Random differences in Szemerédi’s theorem and related results

We extend the interior gradient estimate due to Korevaar-Simon for solutions of the mean curvature equation from the case of euclidean graphs to the general case of Killing graphs. Our main application is the proof of existence of Killing graphs with prescribed mean curvature function for continuous boundary data, thus extending a result due to Dajczer, Hinojosa, and Lira. In addition, we prove the existence and uniqueness of radial graphs in hyperbolic space with prescribed mean curvature function and asymptotic boundary data at infinity.     

Graphs with prescribed Levi form trace

Abstract We prove existence and uniqueness of a viscosity solution of the Dirichlet problem related to the prescribed Levi mean curvature equation, under suitable assumptions on the boundary data and on the Levi curvature of the domain. We also show that such a solution is Lipschitz continuous by proving that it is the uniform limit of a sequence of classical solutions of elliptic problems and by building Lipschitz continuous barriers. Keywords: Levi mean curvature, Quasilinear degenerate elliptic PDE’s, Viscosity solutions, Comparison principle, Global Lipschitz estimates     

On the generalized mean curvature

We study some properties of graphs whose mean curvature (in distributional sense) is a vector Radon measure. In particular, we prove that the distributional mean curvature of the graph of a Lipschitz continuous function u is a measure if and only if the distributional divergence of T u is a measure. This equivalence fails to be true if Lipschitz continuity is relaxed, as it is shown in a couple of examples. Finally, we prove a theorem of approximation in W ^(1,1) and in the sense of mean curvature of C ² graphs by polyhedral graphs. A number of examples illustrating different situations which can occur complete the work.     

Conservation laws for conformally invariant variational problems

We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations,..., etc.) in divergence form. These divergence-free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces. From this form we can recover, without the use of moving frame, all the classical regularity results known for 2-dimensional conformally invariant non-linear elliptic PDE (see [Hel]). It enables us also to establish new results. In particular we solve a conjecture by E. Heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by S. Hildebrandt [Hil1] claiming that critical points of continuously differentiable elliptic conformally invariant Lagrangian in two dimensions are continuous.     

Pseudo-differential operators with involutive double characteristics

Suppose G is a bounded C²domain in IR ⁿ, n ? 2 . We exam¬ine the regularity at the boundary of solutions to a class of quasi-linear elliptic equations having continuous boundary values ? . If ? has a modulus of continuity β , we give a modulus of continunity for the solution which depends on β and the generalized mean curvature of ?G . When the order of non-uniformity of the equation is between 0 and 1 , no curvature condition on ?G is needed.     

Lagrangian mean curvature flow for entire Lipschitz graphs

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in ${{\mathbb R}^{2n}}$ , we show that the parabolic Eq. 1.1 has a longtime solution which is smooth for all positive time and satisfies uniform estimates away from time t?=?0. In particular, under the mean curvature flow (1.2) the graph immediately becomes smooth and the solution exists for all time such that the second fundamental form decays uniformly to 0 on the graph as t → ∞. Our assumption on the Lipschitz norm is equivalent to the underlying Lagrangian potential u being uniformly convex with its Hessian bounded in L ^∞. As an application of this result we provide conditions under which an entire Lipschitz Lagrangian graph converges after rescaling to a self-expanding solution to the mean curvature flow.     

Biharmonic submanifolds with parallel mean curvature vector field in spheres

We present some results on the boundedness of the mean curvature of proper biharmonic submanifolds in spheres. A partial classification result for proper biharmonic submanifolds with parallel mean curvature vector field in spheres is obtained. Then, we completely classify the proper biharmonic submanifolds in spheres with parallel mean curvature vector field and parallel Weingarten operator associated to the mean curvature vector field.     

Non-preserved curvature conditions under constrained mean curvature flows

We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our examples is that they disprove some statements of the previous literature, overshadow a widespread folklore conjecture about the behaviour of these flows and bring out the discouraging news that a traditional singularity analysis is not possible for constrained versions of the mean curvature flow.     

Isoperimetric inequalities for submanifolds with bounded mean curvature

In this paper, we provide various Sobolev-type inequalities for smooth nonnegative functions with compact support on a submanifold with variable mean curvature in a Riemannian manifold whose sectional curvature is bounded above by a constant. We further obtain the corresponding linear isoperimetric inequalities involving mean curvature. We also provide various first Dirichlet eigenvalue estimates for submanifolds with bounded mean curvature.     

Isoperimetric inequalities for submanifolds with bounded mean curvature

In this paper, we provide various Sobolev-type inequalities for smooth nonnegative functions with compact support on a submanifold with variable mean curvature in a Riemannian manifold whose sectional curvature is bounded above by a constant. We further obtain the corresponding linear isoperimetric inequalities involving mean curvature. We also provide various first Dirichlet eigenvalue estimates for submanifolds with bounded mean curvature.     

