Index growth of hypersurfaces with constant mean curvature

In this paper we give the precise index growth for the embedded hypersurfaces of revolution with constant mean curvature (cmc) 1 in (Delaunay unduloids). When n=3, using the asymptotics result of Korevaar, Kusner and Solomon, we derive an explicit asymptotic index growth rate for finite topology cmc 1 surfaces with properly embedded ends. Similar results are obtained for hypersurfaces with cmc bigger than 1 in hyperbolic space. Received: 6 July 2000; in final form: 10 September 2000 / Published online: 25 June 2001     

Embedded constant mean curvature surfaces with special symmetry

We give sharp, necessary conditions on complete embedded CMC surfaces with three ends and an extra reflection symmetry. The respective submoduli space is a two-dimensional variety in the moduli space of general CMC surfaces. Fundamental domains of our CMC surfaces are characterized by associated great circle polygons in the three-sphere. Received: 23 January 1998 / Revised version: 23 October 1998     

Regularity for minimal boundaries in R n ¶with mean curvature in L n

Let be a minimal set with mean curvature in L ⁿ that is a minimum of the functional , where is open and . We prove that if then can be parametrized over the (n−1)-dimensional disk with a C ^α mapping with C ^α inverse. Received: 11 July 1997 / Revised version: 24 February 1998     

On the free boundary of surfaces with bounded mean curvature: the non-perpendicular case

H?lder continuity up to the free boundary is proved for minimizing solutions if they meet the supporting surface in an angle which is bounded away from zero. The problem is localized by proving the continuity of the distance function, a result which is also true for stationary points. Received: 14 April 1998     

A result on large surfaces of prescribed mean curvature in a Riemannian manifold

Plateau's problem (PP) is studied for surfaces of prescribed mean curvature spanned by a given contour in a 3-d Riemannian manifold. We consider the local situation where a neighborhood of a given point on the manifold is described by a single normal chart. Under certain conditions on and the contour, existence of a small -surface to (PP) is guaranteed by [HK]. The purpose of this paper is the investigation of large -surfaces. Our result states: For sufficiently large (constant) mean curvature and a sufficiently small contour depending on the local geometry of the manifold, (PP) has at least two solutions, a small one and a large one. The proof is based on mountain pass arguments and uses – in contrast to results in the 3-d Euclidean space and in order to derive conformality directly – also a deformation constructed by variations of the independent variable. Received November 8, 1995 / Accepted April 29, 1996     

A method to exclude branch points of minimal surfaces

The central problem of this paper is to exclude boundary branch points of minimal surfaces. The method consists in showing that the third derivative of the Dirichlet energy is negative along well-chosen paths in admissible Jacobi field directions, if a “Schüffler condition” is satisfied. Received July 21, 1997 / Accepted October 3, 1997     

On the area of constant mean curvature discs and annuli with circular boundaries

It is still an open question whether a constant mean curvature (CMC) disc which is bounded by a circle is necessarily a spherical cap or a flat disc. The authors together with López [1] recently showed that the only stable CMC discs which are bounded by a circle are spherical caps. In this paper we derive lower bounds for the area of constant mean curvature discs and annuli with circular boundaries in 3-dimensional space forms. Received November 8, 1999; in final form January 18, 2000 / Published online March 12, 2001     

Partitions with prescribed mean curvatures

We consider a certain variational problem on Caccioppoli partitions with countably many components, which models immiscible fluids as well as variational image segmentation, and generalizes the well-known problem with prescribed mean curvature. We prove existence and regularity results, and finally show some explicit examples of minimizers. Received: 7 June 2001 / Revisied version: 8 October 2001     

Compactness theorems and an isoperimetric inequality for critical points of elliptic parametric functionals

We consider parametric variational double integrals with elliptic Lagrangians F depending on the surface normal and prove a compactness theorem for -critical immersions. As a key ingredient for the relevant a priori estimates we use F. Sauvigny's F-conformal parameters adapted to the parametric integrand F. As a by-product of our analysis we obtain an isoperimetric inequality for -critical immersions generalizing the classical isoperimetric inequality for minimal surfaces. Received November 19, 1999 / Accepted February 4, 2000 / Published online July 20, 2000     

