Schwarz operators of minimal surfaces spanning polygonal boundary curves

This paper examines the Schwarz operator A and its relatives Ȧ, Ā and Ǡ that are assigned to a minimal surface X which maps consequtive arcs of the boundary of its parameter domain onto the straight lines which are determined by pairs P _j , P _j+1 of two adjacent vertices of some simple closed polygon . In this case X possesses singularities in those boundary points which are mapped onto the vertices of the polygon Γ. Nevertheless it is shown that A and its closure Ā have essentially the same properties as the Schwarz operator assigned to a minimal surface which spans a smooth boundary contour. This result is used by the author to prove in [Jakob, Finiteness of the set of solutions of Plateau’s problem for polygonal boundary curves. I.H.P. Analyse Non-lineaire (in press)] the finiteness of the number of immersed stable minimal surfaces which span an extreme simple closed polygon Γ, and in [Jakob, Local boundedness of the set of solutions of Plateau’s problem for polygonal boundary curves (in press)] even the local boundedness of this number under sufficiently small perturbations of Γ.     

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     

A dual approach to regularity in thin film micromagnetics

We prove a regularity result in the two-dimensional theory of soft ferromagnetic films. The associated Euler–Lagrange equation is given by a nonlocal degenerate variational inequality involving fractional derivatives. A difference quotient type argument based on a dual formulation in terms of magnetostatic potentials yields a Hölder estimate for the uniquely determined gradient projection of the magnetization field.     

A sufficient condition for a set of calibrated surfaces to be area-minimizing under diffeomorphisms

We extend Choe’s idea in [1] to nonpolyhedral calibrated surfaces and give some examples of polyhedral sets over right prisms and nonpolyhedral calibrated surfaces. Received: 4 October 2004     

On the existence of H-surfaces into Riemannian manifolds

This paper considers the existence of a local minimizer of a conformally invariant functional defined on a space of maps of a closed Riemann surface into a compact Riemannian manifold . The functional is defined for a given tensor on of type (1,2) and we call its extremal an -surface. In fact, we prove that there exists a local minimizer of the functional in a given homotopy class under certain conditions on , and the minimum of the Dirichlet integral of maps of the homotopy class. Received January 21, 1994 / Received in revised form October 24, 1995 / Accepted March 15, 1996     

Orthogonal trajectories on Riemannian manifolds and applications to Plane Wave type spacetimes

We present a result on trajectories of a Lagrangian system joining two given submanifolds of a Riemannian manifold, under the action of an unbounded potential. As an application, we consider geodesics in a class of semi-Riemannian manifolds, the Plane Wave type spacetimes.     

Periodic orbits on Riemannian manifolds under the action of an at most quadratic potential

In this paper we look for periodic orbits for a Lagrangian system in a complete Riemannian manifold under the action of an eventually unbounded potential. An upper bound on the fixed period is obtained by means of variational tools involving penalization arguments and Morse theory.     

Embedded minimal annuli solving an exterior problem

Given a compact, strictly convex body in ³ and a closed Jordan curve ³ satisfying several additional assumptions, the existence of a parametric, annulus type minimal surface is proved, which parametrizes along one boundary component, has a free boundary onX along the other boundary component, and which stays in ³. As a consequence of this and a reasoning developed by W. H. Meeks and S. -T. Yau we find an embedded minimal surface with these properties. Another application is the existence of an embedded minimal surface with a flat end, free boundary onX and controlled topology.This article was processed by the author using the LATEX style filepljourlm from Springer-Verlag.     

A quadratic Bolza-type problem in a Riemannian manifold

A classical nonlinear equation on a complete Riemannian manifold is considered. The existence of solutions connecting any two points is studied, i.e., for T>0 the critical points of the functional with x(0)=x₀,x(T)=x₁. When the potential V has a subquadratic growth with respect to x, J_T admits a minimum critical point for any T>0 (infinitely many critical points if the topology of is not trivial). When V has an at most quadratic growth, i.e., , this property does not hold, but an optimal arrival time T(λ)>0 exists such that, if 0<T<T(λ), any pair of points in can be joined by a critical point of the corresponding functional. For the existence and multiplicity results, variational methods and Ljusternik-Schnirelman theory are used. The optimal value is fulfilled by the harmonic oscillator. These ideas work for other related problems.     

