Holomorphic representation of constant mean curvature surfaces in Minkowski space: Consequences of non-compactness in loop group methods

We give an infinite dimensional generalized Weierstrass representation for spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space R2,1. The formulation is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group SU₂ with SU1,1. The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. We prove that it is defined on an open dense subset, after doubling the size of the real form SU1,1, and prove several results concerning the behavior of the surface as the boundary of this open set is encountered. We then use the generalized Weierstrass representation to create and classify new examples of spacelike CMC surfaces in R2,1. In particular, we classify surfaces of revolution and surfaces with screw motion symmetry, as well as studying another class of surfaces for which the metric is rotationally invariant.     

A characterization of constant mean curvature surfaces in homogeneous 3-manifolds

It has been recently shown by Abresch and Rosenberg that a certain Hopf differential is holomorphic on every constant mean curvature surface in a Riemannian homogeneous 3-manifold with isometry group of dimension 4. In this paper we describe all the surfaces with holomorphic Hopf differential in the homogeneous 3-manifolds isometric to H²×R or having isometry group isomorphic either to the one of the universal cover of PSL(2,R), or to the one of a certain class of Berger spheres. It turns out that, except for the case of these Berger spheres, there exist some exceptional surfaces with holomorphic Hopf differential and non-constant mean curvature.     

An extension of Jellett's theorem

The aim of this work is to show that a star-shaped hypersurface of constant mean curvature into the Euclidean sphere Sn+1 must be a geodesic sphere. This result extends the one obtained by Jellett in 1853 for such type of surfaces in the Euclidean space R³. In order to do that we will compute a useful formula for the Laplacian of a new support function defined over a hypersurface M of a Riemannian manifold .     

Eigenvalue estimates for minimal hypersurfaces in hyperbolic space

Recently Candel [A. Candel, Eigenvalue estimates for minimal surfaces in hyperbolic space, Trans. Amer. Math. Soc. 359 (2007) 3567-3575] proved that if M is a simply-connected stable minimal surface isometrically immersed in H³, then the first eigenvalue of M satisfies 1/4?λ(M)?4/3 and he asked whether the bound is sharp and gave an example such that the lower bound is attained. In this note, we prove that the upper bound can never be attained. Also we extend the result by proving that if M is compact stable minimal hypersurface isometrically immersed in Hn+1 where n?3 such that its smooth Yamabe invariant is negative, then (n−1)/4?λ(M)?n²(n−2)/(7n−6).     

Large time behavior of disturbed planar fronts in the Allen-Cahn equation

We consider the Allen-Cahn equation in Rn (with n?2) and study how a planar front behaves when arbitrarily large (but bounded) perturbation is given near the front region. We first show that the behavior of the disturbed front can be approximated by that of the mean curvature flow with a drift term for all large time up to t=+∞. Using this observation, we then show that the planar front is asymptotically stable in L^∞(Rn) under spatially ergodic perturbations, which include quasi-periodic and almost periodic ones as special cases. As a by-product of our analysis, we present a result of a rather general nature, which states that, for a large class of evolution equations, the unique ergodicity of the initial data is inherited by the solution at any later time.     

Conformal deformations of integral pinched 3-manifolds

In this paper we prove that, under an explicit integral pinching assumption between the L²-norm of the Ricci curvature and the L²-norm of the scalar curvature, a closed 3-manifold with positive scalar curvature admits a conformal metric of positive Ricci curvature. In particular, using a result of Hamilton, this implies that the manifold is diffeomorphic to a quotient of S³. The proof of the main result of the paper is based on ideas developed in an article by M. Gursky and J. Viaclovsky.     

Compactness theorems and the Calabi flow on Kähler surfaces with stable tangent bundle

Abstract. In this paper, we prove some compactness theorems and collapse phenomenon on compact K?hler surfaces with stable tangent bundle. We then apply the results to the Calabi flow. More precisely, we prove, under suitable curvature conditions, the longtime existence and asymptotic convergence for solutions of the Calabi flow on compact K?hler surfaces admitting no nonzero holomorphic tangent vector fields and with stable tangent bundle. We also give some examples where the Calabi flow blows up. Received January 7, 1999 / Revised February 2, 2000 / Published online July 20, 2000     

A new quadratic form on hyperkähler manifolds

Let (M,g,I,J,K) be a 4n-dimensional compact simple hyperkähler manifold. We construct a new quadratic form gM on H⁴(M) and study its properties. In particular, we determine completely its signature on H⁴(M,R) for n=2.     

Extrinsic radius pinching for hypersurfaces of space forms

We prove some pinching results for the extrinsic radius of compact hypersurfaces in space forms. In the hyperbolic space, we show that if the volume of M is 1, then there exists a constant C depending on the dimension of M and the L^∞-norm of the second fundamental form B such that the pinching condition (where H is the mean curvature) implies that M is diffeomorphic to an n-dimensional sphere. We prove the corresponding result for hypersurfaces of the Euclidean space and the sphere with the Lp-norm of H, p?2, instead of the L^∞-norm.     

