. the classification of constant mean curvature spheres) specially challenging, since the previous approaches to this problem in rotationally symmetric spaces do not work here.

In this paper we developed a new method for studying constant mean curvature surfaces in Sol, and we proved that for every H greater than or equal to 3^{-1/2} there exists a unique (up to ambient isometry) sphere in Sol of constant mean curvature H. This sphere is embedded, and has index one. We also reduced the proof of this result for all positive values of H to obtaining  adequate area estimates on the space of index one CMC spheres. These area estimates were subsequently proven by W.H. Meeks in another paper, what finished the solution to Hopf’s uniqueness problem in Sol.

Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space

Isabel Fernández, Pablo Mira
Trans. Amer. Math. Soc. 361 (2009), 5737-5752

In 1910 S. Bernstein proved that any entire minimal graph in Euclidean three-space is a plane. This was a very influential result in many different directions, and nowadays the problem of classifying entire minimal graphs in a Riemannian manifold is called the Bernstein problem in the corresponding space. The Heisenberg space Nil is one of the eight canonical three-dimensional Riemannian model manifolds (the so-called Thurston geometries). After pioneer work by U. Abresch and H. Rosenberg, finding a solution to Bernstein problem in Nil became a popular open problem in the theory of constant mean curvature surfaces.

In this paper we gave a full solution to the Bernstein problem in Nil, by providing a (probably) unexpected formulation for its solution: entire minimal graphs in Nil are in correspondence with holomorphic quadratic differentials in the complex plane or the unit disk. A key ingredient of this theorem was the existence (after Abresch and Rosenberg) of a holomorphic quadratic differential for constant mean curvature surfaces in rotationally symmetric homogeneous three-manifolds. Indeed, we used this holomorphic differential as the classifying object in our description of the solution to Bernstein problem.

The space of solutions to the Hessian one equation in the finitely punctured plane

José A. Gálvez, Antonio Martínez, Pablo Mira
J. Math. Pures Appl. 84 (2005), 1744-1757
In the mid 1950s K. Jörgens wrote two influential papers on solutions of the Hessian one equation D^2(u)=1 in the plane. In the first one, he proved that entire C^2 solutions of this equation were quadratic polynomials. In the second one, he proved that entire solutions to the equation that are C^2 on the plane minus one point were, up to equiaffine transformation, radially symmetric examples. The classification of entire solutions of the Hessian one equation in the plane with a finite number of (at least two) punctures remained open.

In this paper we classified all these solutions, by proving that the corresponding moduli space for n>1 punctures is equivalent to the moduli space of conformal equivalence classes of the complex plane with n disks removed; in particular, this space has dimension 3n-4 for every n>1. For n=0,1 our proof trivially recovered the classical results by Jörgens. For n=2, we wrote down all solutions explicitly in terms of theta functions. A key ingredient of the proof is the classical fact that solutions to the Hessian one equation admit a representation formula in terms of holomorphic data.