Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition

In this work we study the behaviour of compact, smooth, immersed manifolds with boundary which move under the mean curvature flow in Euclidian space. We thereby prescribe the Neumann boundary condition in a purely geometric manner by requiring a vertical contact angle between the unit normal fields of the immersions and a given, smooth hypersurface. We deduce a very sharp local gradient bound depending only on the curvature of the immersions and. Combining this with a short time existence result, we obtain the existence of a unique solution to any given smooth initial and boundary data. This solution either exists for anyt>0 or on a maximal finite time interval [0,T] such that the curvature explodes astT.This article was processed by the author using the LATEX style filepljourlm from Springer-Verlag.     

Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains

In this paper, we study nonparametric surfaces over strictly convex bounded domains in ${\mathbb{R}}^n$, which are evolving by the mean curvature flow with Neumann boundary value. We prove that solutions converge to the ones moving only by translation. And we will prove the existence and uniqueness of the constant mean curvature equation with Neumann boundary value on strictly convex bounded domains.     

Regularity of mean curvature flows with Neumann free boundary conditions

We consider n-dimensional hypersurfaces flowing by the mean curvature flow with Neumann free boundary conditions supported on a smooth support surface. Under assumptions mirroring those for the case of the mean curvature flow without boundary we show that the n-dimensional Hausdorff measure of the singular set is zero.     

Prescribing the Maslov Form of Lagrangian Immersions

We formulate and apply a modified Lagrangian mean curvature flow to prescribe the Maslov form of Lagrangian immersions in ⁿ . We prove longtime existence results and derive optimal results for curves.     

Affine surfaces with constant affine curvatures

In this paper, we study affine non-degenerate Blaschke immersions from a surface M in ³. We will assume that M has constant affine curvature and constant affine mean curvature, i.e. both the determinant and the trace of the shape operator are constant. Clearly, affine spheres satisfy both these conditions. In this paper, we completely classify the affine surfaces with constant affine curvature and constant affine mean curvature, which are not affine spheres.Research Assistant of the National Fund for Scientific Research (Belgium).     

The influence of the boundary behavior on isometric immersions in the hyperbolic space

This paper studies how the behavior of a proper isometric immersion into the hyperbolic space is influenced by its behavior at infinity. Our first result states that a proper isometric minimal immersion into the hyperbolic space with the asymptotic boundary contained in a sphere reduces codimension. This result is a corollary of a more general one that establishes a sharp lower bound for the sup-norm of the mean curvature vector of a Proper isometric immersion into the Hyperbolic space whose Asymptotic boundary is contained in a sphere. We also prove that a properly immersed hypersurface with mean curvature satisfying sup _{p∈Σ ||H(p)|| < 1} has no isolated points in its asymptotic boundary. Our main tool is a Tangency principle for isometric immersions of arbitrary codimension. This work is partially supported by CAPES, Brazil.     

Geometric rigidity of constant heat flow

Let \(\Omega \) be a compact Riemannian manifold with smooth boundary and let \(u_t\) be the solution of the heat equation on \(\Omega \), having constant unit initial data \(u_0=1\) and Dirichlet boundary conditions (\(u_t=0\) on the boundary, at all times). If at every time t the normal derivative of \(u_t\) is a constant function on the boundary, we say that \(\Omega \) has the constant flow property. This gives rise to an overdetermined parabolic problem, and our aim is to classify the manifolds having this property. In fact, if the metric is analytic, we prove that \(\Omega \) has the constant flow property if and only if it is an isoparametric tube, that is, it is a solid tube of constant radius around a closed, smooth, minimal submanifold, with the additional property that all equidistants to the boundary (parallel hypersurfaces) are smooth and have constant mean curvature. Hence, the constant flow property can be viewed as an analytic counterpart to the isoparametric property. Finally, we relate the constant flow property with other overdetermined problems, in particular, the well-known Serrin problem on the mean-exit time function, and discuss a counterexample involving minimal free boundary immersions into Euclidean balls.     