A Dirichlet problem for an H-system with variable H

We give conditions onH, a continuous and bounded real function inR ³, to obtain at least two solutions for the problem (Dir) below.H can be far from being constant in the sense of [9]. Our motivation is a better understanding of the Plateau problem for the prescribed mean curvature equation.     

On Quasiconformal Harmonic Surfaces with Rectifiable Boundary

It is proved that every quasiconfomal harmonic mapping of the unit disk onto a surface with rectifiable boundary has absolutely continuous extension to the boundary as well as its inverse mapping has this property. In addition it is proved an isoperimetric type inequality for the class of these surfaces. These results extend some classical results for conformal mappings, minimal surfaces and surfaces with constant mean curvature treated by Kellogg, Courant, Nitsche, Tsuji, F. Riesz and M. Riesz, etc.     

A note on inverse mean curvature flow in cosmological spacetimes

In 12 Gerhardt proves longtime existence for the inverse mean curvature flow in globally hyperbolic Lorentzian manifolds with compact Cauchy hypersurface, which satisfy three main structural assumptions: a strong volume decay condition, a mean curvature barrier condition and the timelike convergence condition. Furthermore, it is shown in 12 that the leaves of the inverse mean curvature flow provide a foliation of the future of the initial hypersurface.We show that this result persists, if we generalize the setting by leaving the mean curvature barrier assumption out. For initial hypersurfaces with sufficiently large mean curvature we can weaken the timelike convergence condition to a physically relevant energy condition.     

A certain feature of logarithmic spirals

A curve formed by inversion of a logarithmic spiral is called a double logarithmic spiral. The curves in this family possess the following property: there always exists such a spiral with continuous and monotone curvature satisfying any possible boundary conditions (endpoints, tangents, and curvatures). The problem of constructing a spiral with continuous curvature and prescribed curvature elements at the endpoints is thus solved. Bibliography: 6 titles.     

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 2-Spheres in Riemannian 3-spheres

We prove the existence of branched immersed constant mean curvature (CMC) 2-spheres in an arbitrary Riemannian 3-sphere for almost every prescribed mean curvature, and moreover for all prescribed mean curvatures when the 3-sphere is positively curved. To achieve this, we develop a min-max scheme for a weighted Dirichlet energy functional. There are three main ingredients in our approach: a bi-harmonic approximation procedure to obtain compactness of the new functional, a derivative estimate of the min-max values to gain energy upper bounds for min-max sequences for almost every choice of mean curvature, and a Morse index estimate to obtain another uniform energy bound required to reach the remaining constant mean curvatures in the presence of positive curvature.     

On the nonexistence of a hypersurface of prescribed mean curvature with a given boundary

A general method is developed for finding necessary conditions for a given codimension-two submanifold Γ of a riemannian manifold to be the boundary of an immersed hypersurface of prescribed mean curvature. In the simplest case the condition is a comparison of the magnitude of the mean curvature with the ratio of the volume of the projection of Γ into a hyperplane to the volume of the interior of that projection. The method is applied to show that certain recent existence results for surfaces of prescribed mean curvature may not be quantitatively improved.     

The mean curvature flow in Minkowski spaces

Studying the geometric flow plays a powerful role in mathematics and physics. In this paper, we introduce the mean curvature flow on Finsler manifolds and give a number of examples of the mean curvature flow. For Minkowski spaces, a special case of Finsler manifolds, we prove the short time existence and uniqueness for solutions of the mean curvature flow and prove that the flow preserves the convexity and mean convexity. We also derive some comparison principles for the mean curvature flow.     

Some remarks on the level set flow by anisotropic curvature

We show that almost all level sets of the unique viscosity solution for general anisotropic mean curvature flow satisfy a weak form of the flow equation. This generalizes the case of isotropic mean curvature flow studied by Evans and Spruck, in which a relation between the viscosity solution and Brakke's varifold mean curvature flow is established. Received June 4, 1998 / Accepted February 26, 1999     

Uniqueness results for quasilinear parabolic equations through viscosity solutions' methods

In this article, we are interested in uniqueness results for viscosity solutions of a general class of quasilinear, possibly degenerate, parabolic equations set in . Using classical viscosity solutions' methods, we obtain a general comparison result for solutions with polynomial growths but with a restriction on the growth of the initial data. The main application is the uniqueness of solutions for the mean curvature equation for graphs which was only known in the class of uniformly continuous functions. An application to the mean curvature flow is given.Received: 1 December 2001, Accepted: 30 September 2002, Published online: 17 December 2002Mathematics Subject Classification: 35A05, 35B05, 35D05, 35K15, 35K55, 53C44This work was partially supported by the TMR program "Viscosity solutions and their applications."