Variational aspects of Laplace eigenvalues on Riemannian surfaces

We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem and its higher eigenvalue versions via the direct method of calculus of variations. The principal results include the general regularity properties of _λk${\lambda }_k$-extremal metrics and the existence of a partially regular _λ1${\lambda }_1$-maximiser.     

Applications of proximal calculus to fixed point theory on Riemannian manifolds

We prove a general form of a fixed point theorem for mappings from a Riemannian manifold into itself which are obtained as perturbations of a given mapping by means of general operations which in particular include the cases of sum (when a Lie group structure is given on the manifold) and composition. In order to prove our main result we develop a theory of proximal calculus in the setting of Riemannian manifolds.     

Homotopy types for tamely approximable Sobolev maps between manifolds

We define a class ^p (M,N) of Sobolev maps from a manifoldM into a manifoldN, in such a way that each mapu ^p (M, N) has a well defined [p]-homotopy type, providedN satisfies a topological hypothesis. Using this, we prove the existence of minimizers in [p]-homotopy classes for some polyconvex variational problems.     

Viscosity solutions to second order partial differential equations on Riemannian manifolds

We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations F(x,u,du,d²u)=0 defined on a finite-dimensional Riemannian manifold M. Finest results (with hypothesis that require the function F to be degenerate elliptic, that is nonincreasing in the second order derivative variable, and uniformly continuous with respect to the variable x) are obtained under the assumption that M has nonnegative sectional curvature, while, if one additionally requires F to depend on d²u in a uniformly continuous manner, then comparison results are established with no restrictive assumptions on curvature.     

On the regularity of minimal surfaces with free boundaries in Riemannian manifolds

We show c^1,-regularity of minimal surfaces in Riemannian manifolds with a free boundary on C²-hypersurfaces with bounded second fundamental form and a uniform neighborhood on which the nearest point projection is uniquely defined and differentiable. The decisive step is the proof of continuity at the free boundary.partially supported by SFB 72 (Deutsche Forschungsgemeinschaft)     

On the second best constant in logarithmic Sobolev inequalities on complete Riemannian manifolds

We study the second best constant problem for logarithmic Sobolev inequalities on complete Riemannian manifolds and investigate its relationship with optimal heat kernel bounds and the existence of extremal functions.     

Uniqueness of least area surfaces in the 3-torus

Given any -periodic metric g on and a plane through the origin, Bangert [4] shows that there exists a properly embedded surface homeomorphic to which is homotopically area-minimizing w.r.t. g, lies in a strip of bounded width around P, and does not have self-intersections when projected to the 3-torus . For the set of such surfaces, we show the following uniqueness theorems: If P is irrational, i.e., is not spanned by vectors in , the action of on by translations has a unique minimal set. If P is totally irrational, i.e., , then the surfaces in are pairwise disjoint. Received: 8 July 1999 / In final form: 14 February 2000 / Published online: 25 June 2001     

Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter

In this paper, we consider the asymptotic behavior of the fractional mean curvature when $s\to 0^{+}$. Moreover, we deal with the behavior of s-minimal surfaces when the fractional parameter $s\in (0,1)$ is small, in a bounded and connected open set with $C^2$ boundary $\mathrm{\Omega }?{\mathbb{R}}^n$. We classify the behavior of s-minimal surfaces with respect to the fixed exterior data (i.e. the s-minimal set fixed outside of Ω). So, for s small and depending on the data at infinity, the s-minimal set can be either empty in Ω, fill all Ω, or possibly develop a wildly oscillating boundary.Also, we prove the continuity of the fractional mean curvature in all variables, for $s\in [0,1]$. Using this, we see that as the parameter s varies, the fractional mean curvature may change sign.     

Contact symmetries of the Helmholtz form

In this paper, vector fields which are symmetries of the contact ideal are studied. It is shown that contact symmetries of the Helmholtz form transform a dynamical form to a dynamical form which is variational (i.e. comes as the Euler-Lagrange form from a Lagrangian). The case of dynamical forms representing first-order classes in the variational sequence is analysed in detail, which means, by the variational sequence theory, that systems of ordinary differential equations of order ?3 are concerned.