Hardy–Sobolev critical elliptic equations with boundary singularities

Unlike the non-singular case s=0, or the case when 0 belongs to the interior of a domain Ω in (n3), we show that the value and the attainability of the best Hardy–Sobolev constant on a smooth domain Ω,

when 0<s<2, , and when 0 is on the boundary ∂Ω are closely related to the properties of the curvature of ∂Ω at 0. These conditions on the curvature are also relevant to the study of elliptic partial differential equations with singular potentials of the form:

where f is a lower order perturbative term at infinity and f(x,0)=0. We show that the positivity of the sectional curvature at 0 is relevant when dealing with Dirichlet boundary conditions, while the Neumann problems seem to require the positivity of the mean curvature at 0.     

On the extension of the mean curvature flow

Consider a family of smooth immersions ${F(\cdot,t): M^n\to \mathbb{R}^{n+1}}$ of closed hypersurfaces in ${\mathbb{R}^{n+1}}$ moving by the mean curvature flow ${\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)}$ , for ${t\in [0,T)}$ . Cooper (Mean curvature blow up in mean curvature flow, arxiv.org/abs/0902.4282) has recently proved that the mean curvature blows up at the singular time T. We show that if the second fundamental form stays bounded from below all the way to T, then the scaling invariant mean curvature integral bound is enough to extend the flow past time T, and this integral bound is optimal in some sense explained below.     

Sharp Asymptotics for the Neumann Laplacian with Variable Magnetic Field: Case of Dimension 2

The aim of this paper is to establish estimates of the lowest eigenvalue of the Neumann realization of on an open bounded subset with smooth boundary as B tends to infinity. We introduce a “magnetic” curvature mixing the curvature of ∂Ω and the normal derivative of the magnetic field and obtain an estimate analogous with the one of constant case. Actually, we give a precise estimate of the lowest eigenvalue in the case where the restriction of magnetic field to the boundary admits a unique minimum which is non degenerate. We also give an estimate of the third critical field in Ginzburg–Landau theory in the variable magnetic field case. Submitted: June 26, 2008., Accepted: November 28, 2008.     

Non-parametric radially symmetric mean curvature flow with a free boundary

We study the mean curvature flow of radially symmetric graphs with prescribed contact angle on a fixed, smooth hypersurface in Euclidean space. In this paper we treat two distinct problems. The first problem has a free Neumann boundary only, while the second has two disjoint boundaries, a free Neumann boundary and a fixed Dirichlet height. We separate the two problems and prove that under certain initial conditions we have either long time existence followed by convergence to a minimal surface, or finite maximal time of existence at the end of which the graphs develop a curvature singularity. We also give a rate of convergence for the singularity.     

Locating the peaks of the least energy solutions to an ellyptic system with Neumann boundary conditions

We consider a system of the form in Ω with Neumann boundary condition on ∂Ω, where Ω is a smooth bounded domain in and f,g are power-type nonlinearities having superlinear and subcritical growth at infinity. We prove that the least energy solutions to such a system concentrate, as ε goes to zero, at a point of the boundary which maximizes the mean curvature of the boundary of Ω.     

Parabolic Omori–Yau Maximum Principle for Mean Curvature Flow and Some Applications

We derive a parabolic version of Omori–Yau maximum principle for a proper mean curvature flow when the ambient space has lower bound on \(\ell \)-sectional curvature. We apply this to show that the image of Gauss map is preserved under a proper mean curvature flow in euclidean spaces with uniformly bounded second fundamental forms. This generalizes the result of Wang (Math Res Lett 10:287–299, 2003) for compact immersions. We also prove a Omori–Yau maximum principle for properly immersed self-shrinkers, which improves a result in Chen et al. (Ann Glob Anal Geom 46:259–279, 2014).     