Harmonic morphisms from the classical non-compact semisimple Lie groups

We construct the first known complex-valued harmonic morphisms from the non-compact Lie groups SLn(R), SU^∗(2n) and Sp(n,R) equipped with their standard Riemannian metrics. We then introduce the notion of a bi-eigenfamily and employ this to construct the first known solutions on the non-compact Riemannian SO^∗(2n), SO(p,q), SU(p,q) and Sp(p,q). Applying a duality principle we then show how to manufacture the first known complex-valued harmonic morphisms from the compact Lie groups SO(n), SU(n) and Sp(n) equipped with semi-Riemannian metrics.     

The Ricci flow in a class of solvmanifolds

In this paper, we study the Ricci flow of solvmanifolds whose Lie algebra has an abelian ideal of codimension one, by using the bracket flow. We prove that solutions to the Ricci flow are immortal, the ω-limit of bracket flow solutions is a single point, and that for any sequence of times there exists a subsequence in which the Ricci flow converges, in the pointed topology, to a manifold which is locally isometric to a flat manifold. We give a functional which is non-increasing along a normalized bracket flow that will allow us to prove that given a sequence of times, one can extract a subsequence converging to an algebraic soliton, and to determine which of these limits are flat. Finally, we use these results to prove that if a Lie group in this class admits a Riemannian metric of negative sectional curvature, then the curvature of any Ricci flow solution will become negative in finite time.     

Fisher information metric and Poisson kernels

A complete Riemannian manifold X with negative curvature satisfying −b²?KX?−a²<0 for some constants a,b, is naturally mapped in the space of probability measures on the ideal boundary ∂X by assigning the Poisson kernels. We show that this map is embedding and the pull-back metric of the Fisher information metric by this embedding coincides with the original metric of X up to constant provided X is a rank one symmetric space of non-compact type. Furthermore, we give a geometric meaning of the embedding.     

Translation surfaces with constant mean curvature in 3-dimensional spaces

We give the classification of the translation surfaces with constant mean curvature or constant Gauss curvature in 3-dimensional Euclidean space E³ and 3-dimensional Minkowski space E ₁ ³ .The author is supported by the EDU. COMM. of CHINA, the NSF of Liaoning and the Northeastern University.Dedicated to Professor Udo Simon on the occation of his sixtieth birthday     

Pappus type theorems for motions along a submanifold

We study the volumes volume(D) of a domain D and volume(C) of a hypersurface C obtained by a motion along a submanifold P of a space form Mⁿ_λ. We show: (a) volume(D) depends only on the second fundamental form of P, whereas volume(C) depends on all the ith fundamental forms of P, (b) when the domain that we move D₀ has its q-centre of mass on P, volume(D) does not depend on the mean curvature of P, (c) when D₀ is q-symmetric, volume(D) depends only on the intrinsic curvature tensor of P; and (d) if the image of P by the ln of the motion (in a sense which is well-defined) is not contained in a hyperplane of the Lie algebra of SO(n−q−d), and C is closed, then volume(C) does not depend on the ith fundamental forms of P for i>2 if and only if the hypersurface that we move is a revolution hypersurface (of the geodesic (n−q)-plane orthogonal to P) around a d-dimensional geodesic plane.     

Curvature estimates for the Ricci flow I

In this paper we present several curvature estimates for solutions of the Ricci flow and the modified Ricci flow (including the volume normalized Ricci flow and the normalized Kähler-Ricci flow), which depend on the smallness of certain local \(L^{\frac{n}{2}}\) integrals of the norm of the Riemann curvature tensor |Rm|, where n denotes the dimension of themanifold. These local integrals are scaling invariant and very natural.     

Geometry of curves and surfaces through the contact map

We introduce a new approach to the study of affine equidistants and centre symmetry sets via a family of maps obtained by reflexion in the midpoints of chords of a submanifold of affine space. We apply this to surfaces in R³, previously studied by Giblin and Zakalyukin, and then apply the same ideas to surfaces in R⁴, elucidating some of the connexions between their geometry and the family of reflexion maps. We also point out some connexions with symplectic topology.     

Calibrated lifts of minimal submanifolds

A canonical real line bundle associated to a minimal Lagrangian submanifold in a Kähler-Einstein manifold X is known to be special Lagrangian when considered as a subset of the canonical line bundle of X with a natural Calabi-Yau structure. We first verify this result by standard moving frame computation, and obtain a uniform lower bound for the mass of compact minimal Lagrangian submanifolds in CPn. Similar correspondence is then proved for integrable G₂ and Spin(7) structures on the bundle of anti self dual 2-forms and a Spin bundle respectively of a self dual Einstein 4-manifold N constructed by Bryant and Salamon. In this case, analogues of tangent and normal bundles of certain minimal surfaces in N are calibrated, i.e., associative, coassociative, or Cayley.     

Higher order parallel surfaces in the Heisenberg group

We give a classification of k-parallel surfaces in the three-dimensional Heisenberg group. In particular, we prove that every k-parallel surface in the Heisenberg group is a vertical cylinder over a polynomial spiral of degree at most k−1.     

Minimal Legendrian submanifolds of S and absolutely area-minimizing cones

We use minimal Legendrian submanifolds in spheres to construct examples of absolutely area-minimizing cones and we prove a result about Legendrian 2-tori in S⁵.