On the compactness of constant mean curvature hypersurfaces with finite total curvature

In this work we consider a complete submanifold M with parallel mean curvature vector h immersed in a space form of constant sectional curvature c ￡ 0c\leq 0. If M has finite total curvature and |H|² > -c|H|^2>-c, we prove that M must be compact.     

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.     

On the total mean curvature of piecewise smooth surfaces under infinitesimal bending

Variation of the total mean curvature of piecewise smooth surfaces in Euclidean 3-spaces under infinitesimal bending is discussed and reduced to a sum of line integrals of a rotation vector field.     

On complete noncompact submanifolds with constant mean curvature and finite total curvature in Euclidean spaces

In this paper, we consider a complete noncompact n-submanifold M with parallel mean curvature vector h in an Euclidean space. If M has finite total curvature, we prove that M must be minimal, so that M is an affine n-plane if it is strongly stable. This is a generalization of the result on strongly stable complete hypersurfaces with constant mean curvature in Received: 30 June 2005     

On symplectic mean curvature flows

In this paper we review all the main known results about mean curvature flows with initial surfaces symplectic in a Kähler-Einstein surface, including published results and new results obtained recently. We also propose some problems that we think are very interesting.     

An integral formula for the total mean curvature matrix

The notion of total mean curvature matrix of a submanifold in ${\mathbb{R}}^n$ is defined. A kinematic integral formula for the total mean curvature matrix is proved.     

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.     

On a sub-supersolution method for the prescribed mean curvature problem

The paper is about a sub-supersolution method for the prescribed mean curvature problem. We formulate the problem as a variational inequality and propose appropriate concepts of sub-and supersolutions for such inequality. Existence and enclosure results for solutions and extremal solutions between sub-and supersolutions are established.     

On a class of graphs with prescribed mean curvature

We study a class of quasilinear elliptic equations on the unit ball of ℝ ⁿ in the divergence form ∑ _j=1 ⁿ D _j{G(|x|²,|Du|²)D _j u} =H(|x|) and get estimates on the boundary by using a modified barrier-function technique of Bernstein. We establish a maximum principle for the gradients of solutions and get a global gradient estimate. We prove that solutions with constant boundary condition must be radial. Finally, we apply these results to graphs {(x,u(x)):x∈H ⁿ } whereu:H ⁿ →ℝ is a smooth map of then-hyperbolic spaceH ⁿ =B(0,1) with the metric to get the existence of graphs with radial prescribed mean curvature.     

On the problem of isometry of a hypersurface preserving mean curvature

The problem of determining the Bonnet hypersurfaces in R ⁿ⁺¹, for n > 1, is studied here. These hypersurfaces are by definition those that can be isometrically mapped to another hypersurface or to itself (as locus) by at least one nontrivial isometry preserving the mean curvature. The other hypersurface and/or (the locus of) itself is called Bonnet associate of the initial hypersurface. The orthogonal net which is called A-net is special and very important for our study and it is described on a hypersurface. It is proved that, non-minimal hypersurface in R ⁿ⁺¹ with no umbilical points is a Bonnet hypersurface if and only if it has an A-net.     

On the nondegeneracy of constant mean curvature surfaces

We prove that many complete, noncompact, constant mean curvature (CMC) surfaces are nondegenerate; that is, the Jacobi operator Δ_f + | A_f |² has no L² kernel. In fact, if ∑ has genus zero with k ends, and if f (∑) is embedded (or Alexandrov immersed) in a half-space, then we find an explicit upper bound for the dimension of the L² kernel in terms of the number of non-cylindrical ends. Our main tool is a conjugation operation on Jacobi fields which linearizes the conjugate cousin construction. Consequences include partial regularity for CMC moduli space, a larger class of CMC surfaces to use in gluing constructions, and a surprising characterization of CMC surfaces via spinning spheres. R.K. partially supported by NSF grants DMS-0076085 at GANG/UMass and DMS-9810361 at MSRI, and by a FUNCAP grant in Fortaleza, Brazil. J.R. partially supported by an NSF VIGRE grant at Utah. Received: January 2005; Accepted: June 2005     

On mean curvature flow with forcing

Abstract

This paper investigates geometric properties and well-posedness of a mean curvature flow with volume-dependent forcing. With the class of forcing which bounds the volume of the evolving set away from zero and infinity, we show that a strong version of star-shapedness is preserved over time. More precisely, it is shown that the flow preserves the ρ-reflection property, which corresponds to a quantitative Lipschitz property of the set with respect to the nearest ball. Based on this property we show that the problem is well-posed and its solutions starting with ρ-reflection property become instantly smooth. Lastly, for a model problem, we will discuss the flow’s exponential convergence to the unique equilibrium in Hausdorff topology. For the analysis, we adopt the approach developed by Feldman-Kim to combine viscosity solutions approach and variational method. The main challenge lies in the lack of comparison principle, which accompanies forcing terms that penalize small volume.     

On the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we study the behavior of the scalar curvature S of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of S. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, Rigoli, Setti (2005) [17].     

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.     

The mean curvature of the influence surface of wave equation with sources on a moving surface

The mean curvature of the influence surface of the space–time point ( x , t) appears in linear supersonic propeller noise theory and in the Kirchhoff formula for a supersonic surface. Both these problems are governed by the linear wave equation with sources on a moving surface. The influence surface is also called the Σ‐surface in the aeroacoustic literature. This surface is the locus, in a frame fixed to the quiescent medium, of all the points of a radiating surface f( x , t)=0 whose acoustic signals arrive simultaneously to an observer at position x and at the time t. Mathematically, the Σ‐surface is produced by the intersection of the characteristic conoid of the space–time point ( x , t) and the moving surface. In this paper, we derive the expression for the local mean curvature of the Σ‐surface of the space–time point ( x , t) for a moving rigid or deformable surface f( x , t)=0. This expression is a complicated function of the geometric and kinematic parameters of the surface f( x , t)=0. Using the results of this paper, the solution of the governing wave equation of high‐speed propeller noise radiation as well as the Kirchhoff formula for a supersonic surface can be written as very compact analytic expressions. Copyright © 1999 John Wiley & Sons, Ltd.     

On codimension-one foliations of constant mean curvature

The author is partially supported by a grant from the Alexander von Humboldt Foundation