Rotational hypersurfaces of periodic mean curvature

In this paper we discuss rotational hypersurfaces in and more specifically rotational hypersurfaces with periodic mean curvature function. We show that, for a given real analytic function H(s) on , every rotational hypersurface M in with mean curvature H(s) can be extended infinitely in the sense that all coordinate functions of the generating curve of M are defined on all of as well. For rotational hypersurfaces with periodic mean curvature we present a criterion characterizing the periodicity of such hypersurfaces in terms of their mean curvature function. We also discuss a method to produce families of periodic rotational hypersurfaces where each member of the family has the same mean curvature function. In fact, given any closed planar curve with curvature κ, we prove that there is a family of periodic rotational hypersurfaces such that the mean curvature of each element of the family is explicitly determined by κ. Delaunay's famous result for surfaces of revolution with constant mean curvature is included here as the case where n=3 and κ is constant.     

Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant

Using a geometric flow, we study the following prescribed scalar curvature plus mean curvature problem: Let \((M,g_0)\) be a smooth compact manifold of dimension \(n\ge 3\) with boundary. Given any smooth functions f in M and h on \(\partial M\), does there exist a conformal metric of \(g_0\) such that its scalar curvature equals f and boundary mean curvature equals h? Assume that f and h are negative and the conformal invariant \(Q(M,\partial M)\) is a negative real number, we prove the global existence and convergence of the so-called prescribed scalar curvature plus mean curvature flows. Via a family of such flows together with some additional variational arguments, we prove the existence and uniqueness of positive minimizers of the associated energy functional and give a confirmative answer to the above problem. The same result also can be obtained by sub–super-solution method and subcritical approximations.     

On the nonlinear Neumann problem involving the critical Sobolev exponent on the boundary

In this paper we investigate the solvability of the Neumann problem (1.1) involving the critical Sobolev exponents on the right-hand side of the equation and in the boundary condition. It is assumed that the coefficients Q and P are smooth. We examine the common effect of the mean curvature of the boundary ∂Ω and the shape of the graph of the coefficients Q and P on the existence of solutions of problem (1.1).     

Evolution of nonparametric surfaces with speed depending on curvature II. The mean curvature case

We consider an evolution which starts as a flow of smooth surfaces in nonparametric form propagating in space with normal speed equal to the mean curvature of the current surface. The boundaries of the surfaces are assumed to remain fixed. G. Huisken has shown that if the boundary of the domain over which this flow is considered satisfies the “mean curvature” condition of H. Jenkins and J. Serrin (that is, the boundary of the domain is convex “in the mean”) then the corresponding initial boundary value problem with Dirichlet boundary data and smooth initial data admits a smooth solution for all time. In this paper we consider the case of arbitrary domains with smooth boundaries not necessarily satisfying the condition of Jenkins-Serrin. In this case, even if the flow starts with smooth initial data and homogeneous Dirichlet boundary data, singularities may develop in finite time at the boundary of the domain and the solution will not satisfy the boundary condition. We prove, however, existence of solutions that are smooth inside the domain for all time and become smooth up to the boundary after elapsing of a sufficiently long period of time. From that moment on such solutions assume the boundary values in the classical sense. We also give sufficient conditions that guarantee the existence of classical solutions for all time t ≧ 0. In addition, we establish estimates of the rate at which solutions tend to zero as t → ∞.     

Constrained Willmore surfaces

Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy under compactly supported infinitesimal conformal variations. Examples include all constant mean curvature surfaces in space forms. In this paper we investigate more generally the critical points of arbitrary geometric functionals on the space of immersions under the constraint that the admissible variations infinitesimally preserve the conformal structure. Besides constrained Willmore surfaces we discuss in some detail examples of constrained minimal and volume critical surfaces, the critical points of the area and enclosed volume functional under the conformal constraint. C. Bohle, G. P. Peters and U. Pinkall are partially supported by DFG SPP 1154.     

Analysis of constrained Willmore surfaces

This paper investigates the regularity of constrained Willmore immersions into ?^m≥3 locally around both “regular” points and around branch points, where the immersive nature of the map degenerates. We develop local asymptotic expansions for the immersion and its first and second derivatives, given in terms of residues computed as circulation integrals. We deduce explicit “point removability” conditions ensuring that the immersion is smooth. Our results apply in particular to Willmore immersions and to parallel mean curvature immersions in any codimension.