Enclosure theorems for generalized mean curvature flows

We define a generalized notion of mean curvature for regular hypersurfaces in . This enables us to introduce a new class of geometric curvature flows for which we prove enclosure theorems, using methods of Dierkes [D] and Hildebrandt [H]. In particular, we obtain “neck-pinching” results that generalize previous observations by Ecker [E] concerning the classical mean curvature flow. Received: 8 October 2001 / Accepted: 1 March 2002 / Published online: 23 May 2002     

The asymptotic Plateau problem in Gromov hyperbolic manifolds

We solve the asymptotic Plateau problem in every Gromov hyperbolic Hadamard manifold (X,g) with bounded geometry. That is, we prove existence of complete (possibly singular) k-dimensional area minimizing surfaces in X with prescribed boundary data at infinity, for a large class of admissible limit sets and for all . The result also holds with respect to any riemannian metric on X which is lipschitz equivalent to g. Received: 23 January 2001 / Accepted: 25 October 2001 Published online: 28 February 2002     

On uniqueness of vector-valued minimizers of the Ginzburg-Landau functional in annular domains

Consider a class of Sobolev functions satisfying a prescribed degree condition on the boundary of a planar annular domain. It is shown that, within this class, the Ginzburg-Landau functional possesses the unique, radially symmetric minimizer, provided that the annulus is sufficiently narrow. This result is known to be false for wide annuli where vortices are energetically feasible. The estimate for the critical radius below which the uniqueness of the minimizer is guaranteed is obtained as well. Received: 19 July 2000 / Accepted: 23 February 2001/ Published online: 23 July 2001     

Constant mean curvature spheres in Riemannian manifolds

We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points.     

Curvature estimates for graphs with prescribed mean curvature and flat normal bundle

We consider graphs with prescribed mean curvature and flat normal bundle. Using techniques of Schoen et al. (Acta Math 134:275–288, 1975) and Ecker and Huisken (Ann Inst H Poincaré Anal Non Linèaire 6:251–260, 1989), we derive the interior curvature estimate

Angle theorems for the Lagrangian mean curvature flow

We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle for the corresponding Lagrangian submanifold in the cross product space satisfies . If one considers a 4-dimensional K?hler-Einstein manifold of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that is a compact oriented Lagrangian submanifold w.r.t. J such that the K?hler form w.r.t.K restricted to L is positive and , then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. . Received: 11 April 2001 / Published online: 29 April 2002     

On surfaces of prescribed weighted mean curvature

Utilising a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and von der Mosel. Next we study graphs of prescribed weighted mean curvature, for which a quasilinear elliptic equation is proved. Using this equation, we can show height and boundary gradient estimates. Finally, we solve the Dirichlet problem for graphs of prescribed weighted mean curvature.     

Existence of constant mean curvature graphs in hyperbolic space

We give an existence result for constant mean curvature graphs in hyperbolic space . Let be a compact domain of a horosphere in whose boundary is mean convex, that is, its mean curvature (as a submanifold of the horosphere) is positive with respect to the inner orientation. If H is a number such that , then there exists a graph over with constant mean curvature H and boundary . Umbilical examples, when is a sphere, show that our hypothesis on H is the best possible. Received July 18, 1997 / Accepted April 24, 1998     

The Dirichlet problem at infinity for harmonic map equations arising from constant mean curvature surfaces in the hyperbolic 3-space

The purpose of this paper is to study some uniqueness, existence and regularity properties of the Dirichlet problem at infinity for proper harmonic maps from the hyperbolic m-space to the open unit n-ball with a specific incomplete metric. When m=n=2, harmonic solutions of this Dirichlet problem yield complete constant mean curvature surfaces in the hyperbolic 3-space. Received: 25 January 2001 / Accepted: 23 February 2001 / Published online: 25 June 2001